ABSTRACT. We study a random process inspired by the payment execution mechanism of the Lightning Network \cite{ref_poon2016}, the main layer-two solution on top of Bitcoin \cite{ref_nakamoto2008}, represented as a graph in which users correspond to nodes and payment channels to bidirectional weighted edges with capacities. Each channel has a fixed publicly known capacity, while the balance, which specifies how this capacity is distributed between the two endpoints, is private and known only to the channel owners. Each user can make payments directly to adjacent nodes or indirectly through intermediate nodes, where payments will succeed only if all channels along the payment path can handle the required amount. The process we study is as follows: given an undirected graph $G$, where each edge $e$ has a capacity $C_e$ and each of its endpoints $u$ and $v$ has a balance $b_e(u)$ and $b_e(v)$, such that $C_e = b_e(u) + b_e(v)$, with an initial capacity distributed equally between the endpoints. In each round, a payment of one unit is executed by choosing two nodes $u$ and $v$, and then selecting a shortest path among all possible shortest paths between them, both uniformly at random. Our goal is to investigate how long it takes for the first payment failure to occur, depending on the topology of the graph and the channel capacities. We first prove almost tight upper and lower bounds as a function of the number of nodes and the edge capacities when the underlying graph is complete. Then, we show how such a random process is related to the edge-betweenness centrality \cite{ref_girvan2002} measure and we prove upper and lower bounds for arbitrary graphs as a function of edge-betweenness and capacity. Finally, we validate our theoretical results by running extensive simulations over some classes of graphs, including snapshots of the real Lightning Network.

ABSTRACT. We propose simple and efficient schemes for Affine Volterra processes, using integrated kernel quantities and the Inverse Gaussian distribution. The schemes preserve positivity, and can be shown to converge weakly by recasting them as stochastic Volterra equations with a measure-valued kernel. Our method applies to two important examples: Volterra square-root/Heston and Hawkes processes. In the first case, when using a fractional kernel, the scheme with large time steps seems to be more performant as the Hurst index H decreases to -1/2. In the second case, our scheme has deterministic complexity, in contrast with exact methods based on sampling jump times that have random complexity, which opens the door to efficient Monte Carlo methods.

ABSTRACT. We show that the introduction of resetting is able to expedite the first passage of a diffusion process. To this end, we address the problem of minimizing the expected first-passage time (FPT) and the expected first-exit time (FET) of a one-dimensional diffusion process with Poissonian resetting, with respect to the resetting rate $r.$ We first derive a general analytical relationship that expresses the Laplace transform (LT) and the expected value of the FPT (and FET) for the process with resetting in terms of the LT of the FPT (and FET) of the underlying diffusion without resetting. This framework is then applied to determine the optimal resetting rate $r$ that minimizes the expected FPT (and FET). We provide explicit results for drifted Brownian motion and Ornstein-Uhlenbeck (OU) process. For Brownian motion, we extend existing literature by considering the case where the initial position $x$ differs from the resetting position $x _ R$, providing a comprehensive parametric analysis. For the OU process, we provide new insights into the minimization of the expected FPT, a case that has remained largely unexplored. Our results demonstrate how a strategic choice of the resetting rate can effectively regularize and accelerate search processes across one or two boundaries.

ABSTRACT. We study the Lyapunov and moment Lyapunov stability of a class of parabolic SPDEs driven by additive noise, including the stochastic Allen-Cahn equation. To do so, we analyze properties of the associated projective process.

ABSTRACT. This talk is based on the paper [1]. The main focus is pathwise uniqueness for mild solutions to stochastic PDEs with drift given in differential form. The singularity of the drift perturbation allows to achieve novel pathwise uniqueness results for several classes of examples, ranging from fluid-dynamics to phase-separation models, previously studied only in the context of weak uniqueness, see [2,4]. Finally, the technique introduced here also yields significant improvements over the results already known in the non-singular case, see [3].

References:

[1]D. Addona, D. A. Bignamini, C. Orrieri, L. Scarpa, Pathwise uniqueness by noise for singular stochastic PDEs, e-print arXiv:2512.17736, 2025.

[2] Bertacco F., Orrieri C., Scarpa L., Weak uniqueness by noise for singular stochastic PDES, Transactions of the American Mathematical Society 378, 7977-8023 (2025).

[3]G. Da Prato, F. Flandoli, Pathwise uniqueness for a class of SDE in Hilbert spaces and applications, Journal of Functional Analysis 259, 243-267 (2010).

[4]E. Priola, An optimal regularity result for Kolmogorov equations and weak uniqueness for some critical SPDEs, Annals of Probability 49, 1310–1346 (2021).

ABSTRACT. \title{Bismut-Elworthy type formulae for BSDEs with degenerate noise }

% Author name(s) \author{ Davide Addona\inst{2} \and Federica Masiero\inst{1}\thanks{Presenter} %\and %Third Author\inst{2} } % % \institute{Department of Mathematics and Applications, University of Milano Bicocca, Italy, \\ \email{federica.masiero@unimib.it}\\ \and Department of Mathematical, Physical and Computer Sciences, University of Parma, Parma, Italy\\ \email{davide.addona}@unipr.it}

% \maketitle %

\keywords{gradient estimates \and degenerate noise \and backward stochastic differential equations} \\

In this talk we present how to derive Bismut-Elworthy formula under assumptions weaker than non degeneracy of the noise. By Bismut-Elworthy formula we mean a gradient type estimate on the transition semigroup of a stochastic differential equation in a possibly infinite dimensional Hilbert space. \newline We also present a nonlinear version of the Bismut formula for BSDEs, in analogy to what is done in \cite{FT} in the case of non degenerate noise, and we discuss applications to the solution of semilinear Kolmogorov equations.

Our study is motivated by the regularizing properities of the transition semigroup of the stochastic wave equations, studied in \cite{MP}, and of the stochastic damped wave equation, first studied in \cite{AddBig24} and next also in \cite{AddMas}.

\def\sessionnumber{CS124}

\def\sessionname{Infinite Dimensional Analysis and Malliavin Calculus}

\def\firstorganizer{Davide Addona}

ABSTRACT. We propose a novel approach to numerically approximate McKean-Vlasov stochastic differential equations (MV-SDE) using stochastic gradient descent (SGD) while avoiding the use of interacting particle systems (IPS) and the associated simulation costs required to achieve the “propagation of chaos” limit. The SGD technique is deployed to solve a Euclidean minimization problem, obtained by first representing the MV-SDE as a minimization problem over the set of continuous functions of time, and then approximating the domain with a finite-dimensional subspace. Convergence is established by proving certain intermediate stability and moment estimates of the relevant stochastic processes, including the tangent processes. Numerical experiments illustrate the competitive performance of our SGD based method compared to the IPS benchmarks. This work offers a theoretical foundation for using the SGD method in the context of numerical approximation of MV-SDEs, and provides analytical tools to study its stability and convergence

ABSTRACT. Transformers are a central architecture in modern deep learning, forming the backbone of large language models such as ChatGPT. In this talk, I will present a mathematical framework for studying how information—represented as ”tokens”—evolves through the layers of such neural networks. Specifically, we consider a family of partial differential equations that describe how the distribution of tokens—modeled as particles interacting in a mean-field way—changes with depth. Numerical experiments reveal that, under certain conditions, these dynamics exhibit a metastable clustering phenomenon, where tokens group into well-separated clusters that evolve slowly over time. A rigorous analysis of this behavior uncovers a range of open questions and unexpected connections to various fields of mathematics.

ABSTRACT. Transport noise appears in a variety of contexts in applied sciences, especially in fluid dynamics. In the latter, it typically models the effect of small-scale turbulence on large-scale dynamics. As shown in the seminal work of Galeati, and later by Flandoli and Luo, transport noise can provide dissipation effects on the dynamics via a certain scaling limit, thereby leading to regularising phenomena in nonlinear PDEs. The aim of this talk is to provide a guide through the key contributions in this area, up to recent developments in applications to reaction-diffusion equations and the 3D Navier-Stokes equations with small hyperviscosity, in which scaling limits meet and benefit from the maximal L^p-theory of SPDEs.

ABSTRACT. We propose in~\cite{AleCriGhi26} a novel extension of the Indian Buffet Process (IBP) that introduces explicit probabilistic dependence among features in multi-factorial innovation processes. In contrast to classical IBP models \cite{GG06,GG11}, where feature inclusion events are independent, our model allows the inclusion of a feature to influence the inclusion probabilities of others, providing a framework to study interacting latent factors in a mathematically tractable setting.

We develop a rigorous probabilistic formulation and derive asymptotic results for key quantities, including the total number of observed features $D_t$, the averaged number of features per agent/item $\overline{T}_t$, the averaged feature inclusion probability $\overline{P}_t$, the averaged number of agents/items per feature $\overline{K}_t$, as well as feature-specific quantities such as inclusion probability $P_t(j)$ and popularity $K_{t,j}$ for an observed feature $j$. While the asymptotic growth of $D_t$ coincides with that of the classical three-parameter IBP \cite{TG}, the interacting structure induces novel asymptotic phenomena in both averaged and feature-specific quantities, including power-law behavior and non-linear growth in feature popularity, not observed in the standard IBP.

Our theoretical results include strong laws of large numbers and central limit theorems for these quantities, providing probabilistic guarantees and detailed characterization of the stochastic behavior of the system. This model offers a combination of analytical tractability, interpretability, and flexibility, allowing the study of multi-feature systems with explicit interactions, while extending the asymptotic theory of classical IBP models to settings with dependent feature allocations.

ABSTRACT. We will present in this talk sufficient conditions on the kernel and on the coefficients to get the existence of a solution that stays in a convex domain. The underlying tool is an approximation scheme that also stays in this domain. Applications include: a comparison result for scalar SVEs, existence of solutions possibly with a jump component, weak second-order approximation schemes for SVEs with multifactor kernels such as the multi-factor approximation of the rough Heston model.

ABSTRACT. We study a problem of resource extraction cast as a stochastic control problem where the depletion time of the resource is modeled by the hitting time for the controlled dynamics of a random (non-observable) threshold. Such a threshold may represent a tipping point, i.e., a critical level below which we expect a drastic disruption of the underlying source, leading to its extinction. Mathematically, this is formulated as a singular control problem with random time- horizon. The underlying stochastic source X is singularly controlled by the cumulative extraction and it is modeled as a time-homogeneous diffusion process subject to general boundary conditions. The random time horizon is modeled by the first time X drops below a random thresh- old, which is independent of the Brownian motion and distributed according to a cdf F . The problem is cast in a Markovian setting by introducing the running infimum of X as an additional state variable, which leads to a 2-dimensional singular control problem with infinite time-horizon. Under some assumptions on F , we are able to fully characterize the solution of the problem. That is, we show that the optimal strategy consists of extracting resources in such a way that X reflects along a given boundary, which is expressed as a function of the running infimum. Depending on the chosen distribution F , the precise characterization of this boundary requires either solving an auxiliary problem or applying the so-called maximality principle, borrowed from optimal stopping theory, for singular control.

ABSTRACT. We introduce a tractable multivariate pure jump process in which the trading time is described by an additive subordinator. The multivariate process retains the additivity property, and therefore is time inhomogeneous, i.e., its increments are independent but non stationary. We provide the theoretical framework of our process, perform a sensitivity analysis with respect to the time inhomogeneity parameters, and design a Monte Carlo scheme to simulate the trajectories of the process. We then employ the model in the context of option pricing in the FX market. We take advantage of the specific features of currency triangles to extract the joint dynamics of FX log-rates. Extensive tests based on observed market data show that our model outperforms well established pure jump benchmarks. Moreover, we explore applications of our stochastic process to financial optimization problems and propose state-of-the-art derivative-free adaptive sampling algorithms to efficiently compute solutions.

ABSTRACT. Network-structured data are becoming increasingly common across many fields, including the social sciences, biology, physics, and computer science. A central task in network analysis is community detection, which involves partitioning nodes into groups so that nodes within the same group exhibit similar connectivity patterns. A generative model well suited to capturing such communities is the stochastic block model (SBM). Recent work has applied Bayesian nonparametric methods to jointly infer both community structures and the number of communities in the SBM by placing a prior on the number of blocks and estimating block assignments via collapsed Gibbs samplers. However, efficiently incorporating structural community constraints through the prior remains an open challenge. In this work, we address this gap by studying the effect of enforcing weak and strong assortativity as well as core–periphery structure on Bayesian nonparametric community detection for the SBM. We identify scenarios in which these constraints improve performance over the standard SBM and illustrate our results using benchmark datasets.

ABSTRACT. We consider a system of interacting particles with Lipschitz continuous drift functions, driven by additive fractional Brownian motions with H in [1/2 1). For this system, we address the drift parameter estimation problem over a fixed time interval, considering different assumptions for the drift. We propose several estimators, demonstrate their consistency and asymptotic normality as the number of particles tends to infinity, and present a numerical study illustrating our findings.

This talk is based on joint work with Chiara Amorino and Ivan Nourdin, and on ongoing work with Chiara Amorino, Augustin Puel, as well as Yasan Odeh.

ABSTRACT. Since Smoluchowski introduced his well-known coagulation equation in 1917, there has been an active line of research focused on understanding the properties of the solutions to this equation and related models for coagulation. The framework established by Smoluchowski was later extended, allowing particles to have additional properties beyond their mass, such as spatial location. This lead to the introduction of the Smoluchowski coagulation-diffusion PDE, a system of partial differential equations modelling the evolution in time of mass-bearing Brownian particles which are subject to short-range pairwise coagulation. In 2007, Hammond and Rezakhanlou gave a kinetic limit derivation of such equation via a system of microscopic interacting particles moving as Brownian motion in space, see [1]. In this talk we focus on some recent progress in the study of this particle system. We present the approach based on Poisson Point Processes introduced in [2] to study large deviations of the trajectory of such purely coagulating Markov process in the large volume limit and we explain how this approach can be applied to the study of diffusing particles too. We mention as well how this also provides insight into gelation phenomena and phase transitions for the particle system. This talk is based on a series of joint works with W. König, M. Kolodjekzyk, H. Langhammer, E. Magnanini and R.I.A. Patterson.

References 1. Hammond, A., Rezakhanlou, F.: The kinetic limit of a system of coagulating Brownian particles. Arch. Ration. Mech. Anal.,185(1):1–67, (2007) 2. Andreis, L., König, W., Langhammer, H., Patterson, R.I.A.: Spatial particle processes with coagulation: Gibbs-measure approach, gelation and Smoluchowski equation, arXiv preprint arXiv:2401.06668 (to appear in Annals of Probability), (2024)

ABSTRACT. We will introduce the Threshold-driven Streaming Graph model, which is obtained by performing a randomized distributed algorithm, called RAES, over a dynamic graph evolving with the streaming node-churn process. This model captures two key features of modern peer-to-peer networks: a local threshold mechanism that bounds the degree of each vertex, and a node-churn process that regulates how vertices join and leave the network in each round.

Our main result proves good expansion properties of this model, with high probability. As a consequence, we will establish a logarithmic upper bound on the completion time of the well-known PUSH and PULL rumor-spreading protocols. Our analysis will also provide an upper bound to the message-communication overhead, showing that the overall number of exchanged messages at every round t is optimal in expectation and O(log n) with high probability.

ABSTRACT. In recent times, hypergraphs have been frequently used in applications, as for instance in opinion formation [3] and social contagion [2], to describe higher order interactions. For this reason, algorithms to construct hypergraphs with specific characteristics are necessary to better understand, at least numerically, the dynamics on the aforementioned hypergraph. An algorithm for the construction of scale-free directed graphs has been provided in [1]. In particular, the authors obtain a discrete-time stochastic process $(\mathcal{H}_k)_{k \ge 0}$ in the space of directed graphs such that, denoting by $X_{\sf in}^i(k)$ and $X_{\sf out}^i(k)$ the number of nodes respectively with indegree and outdegree equal to $i$, it holds $X_\eta^i(k)=r^i_\eta k +o(k)$ where $r_\eta^i \sim Ci^{-\epsilon_{\eta}}$ and $\eta \in \{{\sf in}, \ {\sf out}\}$. In this talk we first generalize the approach in [1] to a generic setting for stochastic recursive equations in discrete time and then we use the general results to provide algorithms for the constructions of random sequences of directed hypergraphs $(\mathcal{H}_k)_{k \ge 0}$ such that $k^{-1}X_\eta^i(k) \asymp r^i_\eta$ where $r_\eta^i \sim Ci^{-\epsilon_{\eta}}$ and $\eta \in \{{\sf in}, \ {\sf out}\}$. In particular, our algorithm allows to avoid self-loops, hence also covering the case of directed graphs with no self-loops that was missing in [1].

ABSTRACT. The stick-breaking representation is a popular way of defining the Dirichlet process by the associated sequence of probability weights. It is particularly appealing when the discrete random measure is convolved with a suitable kernel: in this context, the stick-breaking construction is often truncated and posterior inference can be performed using a finite number of parameters. Despite its relevance, little is known about the posterior distribution of the weights in a mixture framework.

Assuming that the data are generated by a mixture with the same kernel and $K^*$ components, we deduce some asymptotic properties of the stick-breaking weights. In particular, an interesting phase transition is observed: the posterior assigns mass to the first $K^*$ weights up to the parametric rate, while any further improvement requires a logarithmic (with respect to the size of the dataset $n$) number of components. Thus the model adapts to the correct number of components, but the mixing measure assigns $\mathcal{O}(n^{-1/2})$ mass to additional terms (which can be thought as the price of having a nonparametric specification).

We use such results to shed some light on the clustering properties of Dirichlet process mixtures (e.g. number of clusters) and to provide posterior guarantees for computational methods based on truncation. The mathematical derivations combine prior properties with tools from Bayesian asymptotics and empirical process theory.

ABSTRACT. node2vec random walks are tuneable random walks that come from the popular algorithm node2vec which is used for network embedding. The transition probabilities of the random walks depend on the previous visited node and on the triangles that contain the current and the previous node. In the node2vec algorithm, node2vec random walks are used to sample neighborhoods for each node of the network and by comparing these an embedding of the network into a Euclidean space can be computed. Since the parameters of the random walks can be tuned to create different types of neighborhoods, this approach is very flexible and advantageous over just using simple random walks.

Even though the algorithm is widely used in practice, mathematical properties of node2vec random walks almost have not been investigated and even basic questions such as how the stationary distribution depends on the walk parameters and if the random walk is recurrent are nearly unexplored. In this talk, we study the behavior of node2vec random walks on regular graphs. By going to a higher-order state space, the space of directed wedges, we can prove a simple expression of the stationary distribution on this space which is determined by the transition type of the wedge. We also formalize a pullback mechanism to retrieve the stationary distribution on the original state space. Further, we show that on infinite regular graphs, node2vec random walks are recurrent if and only if the simple random walk is recurrent.

ABSTRACT. Local volatility models are generally seen as insufficient for handling the many nuances of modern derivative markets. By reverse-engineering a family of no-arbitrage call price functions, this research questions a number of claims in this direction.

We introduce a class of continuous Markovian asset pricing models with closed-form option prices, leading -- by construction -- to identifiable risk-neutral marginal distributions, and then specialize to a significant instance where the SDE well-posedness can be shown, the generalized beta local volatility (GBLV) model. The GBLV finite-dimensional distributions coincide with those of a known discontinuous martingale model that exhibits an at-the-money implied volatility skew divergence. These findings contrast with the commonly accepted wisdom that LV is unsuitable for capturing the implied volatility surface's singular behavior as time-to-maturity approaches zero, and that option prices from jump models cannot be fitted to continuous Markov models. Such claims, typically regarded as valid for the \emph{whole} local volatility class, ultimately hinge on auxiliary assumptions, most notably, regularity of the diffusion coefficient at initial time. By directly embedding in the risk-neutral distributions the desirable properties an implied volatility surface should have, an LV model is freed from the constraints that make it unsuitable for capturing certain phenomena. As a consequence, the GBLV model does not suffer from several of the commonly exposed drawbacks of continuous Markovian models.

ABSTRACT. We introduce a new model which incorporates three key ingredients of a large class of wireless communication systems: (1) spatial interactions through interference, (2) dynamics of the queueing type, with users joining and leaving, and (3) carrier sensing and collision avoidance as used in, e.g., WiFi. In systems using (3), rather than directly accessing the shared resources upon arrival, a customer is considerate and waits to access them until nearby users in service have left. This new model can be seen as a missing piece of a larger puzzle that contains such dynamics as spatial birth-and-death processes, the Poisson hail model, and wireless dynamics as key other pieces. We show that, under natural assumptions, this model can be represented as a Markov process on the space of counting measures.

The main results are then two-fold. The first is on the shape of the stability region and, more precisely, on the characterization of the critical value of the arrival rate that separates stability from instability. We show that, for natural values of the system parameters, the implementation of sensing and collision avoidance stabilizes a system that would be unstable if immediate access to the shared resources would be granted. In other words, for these parameters, renouncing greedy access makes sharing sustainable, whereas indulging in greedy access kills the system.

ABSTRACT. The nonsmooth Kraichnan model \cite{Kra1968} is a linear stochastic transport model in which the velocity field is Gaussian, white in time, incompressible, isotropic, and spatially rough. Introduced as a toy model for turbulent transport, it allows explicit computations and has become a benchmark for predictions such as Richardson pair dispersion and intermittency.

In mathematics, it stands out as one of the few transport models exhibiting both well-posedness and spontaneous stochasticity, as shown in the early 2000s by Le Jan and Raimond \cite{LeJRai2002}, and independently by E and Vanden-Eijnden \cite{EVan2001}. More recently, renewed interest has followed rigorous results on anomalous dissipation and anomalous regularization (e.g. \cite{Row2024,GalGroMau2024,DriGalPap2025}), as well as related regularization-by-noise results (e.g. \cite{CogMau2023,BagGalMau2025}).

In this talk, we will review these mathematical developments and highlight key open problems in stochastic turbulent transport.

ABSTRACT. We prove strong well-posedness results for the stochastic 2D Euler equations in vorticity form and generalized SQG equations, with $L^p$ initial data and driven by a spatially rough, incompressible transport noise of Kraichnan type. Previous works addressed this problem with noise of spatial regularity $\alpha\in (0,1/2)$, in a setting where a rougher noise yields a stronger regularization. We remove this limitation by allowing any $\alpha \in (0,1)$, covering the same range of parameters for which anomalous regularization effects are known to occur in passive scalars. In particular, this covers the physically relevant case $\alpha=2/3$, associated with the Richardson-Kolmogorov scaling of energy cascade.

ABSTRACT. The focus of this talk will be Susceptible-Infected-Recovered (SIR) models on dense dynamic random graphs, in which the joint dynamics of vertices and edges are co-evolutionary, i.e., they influence each other bidirectionally. In particular, edges appear and disappear over time depending on the states of the two connected vertices, on how long they have been infected, and on the total density of susceptible and infected vertices. I will present our main results, which establish functional laws of large numbers for the densities of susceptible, infected, and recovered vertices, jointly with the underlying evolving random graphs in the graphon space. The talk will also include numerical illustrations showing that our model exhibits multiple epidemic peaks, as observed in real-world epidemics.

This talk is based on a joint work with P. Braunsteins, F. den Hollander and M. Mandjes.

ABSTRACT. We will discuss Spectral Gap of Stochastic Wave equations with additive noise, and it's relation to the heat equation case. Our methods combine techiques from Hypocoercivity in an infinite dimesional setting, with techniques from singuarl SPDE's.

ABSTRACT. Almost conditional identically distributed (a.c.i.d.) random variables, \cite{bc25}, are generalizations of conditional identically distributed (c.i.d.), \cite{berti2004limit}, and exchangeable, \cite{ald85}, random ones. This class of random variables naturally arises in applications in statistics, such as in recursive algorithms, contamination models and heteroskedastic observations. The definition of almost conditional identially distributed random variables depends on a sequence of parameters that quantifies departure from exchangeability and conditional identical distribution. An alternative definition of these processes is as measure-valued almost supermartingales.

In this talk, I will first introduce this new class of random variables, illustrating some specific examples in statistics. Secondly, I will present new limit theorems that extend those for exchangeable and c.i.d. random variables to the more general setting of a.c.i.d. random variables. Specifically, asymptotic exchangeability, a Strong Law of Large Numbers and three different Central Limit Theorems, involving respectively the predictive, empirical and asymptotic directing distributions of the process, are presented. Also, necessary and sufficient conditions for the asymptotic directing measure of the sequence to be absolutely continuous with respect to a given sigma-finite measure are described. These theorems have statistical applications, especially in Bayesian predictive inference.

ABSTRACT. We introduce the Wick integral ∫f(X)♢dX for a class of stochastic processes X which are not necessarily Gaussian, in the regime of bounded 2>q-variation. The integral is defined for polynomial integrands, and has the property of being centred if X is such. In the case of 1/2 < H-fractional Brownian motion, the Wick integral agrees with the divergence operator in Malliavin calculus. It satisfies a correction formula with the Young integral ∫f(X)dX and an Itô formula which have infinitely many correction terms, given by integration against the cumulant functions of X, and reduce to familiar identities in the Gaussian case. These results are obtained by first developing diagram formulae for Appell polynomials. Our theory applies to a range of processes taking values in bounded Wiener chaos, such as the Rosenblatt process.

ABSTRACT. Propagation of chaos is a well-known technique formally introduced in the physics literature by Marc Ka\v{c} in the 50s to simplify the study of Boltzmann equation and giving rise, for instance, to the mathematically more tractable Vlasov-like equations. In the following years, this approach has been repeatedly applied to both deterministic and stochastic particle systems, and it is nowadays part of the standard tools used in stochastic processes and statistical mechanics to prove Law of Large Numbers results. However, with the more and more interest of the current research in studying complex systems, the assumption of chaotic initial data is too stringent with regards to describing real-world phenomena.

In this talk, I am going to present recent results on Law of Large Numbers of the empirical measure without assuming any hypothesis on the initial datum but the convergence at time zero. The biggest challenge would be to tackle equations with non-linear coefficients and replace the standard topology in the space of probabilities induced by the Wasserstein distance, with a weaker notion of convergence but more suitable for non-chaotic systems.

ABSTRACT. Lambda-quantiles are a generalisation of classical quantiles and have originally introduced in the financial literature by Frittelli et al.~\cite{ref_article1}. They are obtained by replacing the fixed probability level $\lambda \in [0,1]$ in the usual definition of a quantile with a functional parameter $\Lambda \colon \mathbb{R} \to [0,1]$. When $\Lambda$ is decreasing, $\Lambda$-quantiles are known to share many properties with classical quantiles, and they have thus received growing attention in recent years in financial and insurance applications as well as from a decision-theoretic perspective.

In this talk, we advocate the use of general, possibly non-monotonic functional parameters~$\Lambda$. Under minimal assumptions, we examine how the choice of~$\Lambda$ affects the mathematical properties of the resulting functional. In particular, we study aggregation behaviour, weak continuity, mixture representations, and generalised ordinal covariance properties. Additionally, we show that the latter also provides an axiomatic characterisation of a broad class of~$\Lambda$-quantiles, even when the functional parameter is not monotone.

ABSTRACT. The central result of the theory of choice under uncertainty is Von Neumann and Morgenstern's expected utility theorem, stating that an economic agent whose preference relation among discrete probability measures satisfies suitable rationality axioms, is represented by an expected utility, i.e. by the expected value of a monetary utility function of outcomes.

A remarkable extension of expected utility is Gul's (1991) theory of disappointment aversion, based on a slight weakening of the independence axiom of the vNM theory.

Our first contribution is to point out the connection between the representing functional of Gul's preferences and the probabilistic notion of expectile, a one-parameter family of functionals introduced by Newey and Powell (1987) for asymmetric least squares regression. Indeed, it turns out that the Gul's functional is an expectiled utility, depending on two parameters: a vNM utility function $u$ and a disappointment-aversion parameter $\beta$.

Further, we recast Gul's theory in a Savage framework where the preference is defined over acts with general, possibly non-monetary outcomes, relying on the notion of subjective mixture of acts with general outcomes introduced by Ghirardato et al. (2003).

We introduce a novel axiom of disappointment hedging, that is a stronger version of the axiom of ambiguity hedging introduced by Ghirardato et al. (2003), and we show in our main result that a preference relation over Savage acts is probabilistically sophisticated, invariant biseparable, and disappointment hedging if and only if it is an expectiled utility.

ABSTRACT. We introduce and study a unified Bayesian framework for extended feature allocations which flexibly captures interactions -- such as repulsion or attraction -- among features and their associated weights. We provide a complete Bayesian analysis of the proposed model and specialize our general theory to noteworthy classes of priors. This includes novel priors based on (i) determinantal point processes, which yield promising results in a spatial statistics application, and (ii) shot noise Cox processes, illustrated on genetics and ecological examples. Within the general class of extended feature allocations, we further characterize those priors that yield predictive probabilities of discovering new features depending either solely on the sample size or on both the sample size and the distinct number of observed features. These predictive characterizations, known as ``sufficientness'' postulates, have been extensively studied in the literature on species sampling models starting from the seminal contribution of the English philosopher W.E. Johnson for the Dirichlet distribution. Within the feature allocation setting, existing predictive characterizations are limited to very specific examples; in contrast, our results are general, providing practical guidance for prior selection.

ABSTRACT. The Critical 2D Stochastic Heat Flow (SHF) serves as the universal measure-valued solution to the singular 2D stochastic heat equation. This talk focuses on the asymptotic behavior of the SHF in the large-time, large-disorder regime. We establish a sharp form of local extinction, identifying the precise rate at which the distribution collapses. Furthermore, we characterize the spatial scales governing the phase transitions between extinction and averaged behavior, as well as vanishing versus diverging mass. Parallel results are derived for 2D directed polymer partition functions. These findings offer crucial insights into the 2D SHE regularized via space-time discretization. We show that for any regime of supercritical disorder strength β (including fixed β > 0), the solution exhibits superdiffusive fluctuations. The proof relies on, and introduces, novel refinements of change of measure and coarse-graining techniques.

ABSTRACT. In an arbitrage-free simple market, we demonstrate that for a class of state-dependent exponential utilities, there exists a unique prediction of the random risk aversion that ensures the consistency of optimal strategies across any time horizon. Our solution aligns with the theory of forward performances, with the added distinction of identifying, among the infinite possible solutions, the one for which the profile is the actual optimizer of the system of preferences specified a priori.

ABSTRACT. We consider the problem of detecting whether a power-law inhomogeneous random graph contains a geometric community, and we frame this as a hypothesis-testing problem. More precisely, we assume that we are given a sample from an unknown distribution on the space of graphs on $n$ vertices. Under the null hypothesis, the sample originates from the inhomogeneous random graph with a heavy-tailed degree sequence. Under the alternative hypothesis, $k=o(n)$ vertices are given spatial locations and connect following the geometric inhomogeneous random graph connection rule. The remaining $n-k$ vertices follow the inhomogeneous random graph connection rule. We propose a simple and efficient test based on counting normalized triangles to differentiate between the two hypotheses. We prove that our test correctly detects the presence of the community with high probability as $n\to\infty$, and identifies large-degree vertices of the community with high probability.

ABSTRACT. Exponential Random Graphs are a class of network models that can be seen as the gen- eralization of the dense Erdős–Rényi random graph. They are defined, with a statistical mechanics approach, by introducing a Hamiltonian, a function that biases the occurrence of certain features, such as the number of edges or triangles. In this talk we will primarily focus on the so-called edge triangle model, where the Hamiltonian of the system only collects edge and triangle densities, properly tuned by real parameters. Using tools from statistical mechanics and large deviation theory, we establish limit theorems and concentration inequalities for subgraph densities (mainly focusing on edge and triangle density) in the replica-symmetric regime, where the limiting free energy of the model is known together with its phase diagram. Part of the results are concerned with a mean-field approximation, which allows for explicit computations and provides insights into the behavior of the original model in certain parameter region where rigorous results are hardly achievable. A generalization of the model in which vertices are allowed to carry a type will also be discussed. This talk is based on joint work with A. Bianchi, F. Collet, and G. Passuello.

ABSTRACT. In this talk, we will analyze the convergence to equilibrium of a simple random walk on a directed version of the classical Stochastic Block Model with $m$ communities. We show that the mixing behavior of the walk exhibits a trichotomy governed by the parameter $\alpha$, which controls the strength of inter-community interactions. In the subcritical regime (large $\alpha$) the dynamics displays cutoff at at the entropic timescale $T^* \sim \log(n)/\log\log(n)$. In the supercritical regime (small $\alpha$) the mixing is driven by rare inter-community transitions, leading to a metastable behavior. After an abrupt jump at timescale $T^*$, the distance to equilibrium decays smoothly at an exponential rate on the timescale $1/\alpha$. At criticality (when $1/\alpha\sim T^*$), an intermediate behavior emerges, characterized by an interplay between entropic mixing and inter-community transitions. Joint work with G. Passuello and M. Quattropani.

ABSTRACT. This contribution extends the concept of univariate extremiles introduced by Daouia et al.2019 to a robust multivariate framework. Among the possible multivariate generalizations, we adopt the approach based on the multivariate $M$-quantiles proposed by Kokic et al. 2002.

The proposed formulation ensures that multivariate extremiles lie within the convex hull of the data while allowing for different levels of robustness. We prove the main mathematical and statistical properties of these robust multivariate extremiles and assess their empirical performance through a series of examples on artificial data.

In the presence of covariates, the methodology can be further extended to define multivariate extremile regression, providing a flexible and robust tool for multivariate conditional analysis.

ABSTRACT. The study of extreme values of real random variables and their limiting laws is a cornerstone of probability theory [1]. In many probability models, however, the natural state space is high dimensional and only partially ordered. In such settings, most pairs of elements are incomparable [2], and the classical notions of extrema no longer apply. This motivates replacing ordinary minima and maxima with their order-theoretic counterparts, meet and join, that remain meaningful under partial comparability [3].

We study extreme value phenomena for random samples from partially ordered sets. In particular, we consider the majorization and unordered majorization on finite-dimensional probability simplices, where the order is induced by linear stochastic transformations [4]. These partial orders are relevant in many contexts, ranging from economics [5] to thermodynamics and quantum information [6].

Given independent random elements, we consider their meet or join, and how these objects behave in the limit as the sample size and the dimension of the space grows. We describe regimes in which meets and joins typically collapse to extreme (bottom and top) elements, as well as scaling regimes where nondegenerate limits appear, describing the order of fluctuations close to the boundary.

References

1. Majumdar, S. N., Schehr, G.: Statistics of Extremes and Records in Random Sequences. Oxford Graduate Texts. Oxford University Press, (2024) 2. Cunden F.D., Czartowski J., Gramegna G., and de Oliveira Junior A. : Relative volume of comparable pairs under semigroup majorization. Letters in Mathematical Physics 115(4), 79 (2025). 3. Stanley, R. P.: Enumerative combinatorics, Volume 1. 2nd edn. Cambridge studies in advanced mathematics (2011). 4. Marshall, A. W., Olkin, I., Arnold B.C.: Inequalities: theory of majorization and its applications, Springer (1979). 5. Mosler, K.: Majorization in economic disparity measures. Linear Algebra and its applications 199, 91-114 (1994). 6. Gour, G., Müller M.P., Narasimhachar V., Spekkens R.W., Halpern N.Y.: The resource theory of informational nonequilibrium in thermodynamics. Physics Reports 583, 1-58 (2015)

ABSTRACT. Consider a random matrix $X$ whose entries are i.i.d.\ in the cells of a Young diagram (its `shape') and zero elsewhere. When the shape is the dilation by a factor $N$ of a fixed Young diagram $\lambda$, the Wishart-type matrix $XX^*$ (suitably rescaled) has, as $N\to\infty$, a limiting spectral distribution $F^{\lambda}$ characterised by its moments. These moments enumerate $\lambda$-plane trees, a class of directed plane trees with vertex labelling compatible with $\lambda$, for which we provide explicit enumerative formulae. We show that one cannot `hear the shape of a random matrix', in the sense that there exist distinct Young diagrams yielding the same limiting spectral distribution. We establish that the classes of `isospectral' Young diagrams are those with the same diagonal profile.

ABSTRACT. We derive a Gronwall type inequality for mild solutions of non-autonomous parabolic rough partial differential equations (RPDEs). This inequality together with an analysis of the Cameron-Martin space associated to the noise, allows us to obtain the existence of moments of all order for the solution of the corresponding RPDE and its Jacobian when the random input is given by a Gaussian Volterra process. Applying further the multiplicative ergodic theorem, these integrable bounds entail the existence of Lyapunov exponents for RPDEs. We illustrate these results for stochastic partial differential equations with multiplicative boundary noise. This talk is based on a joint work with Mazyar Ghani Varzaneh and Tim Seitz.

ABSTRACT. We consider a family of graphs $\{G_k,\ k=1, \dots, N\}$, each associated to the (discrete or continuous) {\em Laplacian} operator $\mathcal L_k$ acting on the function defined on the vertices (edges) of the graph.

Given a stochastic mechanism of switching the graphs during time, we get that the evolution is lead by an operator $\mathcal L_{X_k}$ (selected from the set $\{ \mathcal L_1, \dots, \mathcal L_N\}$ according to some Markov chain $X_k$) during the (random) time interval $[T_k,T_{k+1})$ \begin{equation} \label{e1} \begin{cases} \partial_t u(t,x) = \mathcal L_{X_k} u(t,x), \qquad t \in [T_k, T_{k+1}), \\ u(0,x) = f(x). \end{cases} \end{equation} We can associate to (\ref{e1}) the (random) evolution operator \[ S(t) = e^{(t - T_n)\mathcal L_{X_n}} \prod_{k=0}^{n-1} e^{(T_{k+1} - T_k) \mathcal L_{X_k}}, \qquad t \in [T_n, T_{n+1}). \]

Our main problem can be stated as follows: \begin{itemize} \item[({\bf P})] under which condition the random evolution operator $S(t)$ converges? towards which limit? \end{itemize}

ABSTRACT. We show weak convergence of the marginals for a re-scaled rough Heston model to a Normal Inverse Gaussian (NIG) Lévy process. In particular, we introduce a scaling technique that does not depend on the Hurst parameter in the fractional kernel. We later extend our approach to the case where the variance is an affine Volterra process with jumps, and establish weak convergence of the finite-dimensional distributions of the integrated variance to a deterministic time-change of the first-passage time process to lower barriers for a more general class of spectrally positive Lévy processes.

ABSTRACT. We are interested in studying the well-posedness of a non-local singular McKean SDE, whose drift coefficient is a function of time taking values in a Besov Space of negative index and its diffusion is unitary. Due to the singularity of the coefficient, we must rely on a notion of solution for singular SDEs that is framed through the rough martingale problem. The solution to a rough martingale problem is a probability measure which corresponds to the law of X, solution to such an SDE. Existence and uniqueness of such a measure relies on the well-posedness of an associated non-local singular non-linear Fokker-Planck PDE. We prove that the solution to the non-local singular McKean SDE is the probabilistic representation of the above mentioned Fokker-Planck PDE.

ABSTRACT. We introduce a canonical way of performing the joint lift of a Brownian motion W and a low-regularity adapted stochastic rough path X, extending Diehl-Oberhauser-Riedel (2015). Applying this construction to the case where X is the canonical lift of a one-dimensional fractional Brownian motion (possibly correlated with W) completes the partial rough path of Fukasawa-Takano (2024). We use this to model rough volatility with the versatile toolkit of rough differential equations (RDEs), namely by taking the price and volatility processes to be the solution to a single RDE. We argue that our framework is already interesting when W and X are independent, as correlation between the price and volatility can be introduced in the dynamics. The lead-lag scheme of Flint-Hambly-Lyons (2016) is extended to our fractional setting as an approximation theory for the rough path in the correlated case. Continuity of the solution map transforms this into a numerical scheme for RDEs. We numerically test this framework and use it to calibrate a simple new rough volatility model to market data.

ABSTRACT. In the setting of stochastic Volterra equations, and in particular rough volatility models, we show that conditional expectations are the unique classical solutions to path-dependent PDEs. The latter arise from the functional Itˆo formula developed by [Viens, F., & Zhang, J. (2019). A martingale approach for fractional Brownian motions and related path dependent PDEs. Ann. Appl. Probab.]. We then leverage these tools to study weak rates of convergence for discretised stochastic integrals of smooth functions of a Riemann-Liouville fractional Brownian motion with Hurst parameter $H \in (0,1/2)$. These integrals approximate log-stock prices in rough volatility models. We obtain the optimal weak error rates of order 1 if the test function is quadratic and of order $\min\{(3H+1/2),1\}$ if the test function is five times differentiable; in particular these conditions are independent of the value of H.

ABSTRACT. Many applications in machine learning involve data represented as probability distributions. The emergence of such data requires radically novel techniques to design tractable gradient flows on probability distributions over this type of (infinite-dimensional) objects. For instance, being able to flow labeled datasets is a core task for applications ranging from domain adaptation to transfer learning or dataset distillation. In this setting, we propose to represent each class by the associated conditional distribution of features, and to model the dataset as a mixture distribution supported on these classes (which are themselves probability distributions), meaning that labeled datasets can be seen as probability distributions over probability distributions. We endow this space with a metric structure from optimal transport, namely the Wasserstein over Wasserstein (WoW) distance, derive a differential structure on this space, and define WoW gradient flows. The latter enables to design dynamics over this space that decrease a given objective functional. We apply our framework to transfer learning and dataset distillation tasks, leveraging our gradient flow construction as well as novel tractable functionals that take the form of Maximum Mean Discrepancies with Sliced-Wasserstein based kernels between probability distributions.

ABSTRACT. We discuss the stochastic sine--Gordon model in 1+1 dimensions from the perspective of interacting (algebraic) field theory, following the approach developed first in [DDRZ22] and then in [BDR24]. The guiding idea is to realize the random field as an element of a suitable algebra of functional-valued distributions, so that tools from microlocal analysis can be systematically employed to control products, singularities, and the emergence of counterterms. Within this framework, renormalization is implemented in an Epstein--Glaser spirit, i.e. by local and causal constructions rather than by choosing a specific regularization scheme.

In the ultraviolet-cutoff theory, we construct correlation functions and moments of the stochastic sine--Gordon field as convergent power series in the coupling, and we analyze their stability under the removal of auxiliary parameters. A key outcome is a robust perturbative construction that also admits a controlled classical limit \hbar -> 0^+, thereby connecting the stochastic dynamics with the corresponding interacting field theory. Finally, we briefly comment on the bosonization link between sine--Gordon and Thirring: while our focus is entirely on the sine--Gordon analysis of [BDR24], this correspondence provides a natural bridge to the spinorial setting investigated in [BCDR24].

ABSTRACT. The Martin-Siggia-Rose (MSR) formalism is a path-integral approach widely used in the physics literature to compute expectation values and correlation functions associated with stochastic differential equations (SDEs). Despite its effectiveness, the formalism has long lacked a fully rigorous mathematical foundation. This issue has been partially addressed in \cite{Bonicelli_Dappiaggi_Drago_2025} by employing techniques from the algebraic approach to quantum field theory, which provides a robust framework for a rigorous treatment of path-integral formulations and for the solution theory of ordinary, partial, and stochastic differential equations. Within this framework, we establish---at the level of perturbation theory---a precise correspondence between correlation functions and expectation values computed either directly from the SDE or via the MSR formalism. Time permitting, we will also discuss a complementary, more analytical approach to the MSR formalism based on the theory of infinite dimensional Fresnel path integrals, following \cite{Bonicelli_Drago_Mazzucchi_2025}.

ABSTRACT. We propose a stochastic description of the time dependent quantum Bose-Einstein condensate at zero temperature, within the context of Nelson stochastic mechanics. We describe an infinite particle limit of interacting diffusions which corresponds to the mean field limit in the related quantum system. We are able to extend the framework of Nelson stochastic mechanics to nonlinear systems in particular to the case of the nonlinear Schr\"odinger equation. We also propose how to extend to this nonlinear case the Guerra-Morato variational approach. Our work can also be seen in the context of a mean field limit of McKean–Vlasov processes in a general situation where the drift is a very singular function depending non-trivially on all the particles.

ABSTRACT. Conformal prediction is a nonparametric method widely applied in regression, classification, and outlier detection, providing valid predictive inference with finite-sample coverage guarantees. Marginal coverage, in particular, is a fundamental objective in conformal inference, ensuring that prediction sets contain the correct label for a predefined proportion of future test points. However, these guarantees rely on the assumption of data exchangeability, which is often violated in real-world applications due to distribution shifts, outliers, and label noise. In Bortolotti et al. (2025), we address the limitations of conformal classification in the presence of label contamination and propose novel adaptive methodologies that automatically adjust for noise to restore marginal coverage. Our theoretical guarantees are derived under the assumption that the contamination mechanism is known. We show how label noise induces a systematic inflation of coverage and, leveraging tools from empirical process theory, we derive correction factors that restore nominal marginal guarantees. The resulting adaptive calibration procedures provide valid and informative prediction sets even in challenging classification settings with many classes or severe class imbalance. To make the framework fully data-driven, we complement our theoretical results with a practical strategy to estimate the contamination process from noisy data. Specifically, we propose a procedure based on the identification of anchor points, i.e., observations for which the conditional probability of a class is close to one. These points allow us to consistently estimate the class-dependent contamination matrix without requiring access to clean data. The estimated contamination mechanism is then plugged into the adaptive calibration step. The effectiveness of our approach is demonstrated through extensive experiments on synthetic and real-world datasets, including CIFAR-10H and BigEarthNet. Our findings highlight the importance of accounting for label contamination in conformal classification and provide a robust framework for reliable predictive inference in noisy settings.

ABSTRACT. This talk studies a problem arising in a Principal-Agent framework, analysed from the Principal’s perspective. The problem is formulated as a finite-horizon optimal stochastic control problem for a possibly degenerate process with absorption at the boundary. The controlled process represents the contract offered by the Principal, whose objective is to maximise over all admissible contracts offered to the Agent. Properties of the value function are obtained using both probabilistic and analytical techniques. In particular, we establish the existence of a classical solution of the related Hamilton-Jacobi-Bellman equation which allows to characterise explicitly the optimal contract offered by the Principal. Finally, we underline properties of the optimal contract and discuss their economic implications.

ABSTRACT. Reliable decision-making relies on predictive sets that capture the true outcome with a specified probability. In this talk, I will explore Conformal Prediction, a statistical approach that delivers rigorous finite-sample guarantees. While standard conformal methods provide valid marginal coverage, they do not ensure coverage conditional on specific inputs. I will present a framework to quantify conditional miscoverage and discuss strategies to improve conditional reliability, including the tradeoff between set size and conditional coverage. The talk will highlight both theoretical insights and practical implications for building trustworthy predictive systems.

ABSTRACT. A simple yet extremely valuable approach to the study of the climate system comes from the use of Energy Balance Models (EBMs). Such models describe the key features of the zonally averaged temperature on the Earth’s surface. The classical EBM can be improved by increasing the vertical resolution. This talk presents a two-layer energy balance model that allows for vertical exchanges between a surface layer and the atmosphere. Considering random perturbations of the model will allow to better study its long-time average behaviour. Thanks to the weak Harris’ theorem, we will establish exponential ergodicity. This is the first step to study the model dependence on different forcing scenarios via response theory.

ABSTRACT. We study the inference-time evolution of token representations in deep residual streams of encoder-only transformers through a mean-field interacting particle system framework. Motivated by recent context-scaling practices in large language models, where the inverse temperature parameter $\beta$ grows with the number of tokens $N$, we analyze the moderate interaction regime and show that the dynamics exhibits a multiscale structure that reconciles several previously observed behaviors into a unified picture.

Starting from the continuous-depth limit of a layer-normalized self-attention dynamics, we analyze the associated continuity equation on $\mathcal P(\mathbb S^{d-1})$ as the inverse temperature $\beta=\beta_N$ diverges with $N$. Our main technical result identifies a fast \emph{alignment phase} on an $O(1)$ timescale: under mild assumptions on the parameter matrices and on the initial particle distribution, the mean-field dynamics converges to a linear transport equation in which the token distribution collapses onto a low-dimensional subspace dictated by the spectral properties of the matrix $V K^t Q$.

We then show that, once aligned, the next-order behavior emerges on an $O(\beta)$ timescale and is governed by a heat flow on the aligned manifold under additional structural assumptions on $(Q,K,V)$. Finally, on exponentially long timescales in $\beta$, we describe a \emph{pairing phase} where clusters sequentially merge along geodesics, captured by an effective finite-dimensional ODE for the closest pair of clusters. Numerical experiments illustrate all three phases and their separation of timescales.

ABSTRACT. In this work we show that rough stochastic differential equations (RSDEs), as introduced by Friz, Hocquet, and Lê (2021), are Malliavin differentiable. We use this to prove existence of a density when the diffusion coefficients satisfies standard ellipticity assumptions. Moreover, when the coefficients are smooth and the diffusion coefficients satisfies a Hörmander condition, the density is shown to be smooth. The key ingredient is to develop a comprehensive theory of linear rough stochastic differential equations, which could be of independent interest.

ABSTRACT. We present a well-posedness result for Fokker-Planck equations, which describe the evolution of the conditional law of McKean-Vlasov SDEs in the presence of common noise. Such an evolution is governed by a nonlinear, nonlocal SPDE in the space of measures. The well-posedness of such SPDEs is a difficult problem, and the best result to date is due to Coghi and Gess (2019), which however comes with dimension-dependent regularity assumptions.

In this talk, we show how rough path techniques can circumvent these entirely. We consider a mixed rough and stochastic setting, which allows us to derive well-posed rough (deterministic) counterparts of the nonlinear Fokker-Planck equations under dimension-independent regularity assumptions. Importantly, the rough Fokker-Planck equations are seen, upon randomisation, to coincide with the classical nonlinear SPDEs. Therefore, and somewhat contrarily to common belief, the use of rough paths leads to substantially less regularity demands on the coefficients when compared to methods rooted in classical stochastic analysis.

Joint work with Peter K.\ Friz and Wilhelm Stannat (arXiv:2507.17469).

ABSTRACT. Real-world decision-making scenarios are characterized by vagueness, plural interpretations, and partially conflicting information. In natural language processing, these features become particularly evident in tasks involving moral or value-based judgments, where annotations reflect heterogeneous and context-dependent perspectives rather than single ground-truth labels. This contribution examines how such forms of structured uncertainty can be formally modeled by integrating probabilistic and logical approaches within Large Language Models (LLMs).

We first analyze moral value classification as a many-valued problem, where labels should not be interpreted as deterministic assignments but as distributions over admissible interpretations [1]. Inter-annotator disagreement is thus treated not as noise, but as an observable manifestation of epistemic variability. To capture this structure, we introduce a generalized agreement metric, F1-kappa, which extends the classical F1-score by normalizing performance against its expected value under a probabilistic baseline, analogously to Fleiss’ kappa [2]. This formulation enables direct comparison between human and model performance while accounting for label multiplicity, annotator diversity, and class imbalance. From a semantic standpoint, the resulting framework can be interpreted as defining a probabilistic layer over a many-valued labeling space, thereby bridging categorical evaluation and distributional reasoning.

In a second step, we address the complementary problem of controlling generative models under structural constraints. While LLMs approximate probabilistic semantics through next-token prediction, they lack intrinsic guarantees of syntactic or logical well-formedness. We propose a hybrid decoding framework in which formal grammars act as symbolic constraints over stochastic generation [3]. Constrained decoding can be formally characterized as restricting the support of the underlying probability distribution to strings belonging to a language defined by a context-free grammar. This hybridization preserves probabilistic flexibility while ensuring compliance with predefined structural or semantic requirements.

Taken together, these two directions illustrate a broader methodological claim: modeling vagueness in AI systems requires moving beyond purely statistical or purely symbolic paradigms. Many-valued annotations, probabilistic normalization of agreement, and grammar-constrained generation provide complementary tools for integrating logical structure with uncertainty quantification. This hybrid perspective offers a principled approach to reasoning and decision-making under vagueness, aligning formal evaluation methods with the intrinsic indeterminacy of complex linguistic and moral domains.

ABSTRACT. We consider a one-dimensional Wiener process with zero drift initially, which changes at some random and unobservable moment, referred to as the disorder time. We observe the evolution of the process in real time with the goal of detecting the disorder time as precisely as possible. Unlike Shiryaev's seminal work from the 1960s on the Wiener disorder problem, which assumes a known and fixed value of the post-disorder drift, we assume that the post-disorder drift is a discrete random variable with a known distribution. This formulation is particularly useful when the post-disorder regime is unknown, but past data and/or expert opinions can be used to construct a prior distribution for the new drift. Under the additional assumptions that (a) the disorder time is exponentially distributed and (b) the disorder time, the initial Wiener process with zero drift, and the post-disorder drift are independent, we show that the solution to our problem can be expressed in terms of a stopping time which minimizes a linear combination of the probability of a false alarm and the expected detection delay since the onset of the disorder. This stopping time can be characterized as the first moment at which the coordinate processes of the posterior probability that the disorder has already occurred - given the observed path of the Wiener process - enter a region shaped by a curved boundary, where the latter is the unique solution to a certain integral equation.

ABSTRACT. In the last few years it was proved that scalar passive quantities subject to suitable stochastic transport noise, and more recently that also vector passive quantities subject to suitable stochastic transport and stretching noise, weakly converge to the solutions of deterministic equations with a diffusion term. In the background of these stochastic models, we introduce stochastic Vlasov equations which give additional information on the fluctuations and oscillations of solutions: we prove convergence to non-trivial Young measures satisfying limit PDEs with suitable diffusion terms. In the case of a passive vector field, the background Vlasov equation adds completely new statistical information to the stochastic advection equation. This talk is based on a joint work with Franco Flandoli, Eliseo Luongo and Yassine Tahraoui

ABSTRACT. This paper studies the optimal timing of annuitization when individual mortality is only partially observable. Annuities provide insurance against longevity risk by converting wealth into a lifelong income stream, but the decision to annuitize is typically irreversible and depends crucially on one's life expectancy. While insurers price annuities using objective mortality tables, individuals base their decisions on a subjective mortality force. We assume that the individual is uncertain about their mortality and instead relies on partial information about their health status.

Building on recent work on optimal annuitization ([3], [2] and [1]), we consider an individual who invests wealth in a financial fund modelled as a geometric Brownian motion and chooses when to irreversibly convert all wealth into a life annuity. The individual’s mortality force follows a two-state piecewise deterministic process, switching from a low to a high level at an unobservable random time that represents a serious and permanent health deterioration. The individual does not directly observe the change in mortality when it occurs. Instead, they receive noisy information about their health status over time. As a consequence, the individual must form and continuously update beliefs about whether the mortality force has already switched from the low to the high state. Mathematically, this translates into including as a state variable the posterior probability of the occurrence of the change in mortality.

The annuitization problem is formulated as an optimal stopping problem under partial information. The stopping region is shown to be connected and free of isolated boundary points, which ensures continuous differentiability of the value function. The optimal strategy is of a threshold type: the state dynamics is two dimensional, and it includes a wealth process and the posterior belief process of the agent about the occurrence of the health shock. Annuitization becomes optimal when wealth crosses a belief-dependent threshold, either from below or from above, depending on the model parameters. We then analyse the qualitative behaviour of the free boundary, studying in particular its monotonicity properties with respect to beliefs about deteriorating health. Our results bridge optimal annuitization and quickest detection theory, highlighting how health uncertainty and learning dynamics significantly shape retirement timing decisions.

[1] - Buttarazzi, M., De Angelis, T., & Stabile, G. (2025). Optimal annuitization with stochastic mortality: Piecewise deterministic mortality force. arXiv preprint arXiv:2509.13091. [2] - De Angelis, T., & Stabile, G. (2019). On the free boundary of an annuity purchase. Finance and Stochastics, 23(1), 97-137. [3] - Hainaut, D., & Deelstra, G. (2014). Optimal timing for annuitization, based on jump diffusion fund and stochastic mortality. Journal of Economic Dynamics and Control, 44, 124-146.

ABSTRACT. This talk addresses the numerical approximation of solutions to martingale problems that encode the dynamics of the formal SDE \[ dX_t=b(t,X_t)\,dt+dW_t, \qquad b\in C_T\mathcal C^{-\beta}(\mathbb R^d),\ \beta\in(0,\tfrac12). \] In this setting, the drift cannot be evaluated pointwise, and standard arguments used to prove convergence of the Euler scheme are not directly applicable. The martingale-problem formulation makes it possible to avoid writing the singular drift term in classical form.

A central feature of the analysis is a construction of time integrals of distributions along martingale-problem solutions, namely quantities of the form \[ \int_{0}^{t}g(s,X_s)\,ds, \qquad g\in C_T\mathcal C^{-\beta}(\mathbb R^d),\ \beta\in(0,\tfrac12). \] Here this construction is used as the main analytic device in the error analysis. In particular, I will discuss quantitative stability estimates for these integral functionals, showing that perturbations of the integrand can be controlled along singular trajectories. This provides a substitute for the pointwise comparison arguments that underlie the classical error analysis for SDE schemes.

The approximation analysis is organised in two separate steps. First, one compares the martingale-problem solution associated with the singular drift \(b\) to the solution associated with a heat-mollified drift \(b^m\). This is the part where the integral-along-solutions construction is used: the stability estimates for these distributional integral functionals provide quantitative control of the error generated by replacing \(b\) with \(b^m\).

Second, for each fixed \(m\), one studies the Euler-Maruyama discretisation of the regularised equation where the drift is smooth and the scheme is classically well defined. The discretisation error is analysed at the level of the regularised dynamics, via stochastic sewing estimates, independently of the singular formulation. The final convergence rate is then obtained by optimising the choice of the mollification scale as a function of the time step.

Under suitable assumptions, this yields quantitative weak and strong convergence bounds with explicit dependence on the spatial regularity of the drift and on the discretisation scale. I will also briefly comment on a randomised time-freezing variant, mainly to indicate how the same approach can be adapted to alternative numerical schemes for specific classes of drifts.

ABSTRACT. We propose a new theoretical framework that exploits convolution kernels to transform a Volterra-type path-dependent (non-Markovian) stochastic process into a standard (Markovian) diffusion process. Remarkably, it is also possible to go back, i.e., the transformation is reversible. We discuss existence and path-wise regularity of solutions for our class of stochastic differential equations.

In the fractional kernel case, when $H \in (0,\frac12)$, where $H$ is the Hurst coefficient, we propose a numerical simulation scheme which exhibits a remarkable strong convergence rate of order $1/2$, which constitutes a bold improvement when compared with the performance of available Euler schemes, whose strong rate of convergence is $H$.

ABSTRACT. I consider stochastic Mean Field Games on networks with sticky transition conditions. In this setting, the diffusion process governing the agent’s dynamics can spend finite time both in the interior of the edges and at the vertices. The corresponding generator is subject to limitations concerning second-order derivatives and the invariant measure breaks down into a combination of an absolutely continuous measure within the edges and a sum of Dirac measures positioned at the vertices. Additionally, the value function, solution to the Hamilton-Jacobi-Bellman equation, satisfies generalized Kirchhoff conditions at the vertices.

ABSTRACT. Arithmetic Random Waves are the eigenfunctions of the Laplacian on the torus in $d\geq 2$ dimensions. Their geometry has been extensively investigated in the last two decades, starting from the seminal papers \cite{ref_article2,ref_article3}.

We will discuss \cite{ref_article1} the correlation of various functionals including nodal length, boundary length of excursion sets, and the number of intersection of nodal sets with deterministic curves in different classes; the amount of correlation depends in a subtle fashion from the values of the thresholds considered and the symmetry properties of the deterministic curves. In particular, we prove the existence of \emph{resonant pairs} of threshold values where the asymptotic correlation is full, that is, at such values one functional can be perfectly predicted from the other in the high energy limit. We focus mainly on $d=2$ with some extensions to $d=3$.

We will also briefly discuss a related problem of independent interest in geometry, concerning the characterisation of certain special classes of curves and surfaces that naturally come into play.

ABSTRACT. We study the fluctuations over time for critical points and Euler characteristic of the excursion sets of general isotropic Gaussian random fields on the sphere.

ABSTRACT. The Stochastic Sandpile Model is an interacting particle system introduced in the physics literature in the '90s to study the concept of self-organized criticality, which describes physical systems that spontaneously evolve toward a critical state without the need to fine-tune their parameters. In this talk, I will present the model and discuss some questions related to its behaviour, such as the critical density at which the phase transition occurs, how to exactly sample from the stationary distribution on finite graphs, and the stationary particle density on the complete graph.

ABSTRACT. Motivated by the intrinsic unpredictability of turbulent flows, it has long been conjectured that solutions of the stochastic Navier–Stokes equations remain random in the double limit of vanishing noise amplitude and vanishing viscosity. Although this fundamental mathematical question remains open, direct numerical verification in the full system is currently out of reach due to the extreme computational cost. In this talk, we investigate this conjecture within a reduced yet dynamically faithful framework. We consider the three-dimensional incompressible Landau–Lifshitz–Navier–Stokes equations on logarithmic Fourier lattices with small-scale additive noise. This setting allows for high-resolution simulations in regimes of decreasing viscosity and vanishing noise amplitude. By analyzing the statistics of individual large-scale Fourier modes, we provide numerical evidence of intrinsic stochasticity in two distinct scenarios: evolution from rough initial data and continuation beyond finite-time blow-up of a strong solution. In both cases, the convergence of probability density functions across different parameter sequences indicates the emergence of a limiting universal stochastic process. All results presented in this talk are reported in a recent paper in collaboration with Erika Ortiz and Alexei Mailybaev.

This work received funding from the French National Research Agency (ANR Project TILT, ANR-20-CE30-0035) and from the European Union’s ERC program (NoisyFluid, Grant No. 101053472).

ABSTRACT. Motivated by continuous-time optimal inventory management, we study a class of stationary mean-field control problems with singular controls. The dynamics are modeled by a mean-reverting Ornstein-Uhlenbeck process, and the performance criterion is given by a quadratic long-time average expected cost functional. The mean-field dependence is through the stationary mean of the controlled process itself, which enters the ergodic cost functional. We characterize the solution to the stationary mean-field control problem in terms of the equilibria of an associated stationary mean-field game, showing that solutions of the control problem are in bijection with the equilibria of this mean-field game. Finally, we solve the stationary mean-field game, thereby providing a solution to the original stationary mean-field control problem.

ABSTRACT. In this talk, we show that geometric functionals (e.g., excursion area, boundary length) evaluated on excursion sets of sphere-cross-time long memory random fields can exhibit fractional cointegration, meaning that some of their linear combinations have shorter memory than the original vector. These results prove the existence of long-run equilibrium relationships between functionals evaluated at different threshold values; as a statistical application, we discuss a frequency-domain estimator for the Adler-Taylor metric factor, i.e., the variance of the field’s gradient. Our results are illustrated also by Monte Carlo simulations.

ABSTRACT. This paper presents a novel optimization framework for the design of fuzzy partitions specifically tailored to data acquired from visual sensors operating in complex environments.\, Visual sensor streams are typically high-dimensional and affected by uncertainty and vagueness arising from illumination changes, occlusions, and sensor noise, which makes their processing within a Fuzzy Logic System (FLS) particularly appealing.\, In such systems, the overall performance, robustness, and interpretability crucially depend on the choice of the membership functions that define the fuzzy partition of the input space.

We address this design problem by introducing an automatic tuning scheme for the parameters of parametric membership functions, focusing on triangular and Gaussian shapes, whose centers and widths are optimized over a prescribed spectral range.\, The proposed method formulates the search for an adequate partition as a continuous optimization problem, where the objective function combines a global sensitivity index with a penalty term controlling the extent of under-threshold zones, i.e., regions in which all memberships fall below a given minimum sensitivity level.\, In particular, we define a total sensitivity functional $\Delta$, based on the integrated squared differences between membership profiles, and an under-threshold multi-interval $E$, obtained as the intersection of the individual under-threshold sets associated with each fuzzy membership.\, The resulting objective $\Psi = \Delta - w_{\varepsilon}\lvert E\rvert$, with $w_\varepsilon$ a weight, enforces a trade-off between maximizing discrimination among fuzzy sets and minimizing poorly covered regions of the domain.

Conceptually, our approach is inspired by the view of biological sensory systems as collections of specialized fuzzifiers, where stimuli are encoded through families of overlapping receptive fields and processed via fuzzy-granular representations.\, In particular, the human visual system exemplifies how a limited number of receptor types can support fine discrimination by means of suitably arranged fuzzy membership functions over the sensory domain.\, Preliminary analytical evaluations of the proposed objective show that, for both triangular and Gaussian memberships, the sensitivity term admits closed-form expressions, which allow an efficient numerical implementation of the optimization procedure.\, Moreover, the explicit characterization of under-threshold zones through a Marzullo-like algorithm provides a controllable mechanism to tune coverage according to application-dependent sensitivity requirements.\, These first results support Gentili's thesis that perception-oriented fuzzy structures can serve as a conceptual blueprint for artificial sensory systems, and they indicate that optimized fuzzy partitions can enhance both discrimination capability and semantic transparency in visual-sensor FLS models.

ABSTRACT. Since the dawn of stochastic chemical reaction network theory over 50 years ago, there have been many general results about (positive) recurrence, especially in the case of mass-action kinetics. One less-explored area is that of mass-action models whose rate constants, rather than being static, are themselves stochastic. Such models have relevance in applications, since biomolecular systems rarely exist in isolation and their rates often depend on time-changing quantities. In this series of two talks, we will study the stability of such models under some linearity assumptions.

Specifically, this second talk will present matrix conditions for positive recurrence and transience in the special case where there are finitely many possible choices of rate constants. These conditions will depend on the specific choice of parameters for the model, which makes it possible to uncover phase transitions where the stability behavior of the model varies. We will see that the speed at which the rate constants are changing plays an important role, with the model behaving as one with averaged rate constants when this speed is high and behaving as though the rate constants were unaveraged when this speed is low. This talk is based on joint work with Daniele Cappelletti and Chuang Xu.

ABSTRACT. Stochastic reaction networks are mathematical models with a wide range of applications in biochemistry, ecology, and epidemiology, and are often complex to analyze. Except for some special cases, it is generally difficult to predict how the abundances of all considered species evolve over time. A possible approach to address this issue is to develop tools to compare the model under study with a similar one whose behavior is better understood. The main contribution of our work is to provide direct and computable conditions that can be used to ensure the existence of an ordered coupling between two stochastic reaction networks and to identify which parameter changes in a given model lead to an increase or decrease in the count of certain species. We also make an algorithm available that implements our theory and we illustrate it with several applications.

ABSTRACT. In this talk I will present the work published in \cite{ref_article1}. This work is related to the more recent work \cite{ref_article2}, also presented in this session.

Since the dawn of stochastic chemical reaction network theory over 50 years ago, there have been many general results about (positive) recurrence, especially in the case of mass-action kinetics. One less-explored area is that of mass-action models whose rate constants, rather than being static, are themselves stochastic. Such models have relevance in applications, since biomolecular systems rarely exist in isolation and their rates often depend on time-changing quantities. In this series of two talks, we will study the stability of such models under some linearity assumptions. In this first talk, I will present structural conditions implying positive recurrence, regardless of the specific choice of parameters of the model. I will further present an algebraic characterization of the stationary distribution in terms of a stochastic recurrence equation, which can be exploited to numerically calculate the conditional stationary distribution and its moments.

ABSTRACT. In the framework of stochastic modelling of some biological dynamics, the fractional calculus is one of the valid tool to insert memory effects in widely applied Markov models (see, for instance, \cite{AbundoPirozzi2021}, \cite{LeonenkoPirozzi2025}). Here, we focus on a class of fractional stochastic neuronal models among them those investigated in \cite{Pirozzi2018}, \cite{PirFracModels} and \cite{PirMittag}.

In particular, we apply techniques of parameter estimation for a generalized neuronal model driven by a fractional dynamic and stochastic input. Specifically, the membrane potential $V=\{V_t\}_{t\ge 0}$ is modeled through the Caputo fractional differential equation \[ D^\alpha V_t = A V_t + b + \eta(t), \qquad \alpha \in (0,1), \] where $D^\alpha$ denotes the Caputo derivative of order $\alpha$, $A,b\in\mathbb{R}$, and the latent input process $\eta$ satisfies the Ornstein--Uhlenbeck-type dynamics \[ d\eta(t) = -\Theta \eta(t)\,dt + \sigma\, dG_t, \] with $\Theta,\sigma>0$ and a driving process $G$ with stationary increments. The framework is intrinsically multidimensional, although the estimation methodology is first developed in the univariate case.

The mild solution of the fractional equation is expressed in terms of the Mittag--Leffler functions as follows: \[ V_t = E_\alpha(t^\alpha A) V_0 + \int_0^t s^{\alpha-1} E_{\alpha,\alpha}(s^\alpha A)\bigl(b+\eta(t-s)\bigr)\,ds, \] which provides short- and long-time asymptotics. These asymptotic expansions constitute the basis of a constructive estimation strategy. First, in the case of $b=0,$ exploiting the behavior of $V_t$ as $ t\downarrow 0$ we derive estimators for the fractional order $\alpha$. Alternative difference-based estimators are proposed to mitigate the slow convergence of bias terms involving $\log t$. Once $\widehat{\alpha}$ is obtained, the same asymptotics and large-time expansions of $E_\alpha$ yield consistent estimators of $A$.

After recovering $(\widehat{\alpha},\widehat{A})$, the latent noise is reconstructed via \[ \widehat{\eta}(t)=D^{\widehat{\alpha}}V_t-\widehat{A}V_t, \] and classical methods for Vasicek-type processes are applied to estimate $(\Theta,\sigma)$. The propagation of estimation error from the fractional stage to the second-step inference is analyzed numerically.

A discretization scheme for the Caputo derivative of order $\alpha$ is implemented, leading to an iterative algorithm of computational complexity $O(n^2)$. Simulation studies demonstrate that accurate estimation of $\alpha$ requires observations on a sufficiently fine grid near zero, while reliable inference for $(\Theta,\sigma)$ necessitates long time series. The results confirm the feasibility of the proposed two-step procedure and highlight the interplay between fractional memory effects and stochastic input estimation in generalized neuronal models.

ABSTRACT. Noise sensitivity for functionals of independent random variables, introduced by Benjamini, Kalai and Schramm in 1999 in the context of Boolean functions, describes the phenomenon where a small perturbation of the underlying randomness leads to an asymptotically independent outcome.

In this talk, we extend classical noise sensitivity criteria beyond the Boolean setting and derive quantitative estimates with optimal rates. We then consider the model of directed polymers in random environments, and apply our results to the regime in which the partition function converges to a universal limit known as the Stochastic Heat Flow, which we show to be independent of the white noise arising from the scaling limit of disorder.

ABSTRACT. The Ising model is a Random Field built on a lattice where, for each site, it is assigned a plus or minus 1 value called spin. This model is used to represent the ferromagnetic phenomena in physics and the spins are organized in such a way that neighbouring spins tend to be aligned. We are interested in the model when it interacts with random external fields: in particular, we analyze the partition function and the free energy density as they encode the observable information of the model. In dimension one the model has been completely solved and it has been proven that the partition function can be expressed as the trace of the product of 2 by 2 random matrices, called transfer matrices. Furthermore, the free energy density can be expressed through the transfer matrices: it is their Lyapunov exponent. The Lyapunov exponent of a sequence of random matrices is the value which describes how fast the logarithm of the norm of their product diverges: it can be considered as the equivalent of a Law of Large Numbers in higher dimension. In this talk we will deal with a generalization of the Ising model proposed in the physical literature and we will explain what role plays the Lyapunov exponent in the analysis of this statistical mechanics model. The main focus will be on the comparison between the discrete model and a continuum one, obtained via a scaling limit: our target is to understand whether the discrete Lyapunov exponent converges to the one in the continuum case. This is based on a joint work with A. Chiarini and G. Giacomin.

ABSTRACT. In the 70s, Berry argued that in the high-energy limit wave functions locally look like random superpositions of independent plane waves, having all the same wavenumber. He introduced a Gaussian random field whose sample paths are generalized Laplace eigenfunctions. The aim of this talk is to present a similar model for the sub-Laplacian on the Heisenberg group, which is the analogue of the Euclidean space in sub-Riemannian geometry. It combines ideas from PDE, representation theory, and stationary random fields.

ABSTRACT. Accurately quantifying tail risks—rare but high-impact events such as financial crashes or extreme weather—is a central challenge in risk management, with serially dependent data. We develop a Bayesian framework based on the Generalized Pareto (GP) distribution for modeling threshold exceedances, providing posterior distributions for the GP parameters and tail quantiles in time series. Two cases are considered: extrapolation of tail quantiles for the stationary marginal distribution under $\beta$-mixing dependence, and dynamic, past-conditional tail quantiles in heteroscedastic regression models. The proposal yields asymptotically honest credible regions, whose coverage probabilities converge to their nominal levels. We establish the asymptotic theory for the Bayesian procedure, deriving conditions on the prior distributions under which the posterior satisfies key asymptotic properties. To achieve this, we first develop a likelihood theory under serial dependence, providing local and global bounds for the empirical log-likelihood process of the misspecified GP model and deriving corresponding asymptotic properties of the Maximum Likelihood Estimator (MLE). Simulations demonstrate that our Bayesian credible regions outperform naïve Bayesian and MLE-based confidence regions across several standard time-series models, including ARMA, GARCH, and Markovian copula models. Two real-data applications—to U.S. interest rates and Swiss electricity demand—highlight the relevance of the proposed methodology.

ABSTRACT. In this work, we investigate the block maxima method in the context of stationary time series. We begin by extending aspects of likelihood asymptotic theory for the estimation of the marginal parameters of the Generalized Extreme Value (GEV) distribution from the case of independence to scenarios involving serial dependence. Once the likelihood framework is established at a suitable level of generality, we shift our focus to its Bayesian counterpart, studying the corresponding asymptotic properties. Frequentist and Bayesian inference is then employed to estimate marginal parameters of the GEV, the extremal index, return levels and extreme quantiles of the underlying stationary distribution.

ABSTRACT. In the framework of (fuzzy) multi-criteria decision making \cite{Electronics2025,Mathematics2026}, we propose a method that allows the decision maker to subjectively approach the problem by suitably modifying the decision matrix. Starting from \cite{Coletti02} and following the recent approach proposed in \cite{SMPS2024}, we use the conditional probability interpretation of membership functions and the operations among conditionals in the framework of conditional random quantities \cite{GiSa2014} to model logical and probabilistic operations among the columns of the decision matrix seen as particular fuzzy sets. We consider a decision problem related to a random quantity $X$ with set of values $\mathcal{X}=\{x_1,x_2,\ldots,x_n\}$. Let $\mathcal{E}$ be a meta-expert who chooses the relevant properties $\{C_1,C_2,\ldots,C_m\}$ of $X$, with $C_i$ logically independent. In this setting, the properties $C_j$ are the criteria of the decision problem and the alternatives are represented by the events $A_i=(X=x_i)$ for $i=1, \ldots, n$. To build the decision matrix, the decision maker has to set the criteria's weights $w_j$ and the scores $a_{ij}$. The criteria's weights $w_j$, for $j=1, \ldots, m$, are seen as the probabilities of the events ``$C_j$ is relevant with respect to the decision problem''. Moreover, given a criterion $C_j$ and alternative $A_i$, the corresponding score is interpreted as $a_{ij}=P(E_{C_j}|A_i)$, that is, the conditional probability assigned to the conditional event $E_{C_j}|A_i$=``$\mathcal{E}$ claims that $X$ satisfies property $C_j$, knowing that $(X = x_i)$''. Then, in this setting we allow logical operations among criteria, by exploiting the conditional probability interpretation. More precisely, when considering the complement, conjunction and disjunction of criteria, we build the complement, intersection and union of the corresponding fuzzy sets. The conditional probability interpretation of the scores helps us find the new scores of the modified decision matrix, which retains all the original criteria, as well as the new columns given by the logical operations considered by the decision maker.\\

This talk is based on a joint work \cite{Mathematics2026} with G. Filippone, G. La Rosa, G. Sanfilippo and M. E. Tabacchi from Università degli Studi di Palermo, Italy.

ABSTRACT. Bayesian multilevel models provide an effective framework to borrow information between different data sources through the sharing of common features. In a nonparametric setting, a classic example is the hierarchical Dirichlet process, whose generative model can be described through a set of latent variables, commonly referred to as tables in the popular restaurant franchise metaphor. The latent tables greatly simplify the expression of the posterior and allow for the implementation of a Gibbs sampling algorithm to approximately draw samples from it. However, managing their assignments can become computationally expensive, especially as the size of the dataset and of the number of levels increase. In this talk, we identify a prior for the concentration parameter of the hierarchical Dirichlet process that (i) induces a quasi-conjugate posterior distribution, and (ii) removes the need of tables, bringing to more interpretable expressions for the posterior, with both a scalable and an exact algorithm to sample from it. This construction extends beyond the Dirichlet process, leading to a new framework for defining normalized hierarchical random measures and a new class of algorithms to sample from their posteriors.

ABSTRACT. In Bayesian multilevel models, the data are structured in interconnected groups, and their posteriors borrow information from one another due to prior dependence between latent parameters. In this work, we develop a general framework for measuring the amount of dependence for parametric and nonparametric models, both a priori and a posteriori. We define an index measuring partial exchangeability that detects exchangeability for common models, is invariant by reparametrization, can be estimated through samples, and, crucially, is well-suited for posteriors. We achieve these properties through the use of Reproducing Kernel Hilbert Spaces, which map any random probability to a random object on a Hilbert space. This leads to many convenient properties and tractable expressions, especially a priori and under mixing.

ABSTRACT. Our work studies the limits of empirical means of open-loop Nash equilibria of linear-quadratic stochastic differential games as the number of players goes to infinity, when the corresponding mean field game is of potential type and may have multiple equilibria. Via weak compactness arguments, the limit points are characterized as optimal trajectories of the related deterministic control problem, thus ruling out some of the mean field equilibria. Our result is obtained by first connecting the finite player game to a suitable control problem, whose optimal trajectories are the empirical means of Nash equilibria of the game, and in which the number of players $N$ becomes a parameter. True convergence to the unique minimizer of the limit control problem then holds for almost every initial mean. In cases of multiple optimizers, we focus on examples to show that some symmetry of the data ensures that the sequence admits a random limit which is distributes uniformly among the minimizers of the potential. Multidimensional examples of the convergence result appear here for the first time, which show the flexibility of our method. We also establish a similar convergence results for the corresponding linear-quadratic potential mean field games with common noise, as the noise vanishes.

ABSTRACT. Finite-width fully connected neural networks with Gaussian initialization deviate from their infinite-width Gaussian limit through non-vanishing higher-order cumulants. In this talk, I present multidimensional Edgeworth expansions of arbitrary order for neural network outputs evaluated on a finite collection of inputs, providing a systematic way to approximate these non-Gaussian effects. Under the assumption that the limiting Gaussian covariance matrix is invertible and that the activation function is polynomially bounded, we obtain upper bounds of order $n^{-m}$ in total variation distance between the true network law and its Edgeworth approximation of order $4m-2$, together with matching lower bounds. Beyond neural networks, the results apply to general sequences of conditionally Gaussian vectors converging to a non-degenerate Gaussian limit. As an application, I discuss quantitative bounds for Bayesian neural networks, measuring the error introduced when replacing the prior distribution with its Edgeworth approximation.

ABSTRACT. This paper develops a unified framework for the robustification of risk measures beyond the classical convex and cash-additive setting. We consider general monotone risk measures on L^p spaces and construct their robust counterparts through families of uncertainty sets that capture model ambiguity. Two complementary mechanisms generate quasi-convex robustified measures: one where quasi-convexity is inherited from the initial risk measure under convex uncertainty sets, and another where it stems from the quasi-convex or c-quasi-convex structure of the uncertainty sets themselves. Building on Cerreia-Vioglio et al. (2011); Frittelli and Maggis (2011), we derive dual (penalty-type) representations for robust quasi-convex and cash-subadditive risk measures, showing that the classical convex cash-additive case arises as a special instance. We further analyze acceptance families and capital allocation rules under robustification, highlighting how model uncertainty affects acceptability and the distribution of capital.

ABSTRACT. We derive a priori estimates for singular differential equations of the form \[ \mathcal{L} \phi = P(\phi,\nabla\phi) + f(\phi,\nabla\phi)\xi \] where $P$ is a polynomial, $f$ is a sufficiently well-behaved function, and $\xi$ is an irregular distribution such that the equation is subcritical. The differential operator $\mathcal L$ is either a derivative in time, in which case we interpret the equation using rough path theory, or a heat operator, in which case we interpret the equation using regularity structures. Our only assumption on $P$ is that solutions with $\xi=0$ exhibit coercivity. Our estimates are local in space and time, independent of boundary conditions, and generalise the results of \cite{MoinatWeber20,MW20_reaction,BCMW22,CMW23,Jin_Perkowski_25}.

Our method is based on rescaling the equation, which differs from the aforementioned works and which makes the role of subcriticality especially transparent. One of our main results is an abstract estimate that allows one to pass from a local coercivity property to a global one using scaling, for a large class of equations. This allows us to reduce the problem of deriving a priori estimates to the case when $\xi$ is small.

This talk is based on the work \cite{CG25}.

ABSTRACT. We study stability of optimizers and convergence of Sinkhorn's algorithm for the entropic optimal transport problem. In the special case of the quadratic cost, our stability bounds imply that if one of the two entropic potentials is semiconcave, then the relative entropy between optimal plans is controlled by the squared Wasserstein distance between their marginals. When employed in the analysis of Sinkhorn's algorithm, this result gives a natural sufficient condition for its exponential convergence, which does not require the ground cost to be bounded. By controlling from above the Hessians of Sinkhorn potentials in examples of interest, we obtain new exponential convergence results. For instance, for the first time we obtain exponential convergence for log-concave marginals and quadratic costs for all values of the regularization parameter, based on semiconcavity propagation results. These optimal rates are also established in situations where one of the two marginals does not have subgaussian tails. Other interesting will be presented in this joint-talk.

ABSTRACT. We study stability of optimizers and convergence of Sinkhorn’s algorithm for the entropic optimal transport problem. In the special case of the quadratic cost, our stability bounds imply that if one of the two entropic potentials is semiconcave, then the relative entropy between optimal plans is controlled by the squared Wasserstein distance between their marginals. When employed in the analysis of Sinkhorn’s algorithm, this result gives a natural sufficient condition for its exponential convergence, which does not require the ground cost to be bounded. By controlling from above the Hessians of Sinkhorn potentials in examples of interest, we obtain new exponential convergence results. For instance, for the first time we obtain exponential convergence for log-concave marginals and quadratic costs for all values of the regularization parameter, based on semiconcavity propagation results. These optimal rates are also established in situations where one of the two marginals does not have sub-Gaussian tails. Other interesting will be presented in this joint-talk.

ABSTRACT. The simple exclusion process is one of the most prominent models of interacting particle systems. In this seminar, we consider a resistor network whose nodes are sampled according to a simple point process on $\mathbb{R}^d$ and are connected by certain random conductances. On top of this resistor network, particles move according to random walks with the rule that there is at most one particle per site. Under soft assumptions on the point process measure and conductances, which include ergodicity, stationarity and certain moment conditions, it is known that the empirical density of particles converges for almost all realisation of the environment to the solution of an heat equation with a certain homogenised diffusivity. In this talk, we examine its equilibrium fluctuations. For $d\geq3$, under the same assumptions that ensure the hydrodynamical limit, we show that the empirical density fluctuation field converges for almost all realisation of the environment, in the sense of finite-dimensional distributions, to a generalised Ornstein-Uhlenbeck process. For $d=2$, if we require some additional regularity on the environment to have Hölder regularity estimates for solutions to parabolic problems, we can show that the same conclusion holds.

ABSTRACT. We consider the simple random walk on the infinite cluster of a general class of percolation models on ℤᵈ, d ≥ 3, including Bernoulli percolation as well as models with strong, algebraically decaying correlations. For almost every realization of the percolation configuration, we obtain uniform controls on the absorption probability of a random walk by certain "porous interfaces" surrounding the discrete blow-up of a compact set A. These controls substantially generalize previous results obtained in "Solidification of porous interfaces and disconnection" (J. Eur. Math. Soc., 2020) for Brownian motion in ℝᵈ and in "Disconnection and entropic repulsion for the harmonic crystal with random conductances" (Commun. Math. Phys., 2021) for random walks on ℤᵈ equipped with uniformly elliptic edge weights to a manifestly non-elliptic framework. This talk is based on the recent work "Solidification estimates for random walks on supercritical percolation clusters" (Potential Anal., to appear) and an ongoing project.

ABSTRACT. We introduce a new class of generative diffusion models that, unlike conventional denoising diffusion models, achieve a time-homogeneous structure for both the noising and denoising processes, allowing the number of steps to adaptively adjust based on the noise level. This is accomplished by conditioning the forward process using Doob's $h$-transform, which terminates the process at a suitable sampling distribution at a random time. The model is particularly well suited for generating data with lower intrinsic dimensions, as the termination criterion simplifies to a first hitting rule. A key feature of the model is its adaptability to the target data, enabling a variety of downstream tasks using a pre-trained unconditional generative model. We highlight this point by demonstrating how our generative model may be used as an unsupervised learning algorithm: in high dimensions the model outputs with high probability the metric projection of a noisy observation $y$ of some latent data point $x$ onto the lower-dimensional support of the data---which we don't assume to be analytically accessible but to be only represented by the unlabeled training data set of the generative model.

ABSTRACT. The Dempster-Shafer theory [3,6] is a well-known mathematical framework for mod- eling situations involving incomplete or partially specified information, that generalizes classical probability theory through completely monotone normalized capacities, the latter known as belief functions. The theory of probability boxes (or p-boxes for short) [5,8] is a distinguished part of this theory, since natural extensions of p-boxes reveal to be special belief functions, whose properties are given by the positional structure of jumps in the related p-boxes. Following [2], we consider the problem of approximating an arbitrary belief function with a “closest” p-box natural extension under some constraints. The resulting approxima- tion seeks to preserve the same information of the p-box induced by the initial belief func- tion and to satisfy given upper bounds on the corresponding lower and upper Value-at-Risk (VaR) risk measures, defined as generalized inverse functions [4]. The quoted approximation problem can be faced through a generalization of the classical optimal transport problem and the related Wasserstein distance [7]. Then, the computation of the approximating p- box can be carried out efficiently through a generalization of the Dykstra’s algorithm by relying on a proper entropic formulation. We apply the described approximation on an ambiguous stop-loss reinsurance problem modeled as a Stackelberg game between a reinsurer, who acts as leader, and an insurer, who acts as follower. More precisely, the reinsurer’s aim is to choose the safety loading to determine the reinsurance premium, while the insurer’s aim, inspired by [1], is to choose the retention level that minimizes the lower VaR of his total loss, given by the sum between the retained loss and the optimistic reinsurance premium. In this formulation, we assume that the leader faces two different kinds of ambiguity: a strategic ambiguity on how the follower will respond and an epistemic ambiguity on how the final reward will be evaluated. While the first one reflects a typical Stackelberg-like ambiguity on agent interaction, the second one captures an uncertain external state. This double notion of ambiguity results in four distinct bilevel optimization problems, corresponding to all combinations of optimism or pessimism on both dimensions.

ABSTRACT. We discuss about random motions moving in higher spaces with a natural number of velocities (also known as telegraph processes, continuous time random walks or run-And-tumble processes). In the case of the so-called minimal random dynamics, under some broad assumptions, we establish an affine relationship between motions moving with different directions and we derive the joint distribution of the position of the motion (for both the inner part and the boundary of the support) and the number of displacements performed with each velocity. Explicit results for cyclic and complete motions are presented as particular cases. We also study some useful relationships between motions moving in different spaces, and we obtain the form of the distribution of the movements in arbitrary dimension. Finally,we present further results concerning the distribution over the singularities of the support of motions governed by non-homogeneous Poisson processes.

ABSTRACT. We consider the task of performing Jaccard similarity queries over a large collection of items that are dynamically updated according to a streaming input model. An item here is a subset of a large universe $U$ of elements. A well-studied approach to address this important problem in data mining is to design \textit{fast-similarity data sketches}. In this paper, we focus on \textit{global solutions} for this problem, i.e., a single data structure which is able to answer both \textit{Similarity Estimation} and \textit{All-Candidate Pairs} queries, while also dynamically managing an arbitrary, online sequence of element insertions and deletions received in input.

In this talk, we introduce and provide an in-depth analysis of a dynamic, buffered version of the well-known $k$-min hash sketch. This buffered version better manages critical update operations thus significantly reducing the number of times the sketch needs to be rebuilt from scratch using expensive recovery queries. We prove that the \textit{buffered} $k$-min hash uses $O(k \log |U|)$ memory words per subset and that its \textit{amortized} update time per insertion/deletion is $O(k \log |U|)$ \textit{with high probability}. Moreover, our data structure can return the $k$-min hash signature of any subset in $O(k)$ time, and this signature is exactly the same signature that would be computed from scratch (and thus the quality of the signature is the same as the one guaranteed by the static $k$-min hash

ABSTRACT. Our previous research (joint with S. Cohen, F. de Feo, J. Sirignano) shows that Hilbert Neural Operators are able to approximate classical solutions of fully nonlinear second-order partial differential equations on Hilbert spaces, such as Hamilton-Jacobi-Bellman and backwards Kolmogorov equations. Based on this result, we propose two actor-critic algorithms for solving Hilbert-valued HJB equations and two algorithms for solving Hilbert-valued backwards Kolmogorov equations. We then apply these algorithms to the control of the stochastic heat equation, a stochastic delay equation, the stochastic Burgers equation, and a mean-field control problem. To the best of our knowledge, these algorithms are the first methods for solving PDEs directly on their whole Hilbert space domain.

ABSTRACT. Derivative-Informed Operator Learning (DIOL), i.e. learning a (nonlinear) operator and its derivatives, is an open research frontier at the foundations of the influential field of Operator Learning (OL). In particular, Universal Approximation Theorems (UATs) of nonlinear operators and their derivatives are foundational open questions and delicate problems in nonlinear functional analysis. We prove the first UATs of non-linear k-times differentiable operators between Banach spaces and their derivatives, uniformly on compact sets and in novel weighted Sobolev spaces for general finite input measures, via OL architectures. Our results are the first complete generalizations of the corresponding influential classical results in [Hornik, 1991] to infinite-dimensional spaces and OL. Our weighted Sobolev spaces are a generalization of classical Gaussian Sobolev spaces in [Bogachev, Gaussian Measures].

We discuss several open areas where DIOL and our UATs find applications: highorder accuracy in OL; fast constrained optimization in Banach spaces (e.g. optimal control of PDEs, inverse problems) via Learn-Then-Optimize; numerical methods for infinitedimensional PDEs (e.g. HJB PDEs from infinite-dimensional optimal control, such as optimal control of PDEs, SPDEs, path-dependent systems, partially observed systems, mean-field control).

We parameterize nonlinear operators via Encoder-Decoder Architectures, classical architecture in OL, which include famous examples such as DeepONets, Deep-H-ONets, PCA-Nets. Based on [de Feo arXiv:2605.15285]; [Cohen, de Feo, Hebner, Sirignano, arXiv:2603.19463]

ABSTRACT. Major pathways for carbon abatement include a large-scale deployment of renewable energy sources (RES) and investment in carbon capture and storage (CCS) technologies. While RES such as solar and wind power offer clean, sustainable energy, significantly expanding their share in the energy mix necessitates heavy infrastructure investment. This is primarily due to issues of intermittency and the need to upgrade or redesign existing electricity grids to ensure stability and reliability. On the other hand, CCS technologies offer a potential solution to decarbonize existing fossil fuel-based infrastructure. However, CCS remains technologically immature and economically un-viable at large scale. Significant research and development (R&D) efforts are required to reach a breakthrough that would make CCS a competitive option. Given limited fiscal capacity, it may be infeasible for societies to simultaneously invest heavily in RES infrastructure and fund foundational CCS research. We explores this trade-off by modeling the problem as a stochastic optimization problem. We analyze the optimal allocation of a constrained research and investment budget over time, under uncertainty about technological breakthroughs and deployment costs. We study the problem using theoretical and numerical methods.

ABSTRACT. In this talk, we present several generalizations of elastic Brownian motion, analyzing three distinct scenarios that extend the process's classical dynamics. First, we consider a model in which the killing rate is governed by an independent continuous-time Markov chain (CTMC), thereby introducing a switching mechanism for the process’s extinction. Next, we introduce non-exponential delays at the boundary; this extension leads to the appearance of non-local operators in time (for instance, fractional derivatives and convolution-type operators). Finally, we study the case in which the process, instead of being killed, restarts inside the domain via jumps (stochastic restart). The latter dynamics are described by non-local spatial operators, as boundary conditions, related to the jump distribution. For each case, we discuss the associated PDEs and the connection with non-local operators, describing the stochastic dynamics and potential physical applications.

ABSTRACT. The success of deep learning in high-dimensional settings is often attributed to low-dimensional structure in real-world data. Many theoretical models assume this structure lies in the target function—mapping otherwise unstructured inputs through a low-dimensional subspace. However, data such as images or text also exhibit strong correlations in the input space itself (e.g., spatial locality). In this talk, we propose a tractable model to study how such spatial correlations affect the sample complexity of learning with gradient descent in shallow neural networks. We further analyze how temporal correlations, relative to the standard i.i.d. training-sample setting, can influence the learnability of certain target functions.

ABSTRACT. Understanding both global and layer-specific group structures is useful for uncovering complex patterns in networks with multiple interaction types. In this work, we introduce a new model, the hierarchical multiplex stochastic blockmodel (HMPSBM), that simultaneously detects communities within individual layers of a multiplex network while inferring a global node clustering across the layers. A stochastic blockmodel is assumed in each layer, with probabilities of layer-level group memberships determined by a node's global group assignment. Our model uses a Bayesian framework, employing a probit stick-breaking process to construct node-specific mixing proportions over a set of shared Griffiths-Engen-McCloskey (GEM) distributions. These proportions determine layer-level community assignment, allowing for an unknown and varying number of groups across layers, while incorporating nodal covariate information to inform the global clustering. We propose a scalable variational inference procedure with parallelisable updates for application to large networks. Extensive simulation studies demonstrate our model's ability to accurately recover both global and layer-level clusters in complicated settings, and applications to real data showcase the model's effectiveness in uncovering interesting latent network structure.

ABSTRACT. Directed polymers in random environments describe a perturbation of the simple random walk given by a random disorder (environment). The partition functions of this model have been thoroughly investigated in recent years, also motivated by their link with the solution of the Stochastic Heat Equation. While classical results focus on space-time independent disorder, we consider a Gaussian environment with (critical) spatial correlations decaying as $|x|^{-2}$ times a slowly varying function. We show that a phase transition, analogous to that in the space-time independent case, still occurs: in the high temperature regime the log-partition function satisfies a central limit theorem, while it vanishes in law in the low temperature regime. Remarkably, the inverse temperature needs to be tuned differently from the independent case, where the scaling constant $\hat{\beta}$ emerges from a nontrivial multi-scale dependence in the second moment computation — the core technical challenge of the work. Based on a joint work with Clément Cosco (Paris Dauphine) and Anna Donadini (Milano-Bicocca).

ABSTRACT. We propose a probabilistic formulation of optimal-stopping mean field games by introducing a new class of BSDEs, termed McKean-Vlasov reflected backward stochastic differential equations. An equilibrium is characterized by a quadruple $(Y,Z,A,L)$, where $L$ is a $[0,1]$-valued, non-increasing càdlàg process, representing a randomized stopping strategy. Two additional Skorokhod-type conditions involving the process $L$ enforce the optimality of the stopping rule at equilibrium. We prove the existence of equilibria $(Y,Z,A,L)$ by applying the Kakutani–Fan–Glicksberg fixed-point theorem to a set-valued best-response map; and, under alternative assumptions, we also obtain existence via Tarski's fixed-point theorem. Furthermore, we establish uniqueness under specifc conditions. We also show that the mean field equilibrium induces an approximate Nash equilibrium for the associated $N$-player stopping game. Finally, we connect our probabilistic formulation to the analytical approach, which is characterized by a system of constrained partial differential equations.

ABSTRACT. We provide sufficient conditions for the existence of mild solutions to stochastic differential inclusions in infinite-dimensional Hilbert spaces driven by a cylindrical Wiener process. The initial condition is described by a prescribed map depending on the behavior of the solution over the whole time interval. The model includes multivalued terms both in the drift and in the diffusion part. This structure allows us to cover a broad range of applications: multivalued terms can represent uncertainty or measurement errors in the data, as well as constraints arising, for instance, in optimal control problems. Under our assumptions, classical initial conditions, such as periodic and multipoint ones, can be treated within a unified framework by means of a single map. This provides a comprehensive setting capable of encompassing a wide class of problems. Assuming suitable growth and upper semicontinuity conditions, we prove the existence of at least one mild solution. The analysis is developed within two complementary frameworks, depending on whether the semigroup generated by the linear part is compact or not. When the underlying semigroup is compact, existence follows from the compactness of the associated solution operator, consistently with the compactness-based approach developed, for instance, by Vinodkumar and Boucherif (2011). In the non-compact case, relying on preliminary results by Angelini, Benedetti and Cretarola (2025), we adopt a weak-topology approach: compactness is replaced by weak sequential compactness in appropriate Bochner spaces, together with weak closedness of the solution multivalued map, in line with the framework investigated by Zhou, Peng and Ahmad (2018). In both frameworks, the argument is completed by applying an appropriate multivalued fixed point theorem. We also extend our results to the half-line and prove the existence of periodic mild solutions. The compactness-based approach is well established in the literature, whereas the weak-topology framework has been less extensively investigated. Moreover, stochastic differential inclusions have been mainly studied in finite-dimensional settings; we refer to the seminal monograph by Kisielewicz (2013). Applications include transport-type stochastic models generated by noncompact shift semigroups, as in the work of Brzeźniak, Priola, Zhai and Zhu (2025), which naturally fit the framework considered here. This occurs, for example, in models for forward curve dynamics in financial mathematics. The abstract setting also applies to climate change modeling, where non-deterministic differential equations are required to describe rapidly varying phenomena such as cyclones, as in Diaz and Diaz (2022). In both contexts, periodicity is crucial to capture seasonal effects and recurrent temporal patterns. The talk is based on a joint work with Irene Benedetti, Lorenzo Guida and Teresa Marino (in preparation).

ABSTRACT. Urn models have found several applications, from adaptive design in medical treatments to random networks and opinion dynamics. Their dynamical evolution is based on reinforced stochastic processes, where the probability of future states depends on the history of the system. In this talk, we review some of the most popular urn models, in particular Polya's and Friedman's urns, from the perspective of associated stochastic processes and present some recent results about interaction and synchronization.

Using the framework of Stochastic Approximation, we first analyze systems of interacting urns where agents are coupled via mean-field or network-based interactions. We survey conditions under which the interplay between reinforcement and interaction leads to almost sure synchronization of the urn proportions and characterize the fluctuations around the limit.

We then extend this review to more general reinforcement mechanisms. First, we discuss recent results on urn models with random multiple drawing and random addition, where the reinforcement matrix is time-dependent and non-balanced. This framework allows for modeling complex sampled populations or clinical trials. Second, we address the phenomenon of ``non-synchronization''. We investigate how non-linear reinforcement functions or competing reinforcement rates (individual versus collective reinforcement) can induce phase transitions, leading to the fragmentation of the system into distinct equilibria.

This talk is based on joint works with I. Crimaldi, P. Dai Pra, I. G. Minelli and M. Mirebrahimi.

ABSTRACT. We consider $L^\infty_t L^p_x$ solutions of the stochastic transport equation with drift in $L^\infty_t W^{1,q}_x$. We show strong existence and pathwise uniqueness of solutions in a regime of parameters $p,q$ for which non-unique weak solutions of the deterministic transport equation exist. When the intensity of the noise goes to zero, we prove that the solutions of the stochastic transport equation converge to the unique renormalized solution of the transport equation in the sense of DiPerna-Lions. Furthermore, we show that the convergence is governed by a Large Deviations Principle in the space $L^\infty_t L^p_x$. Since the space $L^\infty_t L^p_x$ is not separable, the weak convergence approach to Large Deviations by Budhiraja, Dupuis, and Maroulas is not directly applicable.

ABSTRACT. We introduce new classes of Gaussian processes exhibiting distinct memory characteristics, namely Bernstein processes and Hadamard fractional Brownian motion. By applying fractional operators within a white noise framework, we model a variety of memory behaviors. On the one hand, these constructions yield Gaussian processes with explicit Wiener integral representations; on the other hand, the choice of fractional operators determines specific forms of the integrand functions.

ABSTRACT. Motivated by recent advances in AI inspired generative modeling, we investigate the mathematical foundations and universal approximation properties of neural stochastic partial differential equations (SPDEs) of Heath–Jarrow–Morton (HJM) type, whose coefficients are parameterized by function-valued neural networks.

Building on this framework, we then propose a fully data-driven HJM model for the forward interest rate dynamics. Specifically, we consider dynamics driven by linear functionals of the yield curve, such as a finite collection of representative forward rates, possibly augmented by observable macroeconomic factors whose characteristics can be directly estimated from market data. The volatility structure is parameterized via artificial neural networks, naturally leading to a neural SPDE formulation. The neural network parameters are learned from historical yield curve data, yielding an arbitrage-free and data-driven framework for the generation and prediction of yield curves. We demonstrate the proposed deep learning methodology by reconstructing and forecasting the Euro area yield curves.

ABSTRACT. I will discuss limits in low intensity of Poisson--Voronoi tessellations, which we called ideal Poisson--Voronoi tessellations (IPVTs).

In real hyperbolic space of dimension $d\geq 2$, a simple Poissonian description of the cell containing the origin (jewel) allows us to study the fine properties of all cells of the IPVT, each of which is unbounded with an infinite number of bounded faces and a single point at the ideal boundary.

The Poissonian description of the IPVT remains simple in other cases, such as the Cartesian product of hyperbolic planes equipped with the $L^1$ metric, where the properties of the cells are different but can be glimpsed thanks to a two-dial type argument.

Based on a joint work with Nicolas Curien, Nathanaël Enriquez, Russell Lyons, and Meltem Ünel (Ann. Probab.), and on 2412.00822.

ABSTRACT. Point processes are stochastic models for discrete events occurring in continuous space, time, or space–time domains. When each event carries an additional attribute, such as earthquake magnitude or burned area, the process is called a marked point process. Self-exciting point processes represent a natural framework for modeling memory effects in non-Markov stochastic processes, since the occurrence of an event increases the probabil- ity of future events through the dependence encoded in the conditional intensity function [4]. In such models, the intensity is typically decomposed into a background component and a triggering component, the latter capturing the temporal persistence and excitation mechanism. Traditional approaches mainly focus on the spatio-temporal configuration of events, without incorporating additional information. Only recently have models including ex- planatory variables been developed for the analysis of epidemic phenomena [8], seismicity [2], and crimes [10,5]. At the same time, interest in point processes with functional marks has increased. [7] formalized functional marked point processes (FMPPs), where marks are random elements in a (Polish) function space, such as temporal signals or spatial trajectories. In seismology, for example, each earthquake may be associated with a ground-motion waveform or its spectral representation, both naturally treated as functional covariates. Motivated by the will of jointly analysing earthquake locations and their waveform characteristics, [6] introduced local inhomogeneous mark-weighted summary statistics to detect spatial dependence in functional marks and proposed a test to identify regions where the random labelling assumption fails. However, this approach implicitly calls for fully specified FMPP models, which remain underdeveloped. To address this gap, we extend the Epidemic Type Aftershock Sequence (ETAS) model [9] by incorporating functional marks derived from waveform data into the triggering com- ponent, thereby enriching the representation of memory effects driving seismic activity. Following [1], waveforms are summarized through Functional Principal Component Anal- ysis (FPCA) scores, which are included as covariates in the triggering term. Estimation is performed via the Forward Predictive Likelihood (FLP) approach [3]. The goal is to assess the presence of local spatial dependence in the functional marks and to evaluate how the inclusion of waveform-based covariates in the triggering part improves the fit to earthquake sequences.

ABSTRACT. We introduce a class of continuous time finite horizon mean field games where the objective function of the representative player depends on a hidden state, in addition to position, control, and the population distribution. While acting on the position dynamics, the agent has the option to pay for seeing the hidden state. We connect the original formulation of our model with a mean field model of optimal control with discretionary stopping and discuss questions of existence and characterization of solutions. For a class of N-player games with compatible information structure, we show that approximate Nash equilibria can be constructed starting from a solution to the limit model.

ABSTRACT. We consider a discrete-time Markovian random walk with resets on a connected undirected network. The resets, in which the walker is relocated to randomly chosen nodes, are governed by an independent discrete-time renewal process. Some nodes of the network are target nodes, and we focus on the statistics of first hitting of these nodes. In the non-Markov case of the renewal process, we consider both light- and fat-tailed inter-reset distributions. We derive the propagator matrix in terms of discrete backward recurrence time pdfs and in the light-tailed case we show the existence of a non-equilibrium steady state. In order to tackle the non-Markov scenario, we derive a defective propagator matrix which describes an auxiliary walk characterized by killing the walker as soon as it hits target nodes. This propagator provides the information on the mean first passage statistics to the target nodes. We establish sufficient conditions for ergodicity of the walk under resetting. Furthermore, we discuss a generic resetting mechanism for which the walk is non-ergodic. Finally, we analyze inter-reset time distributions with infinite mean where we focus on the Sibuya case. We apply these results to study the mean first passage times for Markovian and non-Markovian (Sibuya) renewal resetting protocols in realizations of Watts-Strogatz and Barabási–Albert random graphs. We show non trivial behavior of the dependence of the mean first passage time on the proportions of the relocation nodes, target nodes and of the resetting rates. It turns out that, in the large-world case of the Watts-Strogatz graph, the efficiency of a random searcher particularly benefits from the presence of resets.

ABSTRACT. In this talk, we introduce a class of two-factor models whose construction is based on subordination of multiparameter Markov processes (see \cite{DMP25}). This approach provides a flexible and mathematically tractable framework for generating dependent factors, where both the dependence structure and jump behavior are driven by a common subordinator. We investigate structural properties of the resulting subordinated processes that are particularly relevant for pricing applications, including their polynomial structure, characteristic function, and integral process representation. Multi-factor models play a central role in financial mathematics, particularly in applications such as electricity price modeling and short-rate dynamics. In many practical settings, two factors are sufficient to capture the key features of observed market behavior; however, the construction extends straightforwardly to higher dimensions. The presented results are relevant in financial mathematics, particularly for asset pricing applications; however, their generality also makes them of independent mathematical interest.

ABSTRACT. Time-inhomogeneous Markov processes are widely used in finance to model asset returns. We propose a construction of time-inhomogeneous Markov processes based on multiparameter stochastic time change ([Barndorff-Nielsen et al., 2001]). The approach consists in subordinating a multiparameter Markov process with an independent multivariate additive subordinator. Since additive subordinators generalize Lévy subordinators by allowing non-stationary increments, this model introduces time inhomogeneity while preserving analytical tractability.

We generalize the results in [Li et al., 2016] and [Mendoza-Arriaga and Linetsky, 2016]. By extending Phillips theorem to the multiparameter and time-inhomogeneous setting, we show that the resulting process is a Feller evolution and we characterize its generator. We derive its pseudo-differential representation and show that the associated symbol admits a Lévy--Khintchine representation.

We then focus on a family of analytically tractable multivariate additive subordinators: multivariate Sato subordinators ([Sato, 1991]) with exponential tempered distributions. We characterize multivariate Sato subordinators by characterizing their Lévy measures, and we focus on a specific dependence structure widely used in finance to include correlations in multivariate models. Then, we build a multivariate time-inhomogeneous Markov process using multivariate Sato-subordination. The construction is designed to obtain a multivariate process with the same dependence structure as the factor-based model in [Luciano and Semeraro, 2010]. Our aim is to keep the flexibility of their dependence structure, to have one-dimensional unit time distributions in given classes, and to include time-inhomogeneous increments. Finally, we consider the case of a multiparameter Ornstein--Uhlenbeck process to incorporate mean reversion, which is an important feature in applications such as energy markets.

ABSTRACT. Sticky diffusion processes on bounded domains can spend finite time (and finite mean time) on the lower-dimensional space given by the boundary. Once the process hits the boundary, then it starts again after a random amount of time. While on the boundary it can stay or move according to dynamics that are different from those in the interior. Such processes may be characterized by a time-derivative appearing in the boundary condition for the governing problem. We use suitable time changes in order to describe fractional sticky conditions and the associated boundary behaviours. We obtain that fractional boundary value problems (involving fractional dynamic boundary conditions) lead to sticky diffusions, strong Markov on the interior, spending an infinite mean time (and finite time) on the boundary. Such a behaviour can be associated with a trap effect from the macroscopic point of view. We provide an example on fractals.

ABSTRACT. We provide a general model for Brownian motions on metric graphs with interactions. In a general setting, for (sticky) Brownian propagations on edges, our model provides a characterization of lifetimes and holding times on vertices in terms of (jumping) Brownian accumulation of energy associated with that vertices. Propagation and accumulation are given by drifted Brownian motions subjected to non-local (also dynamic) boundary conditions. As the continuous (sticky) process approaches a vertex, then the right-continuous process has a restart (resetting), it jumps randomly away from the zero-level of energy. According with this new energy, the continuous process can start (or not) as a new process in a randomly chosen edge. The model well extends to a higher order of interactions, here we provide a simple case and focus on the analysis of earthquakes.

ABSTRACT. We present a model for growth in a multi-species population. We consider two types evolving as a logistic branching process with mutation, where one of the types has a selective advantage. We first study the frequency of the disadvantageous type and show that, once the population approaches the carrying capacity, its evolution converges to a Gillespie-Wright-Fisher diffusion process. We then study the dynamics backward in time: we fix a time horizon at which the population is at carrying capacity and we study the ancestral relations of a sample of individuals. We prove that, provided that the advantageous and disadvantageous branching measures are ordered, this ancestral line process converges to the moment dual of the limiting diffusion. This talk is based on joint work with Julian Kern.

ABSTRACT. In this talk I will explain how the training phase of certain deep neural networks can modeled, in some asymptotic regimes, as a mean-field optimal control problem. I will then explain how this view point allows to address uniqueness and stability properties of the optimal distribution of parameters and what can be said regarding the associated gradient descent. This is based on joint works with F. Delarue.

ABSTRACT. Given an i.i.d. sequence of random variables Xi having a power law distribution with finite mean but infinite variance, consider the random graph process built as follow: at each time step a new vertex is added to the graph with initial degree Xt, and then is attached to the older vertices of the graph following a preferential attachment rule. In this context we show that depending on the support of the random out-degree (whether is 1 or bigger or equal then 2) the resulting graph is a small-world or an ultra-small world, namely it has diameter which is of the order of Clog( t)or Clog(log(t)) for C some constant. This extends the results for the classical preferential attachment model to the one with random initial degrees (the so-called PARID).

ABSTRACT. We study the optimal control of mean-field systems with heterogeneous and asymmetric interactions. This leads to considering a family of controlled Brownian diffusion processes with dynamics depending on the whole collection of marginal probability laws. We prove the well-posedness of such systems and define the control problem together with its related value function. Leveraging tools tailored for this framework, such as derivatives along flows of measures and associated Itô calculus, we establish that the value function for this control problem satisfies a Bellman dynamic programming equation in a L2-set of Wasserstein space-valued functions. To illustrate the applicability of our approach, we present a linear- quadratic graphon model with analytical solutions, and apply it to a systemic risk example involving heterogeneous banks.

ABSTRACT. In an infinite dimensional separable Hilbert space $X$, we study compactness properties and the hypercontractivity of the Ornstein-Uhlenbeck evolution operators $P_{s,t}$ in the spaces $L^p(X,\gamma_t)$, $\{\gamma_t\}_{t\in R}$ being a suitable evolution system of measures for $P_{s,t}$. Moreover, we study the asymptotic behavior of $P_{s,t}$. Our results are produced thanks to a representation formula for $P_{s,t}$ through the second quantization operator. Among the examples, we consider the transition evolution operator associated to a non-autonomous stochastic parabolic PDE.

ABSTRACT. In this talk we investigate abstract integro-differential hyperbolic equations, focusing on the probabilistic representation of their solutions. Our analysis is based on fractional derivatives and non-local operators, which are powerful tools for modeling anomalous behavior and non-Markovian dynamics observed in various phenomena.

We first analyze a time-fractional version of the abstract telegraph equation (involving the Caputo derivative), restricting our analysis to positive self-adjoint operators to leverage spectral theory, which includes key operators in applications, such as the fractional Laplace operator. We derive analytical representations for the solution and provide a stochastic solution to the telegraph-diffusion equation for a specific range of the fractional parameter $\alpha$, thereby generalizing existing results.

Furthermore, we consider the abstract Euler-Poisson-Darboux (EPD) equation, characterized by a singular time coefficient. We demonstrate that the stochastic solution to this EPD equation can be represented in terms of the solution to the abstract wave equation. Crucially, we prove that the solution to the EPD equation admits a representation by means of the Erdelyi-Kober fractional integral.

Finally, this work provides a comprehensive analysis of both time-fractional and singular-coefficient abstract telegraph-type equations, offering new analytical and stochastic representation formulas.

ABSTRACT. It has been conjectured by Liero, Mielke, and Savaré that the Hellinger-Kantorovich distance can be expressed as the metric infimal convolution of the Hellinger and the Wasserstein distances. This statement is quite clear, at least intuitively, if one compares the dynamical representations of the three distances. However, no rigorous proof had yet been provided. We first discuss the infimal convolution between two generic distances, highlighting the difficulties that may arise: in particular, finiteness and triangle inequality may fail. We then sketch the proof of the main result. To prove it, we study with the tools of Unbalanced Optimal Transport the so-called Marginal Entropy-Transport problem that arises as a single minimization step in the definition of infimal convolution. This is a joint work with Nicolò De Ponti and Giacomo Enrico Sodini.

ABSTRACT. I will review the forward-backward stochastic differential equation (FBSDE) approach to the stochastic quantisation of Grassmann measures. This framework allows for the construction of families of weakly coupled super-renormalisable Euclidean fermionic field theories and the study of their correlations. Building on these ideas, I will also consider the extension to many-body interacting states on the lattice, where similar stochastic methods can be applied to construct the corresponding Grassmann measures and study relevant observables. Our preliminary results indicate that the FBSDE formulation provides a flexible and alternative tool for the analysis of interacting fermionic states.

ABSTRACT. In this talk, we address the existence and uniqueness of invariant and reversible measures for a class of stochastic partial differential equations (SPDEs) posed on the full space $\mathbb{R}$, and more generally on $\mathbb{R}^n$. In this setting, the standard approach to proving uniqueness and ergodicity of invariant measures, which is based on establishing the strong Feller or asymptotic strong Feller property of the associated Markov semigroup, typically fails.\\ To overcome this difficulty, we propose a different strategy. We show that any reversible measure for a (sufficiently regular) SPDE on $\mathbb{R}$ must be a Gibbs measure satisfying suitable Dobrushin–Lanford–Ruelle (DLR) equations. Whenever these equations admit a unique solution, it follows that the SPDE admits a unique reversible invariant measure, which is ergodic.

ABSTRACT. The study and approximation of first-passage and hitting times for stochastic processes play a central role in numerous applied fields, as for example in geophysics and finance. We will introduce some techniques allowing to approximate these hitting times without using a time-splitting procedure like the Euler scheme. The idea is to construct non- linear boundaries for which we are able to obtain the explicit form of the distribution of the hitting time. Combining this with the connexion between the Bessel process and the Brownian motion will permit to construct a generic algorithm for both the hitting time (or first passage time) and the corresponding position of the process. These results apply well in some particular cases as we can obtain a path approximation for Bessel processes and some classes of stochastic differential equations. This procedure constructs jointly the sequences of exit times and corresponding exit positions of some well chosen domains. The talk will develop also some new results on the inverse first passage time problem which seeks to determine the boundary corresponding to a given stochastic process and a fixed first passage time distribution. We consider the case of Bessel process and also more general diffusions. Bessel processes are particularly valuable in our study due to the availability of explicit solutions to the direct hitting time problem [3]. These known solutions provide a reliable benchmark for validating numerical methods developed for the inverse problem. Furthermore, we obtain convergence results and present numerical simulations that permit to illustrate the efficiency and accuracy of these methods.

ABSTRACT. The study of symmetries in differential equations, pioneered by Sophus Lie, constitutes a fundamental geometric approach to understand the invariance properties governing a dynamical system. By identifying the groups of transformations that preserve the equation's structure, this framework provides systematic tools for order reduction, the construction of exact solutions, and the simplification of complex problems. Although symmetry analysis stands as a classical pillar for deterministic differential equations (ODEs and PDEs), supported by extensive literature, its extension to Stochastic Differential Equations (SDEs) represents a relatively recent field of research.

Recent literature has also highlighted significant connections with the symmetries of the associated Fokker-Planck or Kolmogorov equations. Furthermore, the application of Lie symmetry theory to SDEs enables the derivation of integration by parts formulas inspired by Bismut’s variational approach to Malliavin calculus, with notable applications to the analysis of the law and regularity of the processes, as well as to the development of a stochastic calculus of variations.

In this talk, we will discuss various notions of symmetry for SDEs, highlighting their connections to established invariance properties of well-known stochastic models. We will analyze how a geometric approach, inspired by Lie’s deterministic framework, enables the development of powerful computational tools for symmetry calculation. Subsequently, we will demonstrate how applying this geometric theory allows for the constructive derivation of an integration by parts formula for SDEs. Finally, we will show how this integration by parts formula acts as a generating identity for well-known probability formulas, and discuss its connections to Stein's identities.

This contribution is based on a joint work with F.C. De Vecchi, P. Morando and S. Ugolini.

ABSTRACT. In Classical Probability, a sequence of random variables is said to be exchangeable if its joint distributions are invariant under all finite permutations. Ryll-Nardzeski’s Theorem establishes that exchangeability is the same as spreadability, the a priori weaker symmetry where all subsequences of the given sequence have the same joint distributions. In the non-commutative setting, it is known that the two symmetries no longer coincide for general quantum stochastic processes. We show that under very natural hypothesis there is an extension of the Ryll-Nardzewski Theorem in the noncommutative setting which covers a wide variety of models. Furthermore we obtain an extended De Finetti’s Theorem for various models including processes based on the CAR algebra, processes based on the infinite noncommutative torus and on parafermion algebras. This talk is based on joint work with Valeriano Aiello and Stefano Rossi.

ABSTRACT. Abstract: Let W be a conservative, ergodic Markov diffusion on some arbitrary state space M, converging exponentially fast to equilibrium. We consider: (1) Systems of up to countably many massive particles in M, with finite total mass. Each particle is subject to an independent instance of the noise W, with volatility the inverse mass carried by the particle. We prove that the corresponding infinite system of SDEs has a unique solution, for every starting configuration and every distribution of the masses in the infinite simplex. (2) Solutions to the Dean--Kawasaki SPDE with singular drift, driven by the generator L of W. We prove that the equation may be given rigorous meaning on M, and that it has a unique `distributional’ solution. This extends Konarovskyi--Lehmann--von Renesse's `ill-posedness vs. triviality' to the case of infinitely many massive particles. (3) Diffusions with values in the space P of all probability measures on M, driven by the geometry induced by L. (4) In the case when M is a manifold, differential-geometric and metric-measure Brownian motions on P induced by the geometry of optimal transportation and reversible for a normalized completely random measure. We show that all these objects coincide. Based on arXiv:2411.14936

ABSTRACT. In the infinite-width limit, deep neural networks induce isotropic Gaussian fields whose covariance structure encodes fundamental information about the network architecture and the choice of activation function.

In this talk, I present a unified theoretical framework, based on three recent works, which reveals a robust three-regime classification that consistently emerges across spectral and geometric descriptors of random networks.

In the first work [1], we introduce the notion of spectral complexity and classify activation functions into three distinct regimes, namely sparse, low-disorder, and high-disorder, according to the asymptotic behavior of the angular power spectrum of the limiting field. This classification reveals deep structural differences in network expressivity, with sparsity emerging prominently in deep ReLU architectures.

In the second work [2], we study the geometry of level set boundaries. For non-smooth activations (e.g., Heaviside), the boundaries exhibit fractal behavior, with Hausdorff dimension increasing with depth. For smoother activations, the boundary volume follows one of three distinct trends, namely contraction, stability, or exponential growth, precisely mirroring the regimes identified at the spectral level.

In the third work [3], we analyze the distribution of critical points of the limiting fields. Under suitable regularity assumptions, we derive asymptotic formulas for the expected number of critical points (at fixed index or above a given threshold), revealing once more the same universal trichotomy: convergence, polynomial growth, or exponential proliferation, depending on the local behavior of the covariance kernel.

We show that this trichotomy is universal and governed by the local behavior of the covariance kernel near its fixed points.

[1] Di Lillo, S.: Critical points of random neural networks (2025), https://arxiv.org/abs/2505.17000 [2] Di Lillo, S., Marinucci, D., Salvi, M., Vigogna, S.: Fractal and regular geometry of deep neural networks (2025), https://arxiv.org/abs/2504.06250 [3] Di Lillo, S., Marinucci, D., Salvi, M., Vigogna, S.: Spectral complexity of deep neural networks. SIAM Journal on Mathematics of Data Science 7(3), 1154–1183 (2025), https://doi.org/10.1137/24M1675746

ABSTRACT. Generalized cumulants provide a powerful framework for the analysis of non-linear statistical quantities, playing a central role in problems involving ratios of quadratic forms, saddlepoint approximations, likelihood expansions, and bootstrap procedures. They naturally arise as intermediate objects between joint moments and joint cumulants, allowing one to express the cumulants of polynomial functions of random variables in a systematic way \cite{ref_article1}. Despite their theoretical relevance and wide range of applications, generalized cumulants remain underused in practice. The main obstacle is the severe computational burden associated with their evaluation, which relies on the enumeration of a specific class of set partitions known as complementary set partitions \cite{ref_article2}.

In practice, the lack of efficient computational tools has led to a widespread reliance on precomputed tables of complementary set partitions, most notably those reported in McCullagh’s monograph \cite{ref_book2}. While these tables remain a fundamental reference, they are necessarily incomplete and become impractical to extend as the order increases. Existing computational approaches are mostly graph-theoretic or algebraic in nature and are typically confined to symbolic software, severely limiting their accessibility in widely used numerical environments such as {\tt R}.

This work addresses these limitations by introducing a novel combinatorial and algorithmic approach for the efficient computation of complementary set partitions. The proposed method is based on two-block partitions and avoids the traditional use of connected graphs, Laplacian matrices, or symbolic algebra. By exploiting simple combinatorial constructions, the algorithm identifies all non-complementary partitions and recovers the complementary ones by set difference. This strategy leads to a procedure that is both conceptually simple and computationally scalable. A new implementation in {\tt R} is developed, filling a gap in the available software landscape. Since alternative methods are not currently implemented in {\tt R}, computational comparisons are carried out in Maple, where the proposed algorithm consistently outperforms existing techniques in terms of execution time.

From a theoretical perspective, we extend the classical definition of generalized cumulants to include more complex dependence structures. This extension is formulated using multiset subdivisions and multi-index partitions, which provide a natural framework for handling powers of random variables. Within this setting, generalized multivariate cumulants are defined as intermediate quantities between multivariate moments and multivariate cumulants, and explicit closed-form expressions are derived in terms of products of multivariate cumulants.

Finally, we propose a novel approach to the unbiased estimation of generalized cumulants. Building on the theory of $k$-statistics and multivariate polykays \cite{ref_article3}, the proposed estimators exploit dummy variables, labeling rules, and tailored transformations between multi-index partitions and set partitions. This strategy substantially reduces computational complexity compared to traditional plug-in or symbolic methods, making the practical use of generalized cumulants feasible in high-dimensional statistical applications.

ABSTRACT. In this talk, we deal with a class of stochastic diffuse interface models driven by conservative noise. More precisely, we introduce the Cahn–Hilliard and the conserved Allen–Cahn equations with logarithmic type potential and conservative noise in a periodic domain. These features ensure on one hand that the order parameter takes its values in the physical range and, on the other, albeit the stochastic nature of the problems, that the total mass is conserved almost surely in time. Existence and uniqueness of probabilistically-strong solutions is discussed, highlighting the key technical points arising from the structure of the noise. Further directions of research will also be presented.

ABSTRACT. In this talk we present the study of the long-time behaviour of a stochastic Allen-Cahn- Navier-Stokes system. The model features two stochastic forcings, one on the velocity in the Navier-Stokes equation and one on the phase variable in the Allen-Cahn equation, and includes the thermodynamically-relevant Flory-Huggins logarithmic potential. We first show existence of ergodic invariant measures. Secondly, we prove that if the noise acting in the Navier-Stokes equation is non-degenerate along a sufficiently large number of low modes, and the Allen-Cahn equation is highly dissipative, then the stochastic flow admits a unique invariant measure which is asymptotically stable with respect to a suitable Wasserstein metric. The talk is based on a joint work with A. Di Primio and L. Scarpa.

ABSTRACT. In this talk, we propose an analysis of a class of exploration processes on large random graphs having a fixed degree distribution, using the « constructing while exploring » approach: The graph is constructed by uniform pairing of half-edges, thus leading to a realization of the configuration model, while simultaneously exploring it.

Under general assumptions, we show how this approach allows to estimate key characteristics of the exploration process, to the large graph limit, by solving a system of ordinary differential equations in a space of measures, obtained as the hydrodynamic limits of a (properly scaled) sequence of point measure-valued continuous-time Markov chains. This procedure thus extends Wormald's differential equation method, to a space of infinite dimension.

We will focus on a particular example to illustrate this methodology: the greedy matching problem on general graphs, using a local matching criterion.

ABSTRACT. In joint work with B. Bhattacharya, A. Ganguly and G. Zucal, we study dense edge-colored exponential random graph models (ERGMs) through the language of probability graphons, building on the large deviation principle developed with G. Zucal in a recent paper. For a finite set of k colors, a probability graphon is a symmetric measurable map $W:[0,1]^2\to\Delta_k$ (the k probability simplex), extending the usual graphon formalism to colored graphs (and, more generally, to random weighted graphs with a prescribed edge-color law).

Within this framework, the asymptotic log-partition function admits a variational characterization that balances the chosen Hamiltonian with an explicit relative-entropy functional, extending the dense-graph ERGM theory to the colored setting. The same formulation yields compactness of maximizers and natural optimality conditions (Euler-Lagrange type) for typical interaction terms built from chromatic subgraph densities. We also discuss qualitative consequences for the typical structure of the model depending on its parameters, including regimes of uniqueness/replica-symmetric behavior and the onset of symmetry breaking as parameters vary.

Finally, a low-temperature (large-parameter) scaling links the probabilistic variational problem to extremal combinatorics: the model concentrates around near-extremizers of the underlying density functional, providing an entropic/probabilistic-method perspective on stability phenomena. We illustrate this connection on rainbow-triangle-type objectives and related extremal colorings.

ABSTRACT. Within a general semimartingale framework, we study the relationship between collective market efficiency and individual rationality. We derive necessary and sufficient condition for the existence of (possibly zero-sum) exchanges among agents that strictly increase their indirect utilities and characterize this condition in terms of the compatibility between agents’ preferences and collective pricing measures. The framework applies to both continuous- and discrete-time models and clarifies when cooperation leads to a strict improvement in each participating agent’s indirect utility.

ABSTRACT. This paper develops a comprehensive theory of collective arbitrage, pricing–hedging duality, and market completeness in a discrete-time multi-agent framework with segmented markets and cooperative risk exchange. Building on the notion of Collective Arbitrage introduced by Biagini et al., Collective arbitrage and the value of cooperation, Finance and Stochastics (2025), we investigate financial markets in which agents trade in distinct submarkets while being allowed to reallocate risk through structured exchange mechanisms. Within this setting, the absence of arbitrage must be reformulated as No Collective Arbitrage (NCA), reflecting the possibility that cooperation itself may generate or eliminate arbitrage opportunities. Wespecifically work with sets of exchanges modelled by finite dimensional vector spaces of zero-sum rando vectors. Our first contribution is a strengthened version of the First Fundamental Theorem of Asset Pricing (CFTAP I), establishing the equivalence between NCA and the existence of equivalent collective martingale measures under minimal integrability requirements. The proof introduces new techniques and relaxes certain assumptions present in earlier work, allowing for heterogeneous filtrations and greater flexibility in the choice of probability measures. Second, we prove a collective pricing–hedging duality: the collective superhedging price of a vector of contingent claims equals the supremum of the aggregated expectations under suitable collective martingale measures. As the exchange space is finite dimensional, we establish closure properties of the relevant attainable sets, obtain dual representations without restrictive integrability conditions, and prove attainment of optimal hedging strategies. Third, we introduce the notion of collective replication and characterize collectively complete markets. We show that a segmented multi-agent market is collectively complete if and only if the set of equivalent collective martingale measures is a singleton (CFTAP II), thereby extending the classical Second Fundamental Theorem of Asset Pricing to cooperative environments.

ABSTRACT. We present a new mathematical approach to obtain posterior contraction rates (PCRs) in Bayesian consistency, based on Wasserstein calculus. The description of the approach is contained in the attached PDF file.

ABSTRACT. Many statistical methods for analyzing the extreme value behavior of a sample of $d$-dimensional random vectors rely on the assumption that the observed vectors are multivariate regularly varying (perhaps after a marginal transformation). Despite its importance, surprisingly few statistical tests for this hypothesis have been proposed and thoroughly analyzed. Taking up an idea from \cite{DM08}, we discuss a general approach to tackle this problem. The proposed test statistics are based on empirical processes, which give further insight about the type of deviation from regular variation if the test rejects the null hypothesis.

ABSTRACT. We propose a framework for approximating Nash equilibria in mean-field games (MFGs) with common noise based on a two-time-scale structure. In our model, the common noise is modeled by a fast variable evolving under ergodic dynamics. The framework applies to several classes of MFGs, including games with regular control and with optimal stopping. The main idea is to avoid solving the full MFG with common noise by approximating it with an “effective” MFG without common noise, whose coefficients are obtained by averaging with respect to the stationary measure of the fast-scale process.

Starting from an equilibrium of the effective MFG, we construct an explicit $\varepsilon$-MFG equilibrium for the original game by introducing randomized control and stopping. To this end, we establish new existence results for MFG equilibria with randomized stopping. Our approach relies on convergence results for two-scale diffusions under various structural assumptions on the MFG, and we show that the time-scale separation parameter controls the error in the Nash equilibrium condition.

ABSTRACT. Score-based generative models (SGMs) have emerged as one of the most popular classes of generative models. A substantial body of work now exists on the analysis of SGMs, focusing either on discretization aspects or on their statistical performance. In the latter case, bounds have been derived, under various metrics, between the true data distribution and the distribution induced by the SGM, often demonstrating polynomial convergence rates with respect to the number of training samples. However, these approaches adopt a largely approximation theory viewpoint, which tends to be overly pessimistic and relatively coarse. In particular, they fail to fully explain the empirical success of SGMs or capture the role of the optimization algorithm used in practice to train the score network. To support this observation, we first present simple experiments illustrating the concrete impact of optimization hyperparameters on the generalization ability of the generated distribution. Then, this paper aims to bridge this theoretical gap by providing the first algorithmic- and data-dependent generalization analysis for SGMs. In particular, we establish bounds that explicitly account for the optimization dynamics of the learning algorithm, offering new insights into the generalization behavior of SGMs. Our theoretical findings are supported by empirical results on several datasets.

ABSTRACT. We develop a spectral approach to Bayesian nonparametric regression on the sphere. The methodology is based on isotropic Gaussian random field priors defined through the eigensystem of the Laplace–Beltrami operator and the associated spherical harmonic decomposition. This representation provides a natural frequency-domain description of both the signal and the prior, and transforms the regression problem into a diagonal Gaussian sequence model. As a consequence, posterior inference admits an explicit characterization and can be interpreted as frequency-wise Bayesian shrinkage. The angular power spectrum plays a central role in determining the regularity properties of the prior and the resulting asymptotic behaviour of the posterior distribution. Connections with Gaussian random fields, harmonic analysis on the sphere and spectral regularization methods will also be discussed.

ABSTRACT. Finding regular transport maps between measures is an important task in generative modelling and a useful tool to transfer functional inequalities. The most well-known result in this field is Caffarelli’s contraction theorem, which shows that the optimal transport map from a Gaussian to a uniformly log-concave measure is globally Lipschitz. Note that for our purposes optimality of the transport map does not play a role. This is why several works investigate other transport maps, such as those derived from diffusion processes, as introduced by Kim and Milman. Here, we establish a lower bound on the log-semiconcavity along the heat flow for a class of what we call asymptotically log-concave measures. We will see that this implies Lipschitz bounds for the heat flow map introduced by Kim and Milman. I will also comment on its implication for stability of these maps.

Based on a joint work with Louis-Pierre Chaintron and Giovanni Conforti.

ABSTRACT. Later-stage matches in sports tournaments, especially semifinals and finals, are often treated as more important and therefore played longer. We ask whether this intuition can be justified from a sequential testing viewpoint. We study a knock-out tournament with $2^n$ players in which each match is modeled by a Brownian motion with an unobservable drift, representing the players' relative abilities. The tournament designer chooses how long each match should be played so that the strongest player wins the tournament with a prescribed probability.

We analyze two design regimes: $(i)$ deterministic designs, where all match lengths are fixed in advance, and $(ii)$ sequential designs, where match duration can adapt to the observed paths. Our main structural result shows that in both regimes, the optimal schedule makes the late-round matches longer than early-round matches, providing a formal statistical justification for common tournament practice.

We then quantify the efficiency gain from allowing sequential decisions, comparing the expected total observation time under optimal sequential designs to that of optimal deterministic schedules achieving the same success probability. We derive explicit bounds on the average reduction in sample size: sequential testing saves at least $36\%$ and at most $75\%$ on average. Moreover, the relative advantage of sequential methods grows as one requires higher precision.

ABSTRACT. A Lorentz process is a model for the motion of a particle among randomly located scatterers, also known as obstacles. It was originally used to describe the transport of electrons through a conductor.

In the classical setting, when the scatterers are distributed according to a Poisson point process, the deterministic dynamics of elastic collisions can be approximated, under the Boltzmann-Grad scaling limit, by a Markovian random flight. The density of this limiting process is governed by the Boltzmann equation. Passing further to the hydrodynamic limit, one recovers Brownian motion as the macroscopic description of the particle’s position.

In this work, we introduce a new class of point processes that generalizes the Poisson process and we investigate the motion of a particle which collides elastically with obstacles distributed according to this distribution. Unlike the classical case, the corresponding limiting random flight process is no longer Markovian. Instead, it exhibits memory effects that lead to superdiffusive behavior. At the macroscopic level, the particle’s position converges to a continuous superdiffusive process.

Within this framework, we derive a non-local analogue of the Boltzmann equation governing the non-Markovian random flight. Moreover, we show that the density of the superdiffusive scaling limit satisfies a fractional heat equation, reflecting the anomalous transport induced by the underlying correlations.

ABSTRACT. We present a recent joint work with G. La Rosa, M. E. Tabacchi [1], where a judgment aggregation argument is pointed out. In particular, we adopt the perspective of a meta-expert who forms its final judgement after consulting a group of experts, giving greater weight to opinions closer to its own.

Expert assessments are modelled using trapezoidal fuzzy numbers for both criteria weights and ratings, while the aggregation process relies on t-norms and t-conorms to combine fuzzy information coherently.

The influence of each expert is determined through a recently introduced distance measure, see [2], for trapezoidal fuzzy numbers, suitably rescaled to quantify the divergence between the meta-experts confidence index and those of the others. A deeper discussion on this distance is provided (existence, nonexistence and characterization). Moreover, the use of this interval distance allows us to consider a neutral, meta-expert that evaluate and aggregate the judgements of other experts and decides to assign lower weights to opinions that are farther from their own and higher weights to those that are closer. Nevertheless, this does not imply ignoring the opinions that are too far away. On the other hand, the trapezoidal fuzzy numbers permits to capture both the inherent epistemic uncertainties in expert judgements and the vagueness associated with linguistic evaluations.

Finally, a benchmark on the seminal engineering problem of Sequoyah nuclear power plant (the aim is the evaluation of the pressure rise inside the containment building of the Sequoyah Nuclear Power Plant) is analysed.

Bibliography: [1] G. Failla, G. La Rosa, M. E. Tabacchi, \emph{An Application of Fuzzy Set Theory and Interval-Based Distances for Expert Judgment Aggregation}, Submitted.

[2] W. He, R. M. Rodr\'iguez, Z. Tak\'a\v{c}, L. Mart\'inez, \emph{Ranking of Fuzzy Numbers on the Basis of New Fuzzy Distance}. Int. J. Fuzzy Syst. \textbf{26}, 17–33 (2024). \doi{10.1007/s40815-023-01571-5}

ABSTRACT. The coalescent is a foundational model of latent genealogical trees under neutral evolution, but suffers from intractable sampling probabilities. Methods for approximating these sampling probabilities either introduce bias or fail to scale to large sample sizes. We identify a class of functionals of the coalescent which describe the variance of estimators from classical importance sampling algorithms, and which have tractable infinite-sample limits. These functionals provide the first mathematical descriptions of the performance of some seminal coalescent inference methods, and reveal that coalescent importance sampling differs markedly from the behaviour of (sequential) importance samplers in more standard settings, with or without resampling.

ABSTRACT. A tandem of two queues sharing a pool of servers, where users need time to switch to the second queue, is used to model a typical pathway through an emergency department (ED), where patients undergo two consultations separated by diagnostic tests. In this paper~\cite{Fayolle2026stability}, explicit conditions for ergodicity, transience and null-recurrence are given and proven via Foster’s criterion, using a linear Lyapunov function. This result is extended to a Jackson network, with the key feature that the nodes share a pool of servers, with a non-idling one-limited service policy and Markovian routing for the servers. Furthermore, delay times for customers to move from one node to another are also taken into account. This covers some of the main features of models for emergency departments, namely priorities (triage) between patients.

In the case of the tandem queue, after scaling the arrival rate and the number of servers by $N$, and dividing the process by $N$, we obtain a renormalized process converging to the solution of an ordinary differential equation (ODE) subject to boundary conditions. We gives some insights of the solution of this ODE in case of ergodicity, mainly we discuss the long time behavior, more precisely convergence to the equilibrium point.

ABSTRACT. We study Mean Field Games (MFG) systems in real, separable infinite-dimensional Hilbert spaces, addressing both general nonlinear formulations and the specific linear-quadratic (LQ) case. In the general setting, the MFG system consists of a second-order parabolic Hamilton-Jacobi-Bellman (HJB) equation coupled with a nonlinear Fokker-Planck (FP) equation, both involving Kolmogorov operators. Solutions are interpreted respectively in the mild and weak senses, and we establish well-posedness via Tikhonov’s fixed point theorem, with uniqueness ensured under separability and Lasry-Lions-type monotonicity conditions. In the LQ framework, we focus on the case where the mean field interaction enters only through the objective functional via the mean of the distribution. This structure allows the reduction of the MFG system to a Riccati equation and a forward-backward system of abstract evolution equations—an approach that is novel in infinite dimensions. Existence and uniqueness are obtained through a refined approximation method, and the theory is applied to a production output planning problem with delayed control.

ABSTRACT. The grass-bushes-trees process is a two-type contact process in which one type (the trees), of infection parameter λ1, can invade the other type (the bushes) of infection parameter λ2. We look to show which graph parameters lead to the possibility of coexistence versus the necessity of competitive displacement, i.e. metastability of both types or fast extinction of the bushes.

ABSTRACT. We study a mean-field control (MFC) problem with singular controls over a finite horizon, allowing for general dependence on the measure argument. To analyze the search for an optimal MFC strategy, we associate to it a mean-field game (MFG), which we refer to as a potential MFG of singular controls. We show that, under suitable convexity assumptions, any solution to this potential MFG yields a solution to the original MFC problem. We apply our results to a mean-field control version of the classical Monotone Follower problem of I.\ Karatzas and S.\ E.\ Shreve (SICON, 1984). The scalar mean-field interaction term is modulated by an interaction-strength parameter, leading to either strategic complementarity or strategic substitutability. The associated potential MFG with singular controls is solved by relying on its connection with optimal stopping problems for the optimization step, and on two distinct fixed-point theorems to handle the two strategic regimes.

ABSTRACT. A natural framework for studying semigroups associated with elliptic operators with unbounded coefficients is given by L^p spaces related to invariant measures. This is the case, for instance, of the classical Ornstein-Uhlenbeck semigroup (P(t)), which enjoys many nice properties in L^p(m), where m denotes the standard Gaussian measure that turns out to be the unique associated invariant measure. One of the most relevant properties of the Ornstein--Uhlenbeck semigroup, proved by Nelson, concerns hypercontractivity; that is, for any 1<p<q (not infinity) there exists t_0>0 such that

||P(t)f||_{L^q(m)} \le ||f||_{L^p(m)},

for all f \in L^p(m) and t > t_0.

The hypercontractivity of P(t) is strictly connected to the validity of the classical logarithmic Sobolev inequality. Moreover, the above estimat allows one to deduce the asymptotic behavior of P(t) as t tends to infinity.

The Ornstein-Uhlenbeck semigroup can be interpreted as a particular case of a generalized Mehler semigroup and, as is well known, in the general case hypercontractivity fails to hold for such semigroups.

In this talk we consider generalized Mehler semigroups on L^p spaces related to invariant measures and investigate their summability-improving properties. We identify natural subspaces of L^p where hypercontractivity-type estimates are satisfied, providing both examples and counterexamples. The results we prove extend and, in some cases, improve the existing theory. This is joint work with Luciana Angiuli (Università del Salento).

ABSTRACT. We consider the stochastic nonlinear Schr\"odinger equation on a $d$-dimensional domain with the polynomial nonlinearity% and multiplicative noise \[ {\rm d} u(t,x)+\left[ \mathrm{i} \Delta u(t,x)+\mathrm{i} \alpha |u(t,x)|^{2\sigma} u(t,x) \right] \,{\rm d}t = \phi(u(t,x)) \,{\rm d} W(t) \] Classical results of global existence are obtained for power $\sigma$ not too large, depending on the spatial dimension $d$ and the parameter $\alpha$ ($\alpha>0$ is the focusing case and $\alpha<0$ is the defocusing case). This is known for the deterministic equation ($\phi=0$) and the stochastic one with an additive or linear multiplicative noise. Higher values of $\sigma$ can give rise to blow-up in finite time.

In our paper we prove that working on the $d$-dimensional torus $\mathbb T^d$, for any power $\sigma \in \mathbb N$ there exists a class of noises such that there exists a unique global solution for {\em any} initial data in $H^s(\mathbb T^d)$ when $s>\frac d2$. The effect of the noise is to prevent blow-up in finite time, differently from the deterministic setting. Moreover, we prove the existence of an invariant measure and its uniqueness under more restrictive assumptions on the noise term.

As an example, one can consider a one dimensional real Wiener process $W$ and diffusion $\phi(u)=[a(1+\|u\|_{L^\infty})^\sigma+\mathrm{i} b(1+\|u\|_{L^\infty})^\sigma]u$ for real values $a,b$ with $a$ large enough. The choice $s>\frac d2$ provides the helpful estimate $\|u\|_{L^\infty(\mathbb T^d)} \le C \|u\|_{H^s(\mathbb T^d)}$, because of the continuous embedding $H^s(\mathbb T^d) \subset L^\infty(\mathbb T^d)$. Therefore the local existence result is a trivial fact. Our proof of global existence relies on a tightness method based on the choice of a suitable Lyapunov function. In particular, the global existence holds in both focusing and defocusing cases.

ABSTRACT. In many real-world domains, knowledge is inherently vague or imprecise --- features that classical ontology languages based on crisp Description Logics (DLs) are unable to capture effectively. This shortcoming poses particular challenges for applications in the Semantic Web and Explainable Artificial Intelligence (XAI), where robust reasoning over vague or graded information is essential. Fuzzy ontologies address this limitation by enriching DLs with fuzzy logic, enabling the expression of partial truth and supporting more nuanced modelling of real-world knowledge. This article presents a complete re-engineering in Python of the \textsf{fuzzyDL} reasoner and the Fuzzy OWL 2 framework. The former is an expressive fuzzy DL reasoner, while the latter allows for defining fuzzy ontologies within OWL 2. Our contribution addresses several shortcomings of the original software, including semantic inconsistencies, rigid architectural design, and limited solver integration. The re-implementation features a modular class hierarchy tailored for extensibility, supports a broader range of Mixed-Integer Linear Programming (MILP) solvers (including open-source alternatives), and corrects URI ambiguities arising from overlapping ontological elements. Furthermore, a dedicated Python library has also been developed to handle OWL 2 annotations in a standards-compliant manner, improving interoperability with existing Semantic Web tooling and resolving URI ambiguities. The resulting framework offers a portable, extensible, and theoretically grounded platform for reasoning with fuzzy ontologies, suitable for both research and deployment in vague-aware systems. The source code and full documentation are publicly available to facilitate community adoption and further development.

ABSTRACT. We develop a functional framework for shallow neural networks based on reproducing kernel Banach spaces. This approach enables a nonparametric treatment of neural networks, in direct analogy with kernel methods. A representer theorem shows that finite networks suffice for empirical risk minimization. Estimation and approximation error bounds can then be derived in linear function spaces. As a byproduct, we obtain universality results and approximation bounds showing that neural networks can adapt to latent structure in the problem. Further, we derive complexity estimates based on the Rademacher complexities of RKBS balls, independent of network size.

ABSTRACT. We develop a unified framework for modeling multiple term structures arising in financial, insurance, and energy markets, adopting an extended Heath-Jarrow-Morton (HJM) approach under the real-world probability measure. We study market viability and characterize the set of local martingale deflators. We conduct an analysis of the associated stochastic partial differential equation (SPDE), addressing existence and uniqueness of solutions, invariance properties and existence of affine realizations.

ABSTRACT. This presentation introduces a conformal prediction methodology for quantifying uncertainty in populations of graph data. While existing literature offers numerous methods for graph prediction, techniques for assessing the uncertainty of these predictions remain scarce. The proposed framework addresses this gap by generating prediction regions for both labelled graphs, which possess a clear correspondence between nodes across observations, and unlabelled graphs, which lack such correspondence.

For unlabelled graphs, the methodology constructs prediction regions embedded within a discrete quotient metric space, referred to as graph space. The approach is model-free and does not rely on distributional assumptions. It achieves finite-sample validity and produces component-wise interpretable prediction regions configured as parallelotopes. Furthermore, the framework incorporates a length modulation mechanism to account for the local variability of specific edge or node attributes.

The theoretical properties and empirical performance of this forecasting technique are evaluated through two simulation studies covering both labelled and unlabelled graph scenarios. Additionally, the practical utility of the method is demonstrated using a real-world dataset of player passing networks from the FIFA 2018 World Cup. This application illustrates the framework's capacity to analyze network topology and quantify prediction uncertainty for football teams categorized by varying performance levels.

ABSTRACT. Predictive inference takes the sequence of one-step-ahead predictive distributions as the primitive object for learning and inference, rather than an explicit model- prior specification. This approach naturally encompasses Bayesian procedures, but also applies to prediction--based learning rules that are only asymptotically exchangeable or arise from computationally motivated approximations.

We study asymptotic inference induced by predictive learning rules that are not necessarily exchangeable but converge almost surely to a random limiting distribution. Our main contribution is a functional Doob--type Bernstein--von Mises theorem for predictive inference. We show that, under suitable regularity conditions, the conditional distribution of the limiting predictive process, centered at the current predictive distribution and suitably rescaled, converges almost surely to a Gaussian law in an appropriate functional space. The associated covariance structure is explicitly characterized in terms of predictive updates, yielding an analytic approximation of the implicit posterior distribution and providing a direct tool for uncertainty quantification and predictive efficiency assessment.

Under i.i.d. observations, we obtain a Bernstein--von Mises theorem for the predictive distribution, showing asymptotic normality of the implicit posterior centered at the predictive mean, with variance determined by the learning dynamics of the predictive rule.

Finally, we discuss extensions of the framework to supervised settings with regressors, where predictive distributions depend on covariates. In this context, functional central limit theorems for predictive distributions with fixed covariate values provide Gaussian approximations for conditional laws, with applications to regression and modern prediction-based learning methods.

ABSTRACT. Mean field games (MFGs), introduced independently by Lasry and Lions and by Huang, Malham\'e and Caines, describe strategic interactions among a large population of agents through the coupled evolution of a value function and of the distribution of a representative player. A fundamental issue in this theory is the uniqueness of equilibria, which is essential both for modelling purposes and for the stability of numerical approximations. In the classical framework, uniqueness is usually obtained under the Lasry-Lions monotonicity condition, which relies on a separable structure of the Hamiltonian with respect to the state and the population distribution.

In this work we study finite-state continuous-time mean field games with distribution-dependent jump intensities, leading to Hamiltonians that are genuinely non-separable. The state of a representative player evolves in a finite set $\Sigma=\{1,\ldots,d\}$ and, when the player is in state $x$, the transition rate towards a different state $y$ is of the form \[ \alpha_y(t,x)+b(x,\mu(t)), \] where $\alpha$ is the control and $b$ is a nonnegative interaction term depending on the population distribution $\mu(t)$. This structure naturally arises in models with congestion, network effects or endogenous transition mechanisms.

For a fixed flow of measures, the associated Hamilton-Jacobi-Bellman equation involves the Hamiltonian \[ H(x,\mu,p)=\sum_{y\neq x}\left(\frac12 (p_y)_{-}^{2}-b(x,\mu)p_y\right), \] which couples the population variable and the finite differences of the value function in a non-separable way. The mean field equilibrium is characterised by a forward-backward system consisting of this Hamilton-Jacobi-Bellman equation and a Kolmogorov equation with distribution-dependent transition rates.

We provide a uniqueness result for this class of non-separable finite-state mean field games, valid on arbitrary finite time horizons.

Uniqueness is established under a combination of a strong monotonicity condition on the running cost, a standard monotonicity condition on the terminal cost, and Lipschitz continuity of the interaction term $b$ with respect to the population distribution. In contrast with the classical Lasry-Lions theory, the monotonicity of the costs alone is not sufficient: the dependence of the dynamics on the distribution generates additional coupling terms which must be controlled by explicit quantitative conditions. Our results highlight the precise balance between cost monotonicity and the strength of the distribution-dependent transition rates required to recover uniqueness in non-separable mean field game models.

ABSTRACT. We consider the problem of node clustering in dynamic networks through an invariance-based probabilistic framework grounded on conditional partial exchangeability. Specifically, we extend stochastic block models by allowing community memberships to evolve according to a temporal hierarchy of dependent species-sampling mechanisms. The resulting construction induces a dynamic partition structure that preserves probabilistic coherence with the network data. A spike-and-slab base measure introduces a persistence mechanism that favors the retention of community memberships and connectivity patterns across time. This yields a flexible non-Markovian network-valued process with both node-level and global temporal dependence, acting on both the partition structure and the connectivity patterns. We derive marginal representations of the model and develop efficient sampling algorithms for posterior inference. The generality of the framework and the relaxation of Markovian assumptions allow the model to be studied not only in terms of clustering performance but also for temporal network prediction. Numerical experiments illustrate the inferential and predictive performance of the proposed methodology.

ABSTRACT. We study optimal control problems for a class of dynamical system of McKean–Vlasov type exhibiting mean-field effects, namely where the coefficients also depend on the joint distribution of the state and control. The controlled system is subject to regime switching driven by a hidden Markov chain, so that the problems under consideration are partially observed. The main contribution of this paper is to show how the distribution dependence can be handled within a change-of-probability framework, leading to a well-posed separated control problem. We derive a controlled Zakai equation with a specific structure for the unnormalized filter, and show that the corresponding value function satisfies a dynamic programming principle. This yields a Bellman equation posed on a convex subset of a Wasserstein space, characterizing the optimal control problem under partial observation. The paper is available as arXiv:2601.09311v1.

ABSTRACT. In this talk, we study a class of dynamically consistent risk measures that robustify a time-homogeneous Markovian reference model by allowing for distributional uncertainty in its transition laws. We start from one-step convex risk evaluations in which ambiguity is captured by penalized worst-case expectations over alternative transition laws. Imposing time consistency then yields a convex monotone semigroup on bounded continuous payoff functions, and this semigroup represents the associated dynamic risk measure. The semigroup is uniquely characterized by its risk generator. Under a lower bound on the family of penalties in terms of suitable optimal transport costs relative to the reference laws, we identify the generator on smooth test functions. For optimal transport bounds with linear small-time scaling, this produces a first-order, drift-type correction given by a convex Hamiltonian acting on the gradient. Under martingale-transport constraints and a different scaling, however, the leading correction is genuinely of second order and is described by a convex monotone functional acting on the Hessian. We illustrate both regimes for Wasserstein and martingale Wasserstein penalizations and derive explicit formulas via convex conjugates of the underlying transport costs.

ABSTRACT. We introduce a general Bayesian framework for graph matching grounded in a new theory of \emph{exchangeable random permutations}. Leveraging the cycle representation of permutations and the literature on exchangeable random partitions, we define, characterize, and study the structural and predictive properties of these probabilistic objects. A novel sequential metaphor, the \emph{position-aware generalized Chinese restaurant process}, provides a constructive foundation for this theory and supports practical algorithmic design. Exchangeable random permutations offer flexible priors for a wide range of inferential problems centered on permutations. As an application, we develop a Bayesian model for graph matching that integrates a correlated stochastic block model with our novel class of priors. The cycle structure of the matching is linked to latent node partitions that explain connectivity patterns, an assumption consistent with the homogeneity requirement underlying the graph matching task itself. Posterior inference is performed through a node-wise blocked Gibbs sampler directly enabled by the proposed sequential construction. To summarize posterior uncertainty, we introduce \emph{perSALSO}, an adaptation of SALSO to the permutation domain that provides principled point estimation and interpretable posterior summaries. Together, these contributions establish a unified probabilistic framework for modeling, inference, and uncertainty quantification over permutations.

ABSTRACT. In the 60’s, Kraichnan proposed a synthetic model for passive scalar turbulence, consisting of a scalar advected by a random Gaussian velocity field, white in time and $\alpha$-Hölder continuous in space. Despite its simplicity, this SPDE displays anomalous dissipation of energy, spontaneous stochasticity and intermittency, which are also expected for more realistic turbulent fluids. At the same time, solutions to the inviscid SPDE are unique and can be recovered by vanishing viscosity and mollification schemes. In this talk I will present some recent further understandings on this model: i) solutions to the transport equation with $L^2$ initial data display anomalous regularisation and almost gain Sobolev regularity $H^{1-\alpha}$, but not better (see [1,2]); ii) solutions to the continuity equation starting from Dirac deltas instantaneously gain Lebesgue integrability, due to the diffusive behaviour of Lagrangian particle splitting, and their variance at small times grows like $t^{1/(1-\alpha)}$ (see [2]).

ABSTRACT. In the 1960s, Robert Kraichnan proposed a synthetic model for passive scalar turbulence, consisting of a scalar advected by a random Gaussian velocity field that is white in time and Hölder continuous in space. Despite its simplicity, this linear SPDE exhibits key features of realistic turbulent flows, such as anomalous dissipation. Renewed interest in this model followed the work of Coghi and Maurelli, which showed that the same transport-type noise restores well-posedness in regimes where the deterministic 2D Euler equations admit non-unique weak solutions. In this talk, we further develop this line of research by investigating additional properties of the solutions constructed by Coghi and Maurelli. In particular, we present new results on anomalous fractional Sobolev regularity and anomalous dissipation of the mean enstrophy for solutions to the 2D Euler equations with rough Kraichnan noise. Time permitting, we will also discuss implications for the well-posedness theory of more singular nonlinear advection models, such as the Surface Quasi-Geostrophic and Incompressible Porous Media equations. This talk is based on ongoing joint work with L. Galeati and U. Pappalettera.

ABSTRACT. We investigate the vanishing-noise limit for stochastic regularizations of Lagrangian trajectories associated with incompressible velocity fields on the two-dimensional torus. Given a divergence-free alpha-Holder drift u, we prove that the solution to the SDE associated to u with noise W, where W is either a Brownian Motion, a fractional Brownian motion or a Levy process, does not have a limit when the viscosity goes to zero. As a consequence, we also obtain non-selection phenomena for vanishing (fractional) viscosity limits of the associated transport–diffusion equations, providing explicit examples where stochastic or viscous regularization does not single out a unique inviscid limit.

ABSTRACT. In this work, we investigate the main drivers of risk-neutral densities of quoted stocks, using the functional principal component analysis (FPCA). To this end, we first construct a historical series of risk-neutral densities corresponding to quoted option prices with fixed time to maturity, using exponential expansions of orthogonal polynomials. Then, we apply the centered log-ratio transformation (CLRT) to the extracted densities and we perform the FPCA in the Bayes–Hilbert space. The CLRT provides an isometric isomorphism between the Bayes space of square log-integrable densities and the classical Hilbert space of square-integrable functions. As a result, the projected data onto the principal component basis correspond to the CLRT-transformed densities, and the application of the inverse CLRT yields proper density functions. Furthermore, by modeling the historical series of FPCA scores as a stochastic process, we exploit the FPCA representation for forecasting purposes. Finally, we discuss extensions of this framework to cross-asset analyses and to the modeling of option price surfaces.

ABSTRACT. We present closed-form solutions to the autonomous trading problems in the model in which the logarithm of dynamics of the asset price is described by the observation process from the extended Kalman-Bucy filtering model with generalised Ornstein-Uhlenbeck processes having mean-reverting levels. One can consider the cases in which the mean-reverting levels are either observable (full information) or unobservable (partial information). The optimal trading times are shown to be the first hitting times of the risky asset price process to either upper or lower either stochastic boundaries depending on the running filtering values (full information) or time-dependent boundaries (partial information). The method of proof consists of embedding the initial problems into optimal double-stopping problems for either two-dimensional time-homogeneous (full information) or one-dimensional time-inhomogeneous (partial information) continuous Markov diffusion processes. The latter are solved as either the equivalent elliptic-type free-boundary problems (full information) or the equivalend parabolic-type free-boundary problems (partial information). We show that the resulting optimal trading boundaries provide unique solutions to the associated systems of nonlinear Fredholm-type integral equations.

ABSTRACT. What happens when multiple randomly translated and rotated copies of a periodic function are superposed? This question was explored visually by American artist Sol LeWitt in a series of works during the second half of the 20th century. Interestingly, the resulting patterns often display "scars": long strands of large-amplitude oscillations. These patterns diverge from the white-noise structure usually displayed by random fields at large scale. Remarkably, similar scar-like structures have been observed in the completely different setting of quantum dynamics, in high-energy eigenfunctions of the Laplace operator on a manifold.

In this talk, I will provide an overview of the phenomenon of (random) scars, highlighting the connection between these seemingly unrelated models, and discuss recent advances that provide statistical evidence for the scar phenomenon, via the analysis of high critical points of the Berry random wave model.

ABSTRACT. The Schrödinger Bridge (SB) problem has become a fundamental tool in computational optimal transport and generative modeling. To address this problem, ideal methods such as Iterative Proportional Fitting and Iterative Markovian Fitting (IMF) have been proposed—alongside practical approximations like Diffusion Schrödinger Bridge and its Matching (DSBM) variant. While previous work have established asymptotic convergence guarantees for IMF, a quantitative, nonasymptotic understanding remains unknown. In this talk, I will present the first non-asymptotic exponential convergence guarantees for IMF under mild structural assumptions on the reference measure and marginal distributions, assuming a sufficiently large time horizon. These results encompass two key regimes: one where the marginals are log-concave, and another where they are weakly log-concave. The analysis relies on new contraction results for the Markovian projection operator and paves the way to theoretical guarantees for DSBM. The talk is based on a joint work with Giovanni Conforti and Alain Durmus [1].

[1] Gentiloni Silveri, M., Conforti, G., Durmus, A.: Exponential Convergence Guarantees for Iterative Markovian Fitting. In Thirty-Ninth Annual Conference on Neural Information Processing Systems (2025).

ABSTRACT. The increasing availability of high-dimensional, heterogeneous, and non-Gaussian data has reinforced the importance of nonparametric inference in modern statistical analysis. In many applied contexts, classical parametric assumptions such as normality, linearity, and homoscedasticity are often violated, potentially leading to biased or misleading conclusions[1]. As a result, rank-based and distribution-free methods have gained renewed attention in the analysis of complex data structures. This contribution focuses on nonparametric methods for multivariate analysis, with particular emphasis on Spearman’s rank correlation coefficient. Using rank differences, the classical expression is ρS = 1 − 6 Pni=1 d2i n (n2 − 1), (1) where di is the difference between the ranks of two variables for the i-th pair of data, and n is the number of pairs of observations, a robust measure of dependence originally introduced by Spearman [2]. ρS captures monotonic relationships by operating on ranked data, making it especially suitable for ordinal variables, skewed distributions, nonlinear associations, and datasets affected by outliers [3]. These features are increasingly common in real-world applications, where strict parametric assumptions are rarely satisfied. Within multivariate frameworks, ρS plays a dual role. First, it provides an interpretable measure of pairwise association that remains stable under deviations from normality. Second, it is an effective diagnostic tool for detecting multicollinearity and near-redundancy among variables, a critical issue in multivariate modeling and variable selection procedures[4]. High rank correlations can be used as thresholds to identify redundant information, improving model parsimony and robustness. The methodological relevance of ρS is illustrated through its application to the analysis of the Italian pension system, a socio-economic system characterized by strong interdependencies between demographic and economic variables. Using official institutional data and a nonparametric correlation-based approach, associations among pension costs, revenue inflows, GDP, employment rates, and retirement indicators are explored without imposing restrictive distributional assumptions. The results reveal extremely strong rank correlations among key economic aggregates, confirming structural dependencies previously highlighted in the literature on pension system sustainability [5,6]. Moreover, the analysis uncovers significant regional heterogeneity across macro-areas, emphasizing the complexity of territorial dynamics. Beyond descriptive analysis, ρS serves as a foundational step for subsequent inferential and forecasting procedures. In particular, it supports informed variable selection prior to the application of time-series models on non-stationary data, enhancing both interpretability and statistical stability.

CS000: Methodological Issues in Multidimensional and Composite Data Analysis organized by Massimiliano Giacalone and Gianfranco Piscopo. References 1. Hair, J.F., Black, W.C., Babin, B.J., Anderson, R.E.: Multivariate Data Analysis: A Global Perspective. Pearson (2010) 2. Spearman, C.: The proof and measurement of association between two things. The American Journal of Psychology 15(1), 72–101 (1904) 3. Bocianowski, J., Wrońska-Pilarek, D., Krysztofiak-Kaniewska, A.,: Comparison of Pearson’s and Spearman’s correlation coefficients values for selected traits. Biometrical Letters (2023)

ABSTRACT. Consider $n$ agents labeled $\{1, \dots, n\}$, each holding an arbitrary initial binary opinion $x_i \in \{0,1\}$. We study the \emph{minority dynamics}, in which, at each round, each agent $i$ samples $k$ opinions uniformly at random from $\{x_1, \dots, x_n\}$, and then replaces $x_i$ with the \emph{least common} value among the sampled opinions. The minority dynamics is of interest in computer science and distributed algorithms due to its connection with the \emph{bit-dissemination problem}, which models information spread in biological systems.

This process was previously analyzed in \cite{sodapaper}, where it was shown that if $k = \Omega(\sqrt{n \log n})$ and $k \le n/2$, the system converges to a unanimous state (all 0's or all 1's) within $O(\log^2 n)$ rounds with high probability.

In this work, we analyze the minority dynamics for \emph{polylogarithmic sample sizes}, i.e., $k = \Omega(\mathrm{polylog}(n))$, and show that consensus is still reached rapidly, in $O(\mathrm{polylog}(n))$ rounds with high probability. The chaotic and non-monotone nature of the minority dynamics makes its analysis depart significantly from that of previously studied consensus dynamics in similar settings, as it precludes the identification of a natural potential function to measure progress toward consensus.

ABSTRACT. Understanding latent cognitive processes underlying decision-making and their neural correlates is a central goal in cognitive psychology and neuroscience. Serial reaction time (SRT) tasks provide a valuable framework for studying these processes, as variations in response times and accuracy reflect differences in underlying cognitive and neural mechanisms, ranging from controlled, deliberative processing to more automatic responses. Drift-diffusion models (DDMs) offer a principled computational framework for analyzing such data by modeling decision-making as a process of evidence accumulation toward a response threshold (Ratcliff et al.(2004), Ratcliff et al. (2008), Nunez et al. (2017), Vanderckhove et al.(2011)). Key parameters of the DDM, including drift rate, decision threshold, starting point, and non-decision time, provide interpretable measures of cognitive efficiency, response caution, and processing delays. Recent advances have sought to integrate behavioral and neural data, such as electroencephalography (EEG), into DDM frameworks to better characterize brain–behavior relationships (Turner et al. (2015), Turner et al. (2017), Sun et al. (2022)). However, existing approaches typically focus on linking neural features to model parameters without explicitly capturing structured heterogeneity across trials, time, brain regions, or individuals. Moreover, they often neglect the full functional dynamics of neural signals and the role of brain connectivity networks in shaping cognitive processes.

Our work is motivated by the need for flexible and tailored statistical models to analyze neuro-behavioral datasets, such as the publicly available SRT task data examined in Reetzke et al. (2018), which combines behavioral reaction times with simultaneously recorded EEG signals. Recent contributions have advanced Bayesian approaches for drift-diffusion modeling in related contexts. In particular, Paulon et al. (2021) proposed a semiparametric Bayesian framework for studying tone learning in adults, enabling inference on key decision parameters such as drift rates and decision boundaries. Building on this framework, Mukhopadhyay et al. (2024) addressed the problem of recovering latent category structure in the absence of additional labeling information, highlighting the potential of Bayesian methods to uncover hidden cognitive states from behavioral data alone.

We propose hierarchical integrative neuro-behavioral models to study brain–behavior relationships in cognitive processes across multiple dimensions, including trials, time, spatial locations (i.e., ERP-measured brain regions), and participant subgroups. By clustering observations across trials and time, the framework captures the dynamic evolution of cognitive processes, such as learning and attentional changes. Identifying participant subgroups further enables the investigation of variability in cognitive and neural function. To our knowledge, fully Bayesian drift-diffusion models integrating these dimensions within a unified framework have not been previously proposed.

ABSTRACT. A central task in the statistical analysis of spatial point patterns is to infer the relationship between the point distribution and a collection of covariates of interest. This talk will present recent theoretical and methodological advances for covariate-based nonparametric Bayesian intensity estimation. We devise a “multi-bandwidth” Gaussian process method, and prove that it achieves optimal and adaptive posterior contraction rates in observation schemes with replicated observations of the point pattern and the covariates. Our result cover the case of “anisotropic” intensity functions, which is common in applications where the covariates have different physical nature. We further show how posterior inference can be implemented in practice via a suitable Metropolis-within-Gibbs sampling algorithm. Lastly, we will illustrate the performance of the method via numerical simulations, and present an application to a Canadian wildfire dataset. Joint work Patric Dolmeta.

ABSTRACT. The Markov modulated Poisson process (MMPP) extends the classical Poisson process by allowing the arrival intensity to evolve according to an underlying continuous–time Markov chain, thus capturing regime-switching behavior and temporal dependence. We study a 2-state Markov modulated Poisson process Nt and provide explicit expressions for the probability distribution by making use of probability generating function techniques. The analysis relies on representations involving special functions. The limiting and asymptotic behavior of the state probabilities are also analyzed, providing insight into the role of switching intensities and transition rates. In particular, limiting regimes are examined, revealing connections with the standard Poisson process and highlighting structural transitions in the distributional behavior. We address stationary and interval-stationary versions of the process, and introduce time-changed versions of Nt obtained through different subordinators, including Poisson, Gamma, a-stable, and inverse a-stable. For each resulting process, explicit expressions for the moment generating function, mean, and variance are obtained, highlighting how the choice of subordinator affects memory properties and variability. These extensions are consistent with broader Markov-modulated Poisson modeling frameworks. Shock models driven by a MMP process provide a natural and effective framework for applications in which systems accumulate damage or experience failures at rates influenced by an unobservable or fluctuating environmental regime. We investigate both extreme and cumulative shock models driven by Nt, in line with recent developments on shock processes governed by mixed Poisson dynamics. In particular, in the cumulative shock model, system failure is assumed to occur when the total damage produced by successive shocks exceeds a threshold, which is assumed to be either deterministic or exponentially distributed. We provide analytical formulas for the failure rate function, whose monotonic decreasing behavior is discussed, as well as closed-form expressions of the mean and variance of the lifetime distribution.

ABSTRACT. In this talk we investigate the asymptotic behavior (as t → ∞) of solutions to some multi-term fractional evolution equations with constant coefficients, employing techniques from Fourier analysis. Furthermore, we provide some insights into the use of pseudo-differential calculus for studying the well-posedness, regularity, and spatial decay (|x| → ∞) of sub-diffusive models featuring variable coefficients. The presentation is based on results obtained in [1], [2] and [3].

[1] D’Abbicco, M., Girardi, G.: Asymptotic profile for a two-terms time fractional diffusion problem. Fract. Calc. Appl. Anal. 25, 1199–1228 (2022) [2] D’Abbicco, M., Girardi, G.: Decay estimates for a perturbed two-terms space-time fractional diffusive problem. Evolution Equations and Control Theory 12(4), 1056-1082 (2023) [3] Coriasco, S., Girardi, G., Pilipović, S.: Representation formula, regularity, and decay of solutions for sub-diffusion equations. https://arxiv.org/abs/2511.04885

ABSTRACT. Quantum trajectories are Markov processes describing the evolution of quantum systems undergoing repeated indirect measurements. They were first introduced in the study of continuously monitored quantum systems and as useful computational tools in the theory of open quantum systems. When the measurement is perfect, namely when no information flows into the system and all the information leaking from the system is observed, the set of pure states is invariant under the dynamics. A natural question is under which conditions the set of pure states is also attractive, in the sense that the state of the system almost surely tends to “purify” at large times, regardless of the initial state. Besides its intrinsic mathematical interest, there are several motivations for studying purification, which will be briefly discussed in this talk. In the case of systems with finitely many degrees of freedom, purification is well understood and an insightful characterization is well known: purification occurs unless the dynamics encounters a family of “dark” subspaces, namely subspaces from which no information leaks out. In this talk, we will present the first steps towards understanding purification in infinite-dimensional systems. In particular, we will exhibit a class of models for which purification fails even in the absence of dark subspaces, showing that the finite dimensional characterization no longer holds in full generality in infinite dimensions. We will discuss the mechanism underlying this class of examples and explain that it is representative of all infinite dimensional situations in which purification fails. If time permits, we will conclude by discussing some classes of systems for which the characterization of purification in terms of dark subspaces remains valid even in infinite dimensions. The presentation is based on joint work with A. Vitale.

ABSTRACT. We discuss analytical results at both the micro and macroscale for diffusions in $\mathbb R^d$ subject to advection driven by a drift which is strongly singular at the origin, such as the Lennard-Jones force. The Lennard-Jones kernel, characterized by the parameters $(a, b) \in \mathbb{R}^2_+$, with $a > b > 0$, and, $\epsilon, R_0\in \mathbb R_+$ is given by: $$K(x) = \epsilon \left( \frac{R_0^a}{|x|^{a+1}} - \frac{R_0^b}{|x|^{b+1}} \right) \frac{x}{|x|},$$ This type of force is frequently used in applications to model pairwise interaction of molecules and particles; however, analytical results are not available in the literature.

We briefly examine the local integrability properties of the force by establishing clear relations between the integrability spaces and the free parameters $a$ and $b$. Then, a more probabilistic argument follows. At the microscale we address the existence of a pathwise unique strong solution to the McKean-Vlasov SDE $$dX_t = (K \ast u)(t, X_t) dt + \sqrt{2} dW_t, \quad 0 < t \le T,$$ where $\mathcal{L}(X_t) \sim u(t, \cdot) dx$. At the macroscale the marginal density of $X_t$ is identified as the mild solution to a corresponding Fokker-Planck PDE. Thus, at the microscale we consider the dynamics of a typical Brownian particle interacting with a mean field that evolves at the macroscale, governed by the associated diffusion-advection PDE. At the microscale we further consider a system of a finite number $N \in \mathbb{N}$ of Brownian particles that pairwise interact at a mesoscale. The link between these different scales is proved by showing the convergence in probability of the empirical particle density associated with the particle system to the unique mild solution of the Fokker–Planck equation. This is achieved via a mesoscale regularization approach of prescribed order $\alpha \in (0,1)$, under the assumption that particles interact moderately. A law of large numbers is established by restricting the range of the mesoscale in a appropriate way. We discuss the relationship between the mesoscale regularization parameters and the rate of convergence of the law. In particular, we identify suitable functional spaces associated with the order of singularity of the Lennard-Jones force.

ABSTRACT. We study an ergodic singular stochastic control problem for a one-dimensional compound–Poisson jump diffusion under model ambiguity. Ambiguity affects both the drift and the jump intensity and is modeled via a $(\kappa,\lambda)$-ignorance framework, leading to a robust control problem formulated as a min–max optimization over admissible controls strategies.

We show that the associated robust Hamilton–Jacobi–Bellman equation admits a reduction to a non-ambiguous formulation in which the worst-case drift and jump intensity are of bang-bang type. Under an infinite-horizon average-cost criterion, optimality is characterized by a free-boundary problem with gradient constraints for which we establish a verification theorem.

Focusing on negative and exponentially distributed jump sizes, we obtain a more explicit expression for the bang-bang regions for the drift and the jump intensity. We derive an integro-differential free-boundary problem that can be reduced to piecewise system of ordinary differential equations whose solutions have to satisfy local and global regularity constraints

We propose a two-stage numerical scheme combining closed-form expressions with a root-finding procedure to compute the solution. Numerical experiments illustrate the qualitative effects of ambiguity on the optimal policy and confirm the analytical findings.

The associate paper is still in preparation, it will be soon available on arXiv.

ABSTRACT. In this talk, we introduce the Volterra signature -- an extension of Chen's path signature that incorporates memory kernels in a principled way. Formally, it is defined as the collection of iterated integrals arising from Picard expansions of linear controlled/stochastic Volterra equations, and thus plays the role of the resolvent associated with such equations. The additional flexibility provided by the kernel yields a powerful, memory-aware feature map for machine-learning applications to path and time-series data. In the first part of the talk, we leverage analytic and algebraic properties to prove learning-theoretic results, including universal approximation theorems for continuous functionals on path spaces and PDE-based kernel tricks for the associated reproducing kernel Hilbert space (RKHS). Moreover, to exploit these learning guarantees, we develop practical algorithms to compute Volterra signatures for time series across a broad class of kernels, relying on the fundamental Volterra--Chen relation. Finally, we present first applications on synthetic and real data, showing promising performance in learning tasks with complex memory dependence. In the second part, if time permits, we discuss ongoing research on stochastic Volterra signatures, including explicit expected signature formulas, stochastic Taylor expansions, and Wong--Zakai type of approximations.

ABSTRACT. First exit times of stochastic processes are fundamental in many applications. In mathematical finance, they are used to quantify default risk in path-dependent derivatives; in neuroscience, they describe interspike interval distributions. Diffusion processes, as solutions of stochastic differential equations, form a central class of models, making the accurate approximation of their exit times a problem of broad interest.

We consider the multidimensional setting and study the numerical approximation of the first exit time \(\tau_{\mathcal D}\) of a \(d\)-dimensional diffusion process \((X_t)_{t \ge 0}\) from a bounded, regular domain \(\mathcal D\). The process satisfies \[ dX_t = \nabla \mathcal{U}(X_t,t),dt + dB_t, \qquad X_0 \in \mathcal D, \] where \((B_t)\) is a \(d\)-dimensional Brownian motion and the drift term may depend on both space and time. Our objective is to design an efficient alternative to the classical Euler scheme, which requires small time steps to ensure accuracy near the boundary.

In the Brownian case, the Random Walk on Spheres (WOS) algorithm exploits isotropy to perform large spatial jumps, leading to a mean number of steps proportional to \(|\log(\varepsilon)|\), where \(\varepsilon\) is the boundary layer parameter. Extensions based on spheroids allow the joint approximation of exit position and exit time.

We generalize this approach to multidimensional diffusion processes with drift. The proposed method relies on an acceptance–rejection procedure applied to random walk trajectories and introduces truncated spheroids to account for nonzero drift. This construction preserves the efficiency of large spatial displacements while incorporating the effect of the drift term. The performance of the algorithm is supported by theoretical results and illustrated through numerical experiments.

ABSTRACT. Modified logarithmic Sobolev inequalities characterise exponential speed of convergence to equilibrium for Markov processes. In this talk, I will show how to derive such inequalities for point processes, beyond the Poisson case. Our approach relies on (non-optimal) transport maps from Poisson to the target process, and yields sufficient conditions in the spirit of the celebrated Bakry–Émery criterion on manifolds.

Joint work with Baptiste Huguet and Pablo López Rivera.

ABSTRACT. We consider the setting where the state dynamics at each node in a network depend on interactions with its neighbors. We model this using the general framework of Network Stochastic Differential Equations (N-SDEs). The evolution at each node arises from three components: intrinsic dynamics (a momentum term), feedback from adjacent nodes (a network term), and a stochastic volatility component driven by Brownian motion. Our goals are twofold: parameter estimation for N-SDE systems and recovery of the underlying graph. The main motivation is to handle very high-dimensional time series by exploiting sparsity in the network structure. We study two settings. i) Known network structure: the graph is given, and we provide identifiability conditions for the parameters, accounting for the fact that the parameter dimension grows with the number of edges. ii) Unknown network structure: the graph must be learned from data; for this case, we propose an iterative procedure based on adaptive Lasso, developed for a particular class of N-SDE models. We focus on oriented graphs, which supports applications to causal inference by allowing the investigation of directed cause–effect relationships in dynamical systems. Using simulations and real data, we illustrate the performance of the proposed estimators across several graph topologies in high-dimensional regimes. We establish non-asymptotic bounds for parametric estimation when the system dimension is large, in two observation schemes: (1) high-frequency data from an ergodic diffusion, and (2) continuous observation in a small-diffusion, not necessarily ergodic, setting. Based on joint works with S.M. Iacus and N. Yoshida.

ABSTRACT. In this talk we present a new approach to the following general instance of infinite dimensional moment problem: when can a linear functional on an infinitely generated algebra $A$ be represented as an integral with respect to a Radon measure on the space $X(A)$ of all characters of $A$?

Our approach is based on projective limit techniques, which allow us to exploit the results for the classical finite dimensional moment problem in the infinite dimensional case. In fact, we prove that under the so-called Prokhorov condition, the infinite dimensional moment problem on $A$ is solvable if and only if for any finitely generated subalgebra $S$ of $A$ the corresponding finite dimensional moment problem is solvable.

Among other applications, we present a new characterization of all linear functionals $L$ on $A$ representable as an integral w.r.t. a compactly supported Radon measure solely in terms of a growth condition on $L$, that permits to exactly identify the compact support. This is particularly surprising as the other characterizations available in the literature only show that the support of the representing Radon measure is contained in a compact set and so do not provide exact support descriptions.

ABSTRACT. We investigate geometric and dynamical aspects of hyperbolic lattices arising from regular tilings with 1/p+1/q<1/2. We first characterize finite shapes with minimal perimeter and show that the ratio of perimeter to volume converges to the isoperimetric constant. We also construct a family of regular layered balls that achieve this constant for any fixed volume. We then study the Ising model on finite subgraphs with minus boundary conditions and a positive external field h. For a suitable range of h, we prove the presence of metastable behavior, identify the metastable state, and characterize the exit time. Finally, we describe the energy landscape and analyze the nucleation mechanism for all positive values of h, including beyond the metastable regime.

ABSTRACT. The stochastic Burgers equation stands an important role in fluid dynamics and has been studied by several authors. We mainly refer to the works [3], [4], [5] and [1], where the existence and uniqueness of the global solution as well as the existence and uniqueness of the invariant measure has been established in the "classical" case of space-time white noise.

We discuss the stochastic Burgers equation driven by "rougher" space-time white noise "B dW(t)", where B is the negative Laplacian operator with power gamma in [0, 1/4). We follow the approach of [2], where the polynomial moment estimates of the solution were found in the case gamma = 0. We generalize the results for gamma in [0, 1/4) and we improve the polynomial moment estimates to the exponential moment estimates.

In the second part of our talk, we establish the existence and uniqueness of the invariant measure to our system and we also show its exponential integrability.

References:

[1] Da Prato, G.: Kolmogorov Equations for Stochastic PDEs. Advanced Courses in Mathematics - CRM Barcelona, Springer Basel AG, Basel (2004)

[2] Da Prato, G., Debussche, A.: m-dissipativity of Kolmogorov operators corresponding to Burgers equations with space-time white noise. Potential Analysis 26, 31-55 (2007)

[3] Da Prato, G., Debussche, A., Temam, R.: Stochastic Burgers' equation. Nonlinear Differential Equations and Applications NoDEA 1, 389-402 (1994)

[4] Da Prato, G., Gatarek, D.: Stochastic Burgers equation with correlated noise. Stochastics and Stochastic Reports 52, 29-41 (1995)

[5] Da Prato, G., Zabczyk, J.: Ergodicity for Infinite Dimensional Systems. Vol. 229, Cambridge University Press, Cambridge (1996)

ABSTRACT. Consider the semi-discrete torus $\mathbb{T}_n = [0,1) \times \{0,1,\ldots,n-1\}$, representing unit-length strings running in parallel. A bead configuration on $\mathbb{T}_n$ is a point process on $\mathbb{T}_n$ with the property that between every two consecutive points on the same string, there lies a point on each of the neighbouring strings. In \cite{bead}, we develop a continuous version of Kasteleyn theory to show that partition functions for bead configurations on $\mathbb{T}_n$ may be expressed in terms of Fredholm determinants of certain operators on $\mathbb{T}_n$. We obtain an explicit formula for the volumes of bead configurations on $\mathbb{T}_n$, and show that the asymptotic correlations match those obtained by Boutillier \cite{boutillier}.

The asymptotics of the volume formula confirm a recent prediction due to Shlyakhtenko and Tao \cite{ST} in the free probability literature. We use these asymptotics to prove a large deviation principle for the macroscopic shape of Gelfand--Tsetlin patterns \cite{JP}.

ABSTRACT. We introduce the nested stochastic block model (NSBM) to cluster a collection of networks while simultaneously detecting communities within each network. NSBM has several appealing features including the ability to work on unlabeled networks with potentially different node sets, the flexibility to model heterogeneous communities, and the means to automatically select the number of classes for the networks and the number of communities within each network. This is accomplished via a Bayesian model, with a novel application of the nested Dirichlet process (NDP) as a prior to jointly model the between-network and within-network clusters. The dependency introduced by the network data creates nontrivial challenges for the NDP, especially in the development of efficient samplers. For posterior inference, we propose several Markov chain Monte Carlo algorithms including a standard Gibbs sampler, a collapsed Gibbs sampler, and two blocked Gibbs samplers that ultimately return two levels of clustering labels from both within and across the networks. Extensive simulation studies are carried out which demonstrate that the model provides very accurate estimates of both levels of the clustering structure. We also apply our model to two social network datasets that cannot be analyzed using any previous method in the literature due to the anonymity of the nodes and the varying number of nodes in each network.

ABSTRACT. Two-sample testing assesses whether two populations differ by comparing their probability distributions, with the Kolmogorov–Smirnov test as a classic example. While numerous extensions address multivariate data, modern applications increasingly involve complex objects such as probability distributions themselves. This leads to the problem of testing the equality of laws of random probability measures. We propose a distance-based twosample test for distinguishing laws of random probability measures using optimal transport theory, and leverage tools from empirical process theory to establish nonparametric theoretical guarantees. Empirically, we benchmark our method against existing approaches on simulated datasets and apply it to a mortality dataset.

ABSTRACT. We focus on semistable Lévy processes that appear as limits of normalized sums of iid random variables when the sample size grows geometrically instead of linearly. This generalizes the class of stable Lévy processes in having a weaker scaling property such that the power law behavior of the tail of the Lévy measure can additionally be disturbed by a log-periodic function. The probability density functions of semistable Lévy processes solve a space-fractional diffusion equation, where the fractional derivative of Marchaud-Weyl form can be represented by a Grünwald-Letnikov type formula by using a Fourier series approach for the periodic perturbations. A solution to the corresponding time-fractional differential equation can be given by the densities of an inverse semistable subordinator and is connected to the space-fractional equation by Zolotarev duality. The time-fractional operator of Caputo type is intimately connected to self-similar Bernstein functions and can be seen as a generalized fractional derivative in the sense of Kochubei. These space-fractional and time-fractional processes and also their composition as a space-time-fractional solution serve as models for anomalous diffusion with log-periodic perturbations and appear as limits of certain continuous-time random walks.

ABSTRACT. Traditional corporate compensation schemes inherently discourage sustainable operations, as managerial incentives remain strictly aligned with financial returns rather than environmental outcomes. However, when a firm is backed by a fully informed "green" investor, this misalignment can be overcome. Building on the framework introduced in [1], we investigate the first-best benchmark of this principal-agent interaction. In our setting, the investor can perfectly deduce the manager’s actions and threatens heavy penalties for any deviation from the socially optimal policy. Under this threat, the investor effectively acts as a social planner, directly implementing the optimal greening and investment strategies.

We formulate this benchmark as a two-dimensional singular optimal control problem. The firm’s state is primarily characterized by its production capacity, X, alongside the accumulated abatement effort, R. The investor controls the firm’s dynamics through two forces: injecting external capital (ν) when production capacity is deemed too low, and enforcing abatement (η). Crucially, abatement operates through a pure substitution effect—the cost of greening is fully internalized as a direct reduction in the production capacity X.

In this talk, we formalize the principal’s optimization criterion as a two-dimensional singular stochastic control problem and analyze the properties of the value function via its associated Hamilton-Jacobi-Bellman variational inequality. The core mathematical challenge arises from the interplay between a degenerate diffusion and an oblique reflection driven by the substitution effect. By exploiting the optimal policies derived under a deterministic setting, we explore the geometry of the free boundaries that partition the state space into continuation and action regions. Our preliminary results reveal that the optimal intervention takes the form of a Skorokhod-type reflection along moving boundaries, which are monotonically increasing with respect to the firm’s accumulated abatement effort.

ABSTRACT. The Föllmer process is a Brownian motion conditioned to have a pre-specified law at time 1. This process can be interpreted as an "augmented" time-compression of the reverse stochastic differential equation (SDE) corresponding to the denoising diffusion probabilistic model (DDPM). While this fact has been indirectly used to analyze DDPM sampling errors via discretization of the reverse SDE, implications of directly discretizing the Föllmer process have not yet been fully explored. This talk aims to clarify these implications while surveying relevant results from existing work.

ABSTRACT. We examine Kraichnan's passive scalar model on the \(d\)-dimensional torus. On \(\mathbb{R}^d\), this model exhibits anomalous dissipation and anomalous regularization. We establish that the model on the torus enjoys a similar anomalous regularization property. The added difficulty in this setting is the absence of translation invariance, and to get around this, we use a scheme relying on an expansion of the correlation function of the driving noise. We establish a similar result for the Kazantsev-Kraichnan model on the torus. Based on joint work with Lucio Galeati and Mario Maurelli.

ABSTRACT. Score-based generative modeling, implemented through probability flow ODEs, has shown impressive results in numerous practical settings. However, most convergence guarantees rely on restrictive regularity assumptions on the target distribution—such as strong log-concavity or bounded support. This work establishes non-asymptotic convergence bounds in the 2-Wasserstein distance for a general class of probability flow ODEs under considerably weaker assumptions: weak log-concavity and Lipschitz continuity of the score function. Our framework accommodates non-log-concave distributions, such as Gaussian mixtures, and explicitly accounts for initialization errors, score approximation errors, and effects of discretization via an exponential integrator scheme. Bridging a key theoretical challenge in diffusion-based generative modeling, our results extend convergence theory to more realistic data distributions and practical ODE solvers. We provide concrete guarantees for the efficiency and correctness of the sampling algorithm, complementing the empirical success of diffusion models with rigorous theory. Moreover, from a practical perspective, our explicit rates might be helpful in choosing hyperparameters, such as the step size in the discretization.

ABSTRACT. We introduce the resilience rate as a measure of financial resilience. It captures the expected rate at which a dynamic risk measure recovers, i.e., bounces back, when the risk-acceptance set is breached. We develop the corresponding stochastic calculus by establishing representation theorems for expected time-derivatives of solutions to backward stochastic differential equations (BSDEs) with jumps, evaluated at stopping times. These results reveal that the resilience rate can be represented as a suitable expectation of the generator of a BSDE. We analyze the main properties of the resilience rate and the formal connection of these properties to the BSDE generator. We also introduce resilience-acceptance sets and study their properties in relation to both the resilience rate and the dynamic risk measure. We illustrate our results in several canonical financial examples and highlight their implications via the notion of resilience neutrality.

ABSTRACT. Noise of transport type enjoys high popularity in stochastic fluid dynamics for its modelling abilities and regularising properties. In this talk, we will explore it in the context of natural convection: a fluid confined between two horizontally aligned plates will be heated from below and cooled from above. At a high temperature difference at the boundaries, the fluid will be turbulent and it is a longstanding challenge in physics and engineering to characterise the average heat flux (the Nusselt number Nu) in terms of said temperature difference. In the mathematical literature various results in this direction have been obtained in the form of upper bounds on Nu confirming physicists' predictions. In this talk, I will present a novel viewpoint on a treatment of this problem via stochastic parametrisation: informed by the evolution equations of convection, the stochastic velocity field acts as a transport noise on the temperature and provides a physically relevant upper bound for Nu.

ABSTRACT. Chemical reaction networks (CRNs) are commonly analyzed through deterministic or stochastic models that track molecular populations over time. In regimes with large molecule counts, stochastic dynamics are typically approximated by deterministic mass-action kinetics. We present a CRN that defies this expectation: while the deterministic system is unstable, exhibiting finite-time blow-up of trajectories within the interior of the state space, its stochastic counterpart is positive recurrent.

ABSTRACT. Approximate Bayesian Computation (ABC) is a family of methods that allow sampling from an approximate posterior even when the likelihood is intractable, provided one can simulate from the model and quantify the discrepancy between simulated and observed data. While this discrepancy has traditionally been defined through summary statistics, recent developments in ABC leverage distances between empirical distributions, with the Wasserstein distance emerging as an interpretable and principled choice. However, it has been shown that Wasserstein ABC can be highly sensitive to outlier contamination. We identify that this sensitivity arises from the choice of the cost function rather than from the Wasserstein distance itself. We then propose to replace the usual Euclidean cost with a kernel-based cost, leading to a kernel Wasserstein distance that substantially enhances robustness while preserving ABC posterior concentration under broad conditions. This provides a flexible and theoretically grounded alternative to classical Wasserstein ABC.

ABSTRACT. We are interested in the random pinning model, which depicts a physical system made of a polymer chain interacting attractively with a defect line. The model undergoes a phase transition as the strength of the interaction increases, from a delocalized regime where the polymer touches the defect line finitely many times, to a localized regime where the number of contact points is proportional to the length of the chain. This phase transition has been extensively studied in the case where the interactions between the sequence of monomers and the line are homogeneous, or when they are given by a random i.i.d. sequence called the disorder or the environment. In this work we are interested in the case where the environment is not independent, but on the contrary displays long-range correlations. In this talk we focus on the localized regime and discuss several quantities such as the length of the longest loop between successive contact points, or the asymptotics of the total number of contacts.

ABSTRACT. This talk will focus on introducing new classes of predictive distributions which make use of sample quantities. Theoretical properties of these models will be discussed and compared with a recent proposal based on copulas. We will show some illustrations of their use in the context of predictive resampling.

ABSTRACT. This talk will provide an overview of intermediate interactions in particle systems and their applications to fluid dynamics and biological modeling. The discussion will begin with an introduction to scaling limits, highlighting the distinctions between Mean Field, local, and intermediate interaction cases, which serve as bridges between microscopic particle dynamics and macroscopic PDE formulations. Next, I will explore two applications of this type of interaction: a microscopic approach to the Vlasov-Fokker-Planck-Navier-Stokes equations, which model particle-fluid interactions, and a PDE model for cell-cell adhesion, with a focus on biological aggregation phenomena. The main results will illustrate the convergence properties of empirical measures, achieved through energy estimates and tightness conditions, while addressing challenges in particle-fluid coupling and stochastic modeling. Finally, I will conclude with a discussion of open questions, particularly those related to the problem of fluctuations.

ABSTRACT. The aim of our work is to compare the optimal consumption path under various perceptions of the environment: either only the local environmental quality is considered, either both local and the whole environmental quality are taken into account. A benchmark model for which each representative agents takes into account in her welfare function only local environmental quality is studied. When representative agent in each locality is only concerned by his own local environmental quality, the optimal trajectory follows a balanced growth path solution, that is environmental quality grows at a constant rate all over the time. The consequences on global environmental quality in this benchmark model are then considered. We then study whether taking into account global environmental quality may better local environmental quality. To this aim we study a N-player game where both local and global amenity value of the environmental quality are considered. Some remarks on the behaviour are deduced when the number of the agents tends to infinity. We show that local environmental quality may not necessarily be better by this consideration.

ABSTRACT. We extend a previously introduced one-dimensional diffusion model on probability measures, defined via the rearranged stochastic heat equation, by penalizing the dynamics with an additional entropy-driven gradient-descent term. By means of a splitting argument, we prove that despite the opposite effects of rearrangement and entropy minimization, the resulting penalized stochastic heat equation is well defined. We study several properties of the associated dynamics and show, in particular, that solutions admit a density satisfying a corrected version of the Dean--Kawasaki equation. Moreover, smoothing properties established for the stochastic heat equation are shown to persist, which, together with the existence of a density, leads to regularization results for mean-field models depending on the pointwise value of the density.

ABSTRACT. The Critical Stochastic Heat Flow (SHF) is a measure valued stochastic process that defines a non-trivial solution to the two-dimensional stochastic heat equation with multiplicative space-time noise. Its one-time marginals are a.s. singular with respect to the Lebesgue measure, meaning that the mass they assign to shrinking balls decays to zero faster than their Lebesgue volume. In this talk, we explore the intermittency properties of the Critical 2d SHF by studying the asymptotics of the h-th moment of the mass that it assigns to shrinking balls of radius ϵ. Using a similar method, we also study collision local times of random walks, and identify √ log N as the critical scale for independence of pairwise collision local times of random walks, with lower and upper bound results up to double-logarithmic corrections. Such critical scale was introduced in [1], and this is a refinement of their result.

ABSTRACT. In this talk, we present recent results on strong well-posedness for kinetic SDEs. In particular, we focus on a model with autonomous diffusion driven by a symmetric α-stable process under Hölder regularity conditions for the drift term. We partially recover the thresholds for the Hölder regularity that are optimal for weak uniqueness. In general dimension, we only consider α=2 and need an additional integrability assumption for the gradient of the drift: this condition is satisfied by Peano-type functions. In the one-dimensional case, we do not need any additional assumption. The results presented are based on the joint work with Stéphane Menozzi and Stefano Pagliarani “Strong regularization by noise for a class of kinetic SDEs driven by symmetric α-stable processes”, Stochastic Process. Appl. 189 Paper No. 104691, 19 pp. (2025).

ABSTRACT. Tree-based priors for probability distributions are usually specified using a predetermined, data-independent collection of candidate recursive partitions of the sample space. To characterize an unknown target density in detail over the entire sample space, candidate partitions must have the capacity to expand deeply into all areas of the sample space with potential non-zero sampling probability. Such an expansive system of partitions often incurs prohibitive computational costs and makes inference prone to overfitting, especially in regions with little probability mass. Thus, existing models typically make a compromise and rely on relatively shallow trees. This hampers one of the most desirable features of trees, their ability to characterize local features, and results in reduced statistical efficiency. Traditional wisdom suggests that this compromise is inevitable to ensure coherent likelihood-based reasoning in Bayesian inference, as a data-dependent partition system that allows deeper expansion only in regions with more observations would induce double dipping of the data. We propose a simple strategy to restore coherency while allowing the candidate partitions to be data-dependent, using Cox’s partial likelihood. Our partial likelihood approach is broadly applicable to existing likelihood-based methods and, in particular, to Bayesian inference on tree-based models. We give examples in density estimation in which the partial likelihood is endowed with existing priors on tree-based models and compare with the standard, full-likelihood approach. The results show substantial gains in estimation accuracy and computational efficiency from adopting the partial likelihood.

ABSTRACT. A flexible, distribution-free framework for collective outlier detection and enumeration is introduced, targeting situations in which the presence of outliers can be detected powerfully even though their precise identification may be challenging due to the sparsity, weakness, or elusiveness of their signals. The methodology builds on recent advances in conformal inference and integrates classical ideas from multiple testing, locally most powerful and adaptive rank tests, and nonparametric large-sample asymptotics.

ABSTRACT. We establish central and non-central limit theorems for sequences of geometric functionals of the limiting Gaussian output of random neural networks on the sphere. We show that, as the depth increases, the asymptotic behaviour is determined by the fixed points of the covariance kernel and leads to three possible regimes: convergence to the same functional evaluated at a limiting Gaussian field; convergence to a Gaussian distribution; or convergence to a spherical Rosenblatt/Hermite-type distribution.

More generally, we prove that the transition between these behaviours is governed by the uniform order of integrability (up to controlled errors) of the renormalized covariance function. This mechanism is closely related to what occurs for Gaussian fields with regularly varying covariances at infinity in the Euclidean setting, and reveals an analogous structure on the sphere. Based on a joint work with S. Di Lillo and D. Marinucci.

ABSTRACT. For k ≥ 4, we establish that p = (e/n)^(1/k) is a sharp threshold for the existence of the k-th power H of a Hamilton cycle in the binomial random graph model. Our proof builds upon an approach by Riordan based on the second moment method, which previously established a weak threshold for H. This method expresses the second moment bound through contributions of subgraphs of H, with two key quantities: the number of copies of each subgraph in H and the subgraphs’ densities. We control these two quantities more precisely by carefully restructuring Riordan’s proof and treating sparse and dense subgraphs of H separately. This allows us to determine the exact constant in the threshold.

ABSTRACT. We consider the open problem concerning the possible lack of concentration of the degree distribution in preferential attachment graphs with random initial degree, when its distribution is characterized by extremely heavy tails of power-law type. We show that the addition of such a large number of edges causes a significant upset of the degree distribution, leading to its non-concentration. Furthermore, we show that the smallest value of the exponent for which the degree distribution exhibits concentration is 2.

ABSTRACT. We study Stackelberg differential games between an insurer and a reinsurer in an unobservable Markov-modulated Poisson compound risk model, where the intensity is not known but has to be inferred from the observations of claims arrivals. We consider two different games in which the reinsurance is proportional and the reinsurer adopts an intensityadjusted variance premium principle. In the first games, both insurer and reinsurer aim to maximize the expected exponential utility of their terminal surplus. In the second game, both insurer and reinsurer seeks to maximize the expected terminal surplus penalized by a quadratic term that discourages extreme values of the protection level – either too small, resulting in excessive risk retention, or too large, leading to over-reliance on reinsurance. We characterize the equilibrium of the game and the corresponding value functions under partial information and full information for comparison reasons. Moreover, we numerically investigate the effect of the unobservable stochastic factor that modulates the claim arrival process on the game equilibria.

ABSTRACT. We present Wasserstein Least Squares Regression (WLSR), a model that canonically extends least squares regression in the presence of vector-valued covariates and distribution-valued responses. Unlike competing proposals, which focus on the linear structure in the space of probability measures, ours works directly with the functional form of linear regression. In this talk, we will delve into the geometry of WLSR and draw methodological connections with regression models in Eucledian space.

ABSTRACT. Classical BSDE theory provides an elegant probabilistic representation of stochastic control problems and their associated PDEs. However, this correspondence breaks down in the mean field (McKean–Vlasov) setting, where the law of the state process enters the dynamics intrinsically, rendering the standard framework inadequate. Following [Djete, 25], we discuss a class of “BSDEs” on the Wasserstein space of probability measures, whose solutions are intrinsically measure-dependent. A key feature of this definition is its correspondence with mean field control problems and with PDEs on the Wasserstein space, providing a fully probabilistic counterpart to analytic approaches based on the master equation. Despite these promising applications, well-posedness results for such BSDEs have been proven in [Djete, 25] only in a few particular cases, such as for generators with linear or quadratic growth in the z-variable.

Our main contribution is an existence and uniqueness result for this class of BSDEs, which goes beyond previously studied settings. Our key argument is based on an alternative representation of the solution, which is of independent interest, especially for numerical applications. This talk is based on ongoing joint work with Mao Fabrice Djete.

ABSTRACT. Kernel methods has been used in Bayesian inference for construction adaptive quadrature schemes, and/or adaptive proposal densities and/or as possible emulators of noisy and costly posteriors. Furthermore, in Bayesian inference, the choice of prior distributions plays a central role, and a large body of literature has investigated constructions in which priors are linked, either directly or indirectly, to the likelihood function or to the observed data. These approaches are often motivated by the desire to reduce subjectivity in prior specification while retaining coherence with the underlying statistical model. These ``non-informative'' specifications are determined by the model structure rather than by expert knowledge. Some well examples are given by (a) the empirical Bayes approach for prior parameter tuning, (b) Jeffreys priors (which are derived from the Fisher information) and (c) reference priors, to name a few. More generally, reference priors were developed as a formal framework to maximize the expected information gain from data, providing a principled way to construct priors. In this work, we aim to extend the methodologies developed for the so-called partial, intrinsic, and fractional Bayes factors, along with related approaches. We also show the relationship with the use some improper priors and the application of the proposed approach for model selection purposes.

ABSTRACT. The transmission of Mpox, a zoonotic Orthopoxvirus with rodents as primary reservoirs, exhibits marked clustering during mass gatherings and superspreader events, a feature overlooked by existing models \cite{Rahman2025}. We introduce a stochastic compartmental model incorporating Hawkes processes \cite{Hoks} to capture these self-exciting dynamics in human populations, complemented by Brownian noise for environmental fluctuations in both human and rodent compartments \cite{DiNunno}. \par We prove global existence, uniqueness, and positivity of solutions. Furthermore, we derive the basic reproduction number and establish explicit persistence-in-the-mean conditions for both infected rodents and humans. Numerical simulations are provided to illustrate the impact of self-exciting jumps on epidemic trajectories, highlighting how Hawkes dynamics significantly enhance the predictive capacity of Mpox modeling compared to classical stochastic approaches. Our results suggest that incorporating temporal dependence in jump processes is essential for evaluating the effectiveness of public health interventions, such as quarantine and public awareness campaigns, in the face of clustered transmission patterns.

ABSTRACT. We propose a unified mathematical framework for defining indices of dependence for random probability measures by embedding them into a Hilbert space and applying correlation indices to the resulting Hilbert-valued random variables. This approach overcomes the lack of a linear structure in the space of probability measures and relies on two sources of variability: the choice of the correlation index and the choice of the embedding. We consider Canonical Correlation, Centered Alignment, and Trace Correlation, combined with a Wasserstein-based embedding and two kernel-based embeddings, allowing us to reinterpret existing dependence measures and extend Kernelized Canonical Correlation and Centered Kernel Alignment to distribution-valued data. We characterize the extremal behavior of the proposed indices under independence, almost sure equality, and equality up to linear push-forward transformations, providing theoretical guarantees and interpretability. Numerical experiments on synthetic data and an application to hierarchical clustering of cortical regions in functional brain imaging illustrate the practical relevance of the framework.

ABSTRACT. We consider a class of inhomogeneous random graphs G_n(α, ε) where n vertices carry i.i.d. Pareto weights (Wi)i∈[n] with tail index α > 0. Conditionally on the weights, edges are drawn independently with probability pij = min(εWiWj , 1), where ε = ε_n controls sparsity. The behaviour of the model is driven by the tail index α, with a sharp structural change at the boundary α = 1. The infinite-mean and finite-mean regimes lead to fundamentally different emerging landscapes. Building on recent work of L. Avena, D. Garlaschelli, R.S. Hazra and M. Lalli (Journal of Applied Probability 2025), we analyze the degree asymptotics across the full range α > 0 and identify the relevant scalings of ε_n in each regime for the convergence in distribution of the typical degree. We then characterize the connectivity threshold. In the infinite-mean case α ≤ 1, connectivity is hub-driven and forces a collapse of the diameter to at most two. In the finite-mean regime α > 1, connectivity emerges through a collective mechanism at a density scale distinct from that of ultra-small-world behaviour. This is joint ongoing work with Luisa Andreis, Luca Avena and Rajat Hazra.

ABSTRACT. Diffusions whose dynamics are perturbed at an interface point (through coefficient discontinuities or boundary effects such as reflection, skewness, or stickiness) arise naturally in short-rate and volatility modeling. In this talk we focus on one-dimensional singular SDEs with interfaces and singular diffusion coefficient.

We discuss pathwise (strong) existence and uniqueness under assumptions that allow generalized drift components, including local time terms at the interface, and non-uniformly elliptic diffusion coefficient. As a motivating example, we consider a threshold Cox-Ingersoll-Ross model, where the drift and the diffusion coefficients change across a prescribed level.

We also comment on parameter estimation and simulation of the resulting dynamics, highlighting how first-passage times enter both in the well-posedness analysis and in the simulation and estimation questions.

This talk is mainly based on the recent preprint in collaboration with Benoît Nieto. The estimation aspects are based on joint works with Benoît Nieto and Paolo Pigato. The simulation aspects are object of the PhD thesis of Julia Budzinski.

ABSTRACT. Quantum neural networks (QNNs) constitute the quantum version of deep neural models, where the generated functions are defined by the expectation values of quantum observables measured on the output of parametric circuits. A fundamental breakthrough in the theory of classical deep learning has been the proof that, in the limit of infinite width, the probability distribution of the function generated by a neural network converges to a Gaussian process.

In this presentation, I will explore the extension of these properties to the quantum domain. While recent advancements have established this convergence qualitatively, we provide a rigorous quantitative proof. Using Stein's method for normal approximation, we establish explicit upper bounds on the Wasserstein distance of order 1 between the distribution of a finite-width QNN and the limiting Gaussian process. Furthermore, I will analyze the training dynamics under gradient flow, proving that these quantitative bounds remain valid throughout the optimization process and are uniform in time. This analysis confirms that large-width QNNs preserve their Gaussian characteristics even for infinite training time, providing a solid theoretical foundation for understanding the behavior and stability of overparameterized quantum machine learning models.\\ \noindent \textit{This talk is based on joint works with F. Girardi, D. Pastorello, and G. De Palma.}

ABSTRACT. We study a class of spatially structured stochastic networks that couple queueing-type communication dynamics with sensing-type state estimation. Network nodes are distributed according to a stationary homogeneous Poisson point process $\mathrm{\Phi} \subset \mathbb{R}^2$. Around each node, secondary agents evolve according to localised random motions, yielding a dynamic marked point process. Interactions are induced through shot-noise interference generated by full spectrum reuse, so that both communication and sensing performances depend on the same underlying random field. Communication dynamics are modelled as spatially indexed queues whose service rates are monotone functionals of the instantaneous signal-to-interference-plus-noise ratio (SINR). Sensing dynamics are described by a partially observed stochastic process whose observation noise covariance is itself a functional of the same interference field, leading to a state-dependent filtering problem. This construction induces a non-trivial coupling between a queueing network in random environment and a family of stochastic estimators driven by spatial shot noise. We define system-level performance metrics under the Palm distribution of $\mathrm{\Phi}$. Our main result establishes the association property between communication and sensing functionals at the typical node. Under general shot-noise interference model, we prove that the queue workload process and the filtering error process are associated, in the sense of increasing functionals. The proof relies on coupling constructions, stochastic monotonicity, and comparison arguments for interacting particle systems in random environment. The results suggest that operating regimes that improve communication performance also improve sensing accuracy. More broadly, this framework provides a probabilistic foundation for the analysis of spatial networks with coupled service and estimation mechanisms. It illustrates how tools from stochastic geometry, interacting particle systems, and queueing theory can be combined to analyse large-scale systems where geometry and flow dynamics are intrinsically intertwined.

ABSTRACT. We introduce a nonlocal α-size–biased transform for nonnegative random variables, which recovers the classical size bias in the limit α→1. The transform admits a clear sampling interpretation: it corresponds to an infinite-horizon renewal inspection scheme in which the observation mechanism is α-dependently power-biased toward longer waiting-time gaps. This biasing viewpoint provides a direct link to renewal-based CTRW models of anomalous diffusion, where different observation protocols induce biased waiting-time statistics. We characterize the transform via a Stein identity based on the Riemann–Liouville integral and derive a one-sided concentration inequality as an application. Joint work with Antonio Di Crescenzo.

ABSTRACT. We extend the classical Arbitrage Pricing Theory to a setting where N agents are investing in their respective security markets and additionally are allowed to cooperate through a zero-sum risk exchange mechanism, where no money is injected or taken out of the overall system. Cooperation and the multi-dimensional aspect are the new key features of our setting. In the case of only one agent, the collective theory reduces to the classical Arbitrage Pricing Theory. Within this framework, we introduce the novel notion of Collec- tive Arbitrage. We study the connection between collective and classical arbitrage in our market, and provide various collective versions of the First Fundamental Theorem of Asset Pricing. Secondly, we extend the classical notion of super-replication to the notion of Collective Super-Replication. Collective Superreplication for a given vector of contingent claims, one for each agent in the system, allows for cooperation through risk exchanges among the agents which reduces the overall cost compared to classical individual super-replication. We describe the main properties of the Collective Super-replication functional and its dual representation and discuss the fairness of the cost allocation associated with the Collective Super-replication procedure.

ABSTRACT. An efficient foraging strategy is vital for all living beings \cite{benichou_et_al2024}. Often such search problems can be described by evanescent random walkers (searchers) aiming to hit targets containing the resources which they need for their survival \cite{target_hitting_2024}. % %

In the first part, we consider the target hitting counting process (THCP) of an immortal Markov walker navigating in an ergodic network \cite{fractional_book2019}. We analyze the THCP of an arbitrary stationary set ${\cal B}$ of target nodes. We associate the THCP with an integer counting variable ${\cal N}_i(t;{\cal B}) =\{0,1,2,\ldots\}$ (with ${\cal N}_i(0;{\cal B})=0$ where $i$ is the departure node). This non-decreasing counting variable is increased by a unit when a target node $j\in {\cal B}$ is hit by the walker. In general, the THCP is not a renewal counting process apart of the distinguished cases, in which (a) target ${\cal B}$ consists of a single target node coinciding with the departure node; and (b) for stationary Markov chains, where the THCP boils down to a Bernoulli counting process. We highlight connections with the literature \cite{Noh_Rieger2004}. % %

Then we connect the THCP with the survival statistics of a mortal walker performing Markov steps in an ergodic network \cite{MRW_Mi_Ria2025}. The survival of the walker requires a positive "budget". Each step reduces the budget by one unit. The budget is reset at target hitting times to an IID copy of its initial value, highlighting the connection with stochastic resetting \cite{Evans_Mujamdar2011,Mi_Dono_Poli_Ria_resetting_Chaos2025}. The walker dies when the budget reaches null for the first time. We obtain analytically the evanescent propagator matrix, the survival probability of the walker, the mean residence time on a set of nodes during the walker’s lifetime, and the expected lifetime. The results also include the number of target hits (budget renewals) in a walker's lifetime. We identify analytically and numerically three pertinent scenarios: (i) the forager scenario, in which frequent encounters with target nodes extend the walker’s lifetime, (ii) a detrimental scenario, where frequent encounters instead reduce it, and a neutral scenario (iii) where the frequency of target node hits has no effect on the lifetime. We corroborate our analytical results with random walk simulations on Barabási–Albert graphs. The model has cross-disciplinary applications in finance, gambling, population dynamics, epidemic spreading, chemical reactions, and others \cite{Pastor-Satoras_Vespiani2001,Granger_etal_2024}. Extensions of our model include mortal walkers subjected to stochastic resetting, which sensitively modifies the dynamics \cite{future_paper}.

ABSTRACT. see pdf file

ABSTRACT. It is increasingly common in machine learning to use learned models to label data and then employ such data to train more capable models. The phenomenon of weak-to-strong generalization exemplifies the advantage of this two-stage procedure: a strong student is trained on imperfect labels obtained from a weak teacher, and yet the strong student outperforms the weak teacher. In the talk, I will start by considering ridgeless, high-dimensional regression, and I will provide a sharp characterization of the risk of the target model when the surrogate model is either arbitrary or obtained via empirical risk minimization. This shows that weak-to-strong training, with the surrogate as the weak model, provably outperforms training with strong labels under the same data budget, but it is unable to improve the scaling law. Next, I will show that the scaling law can improve when both the student and the teacher are trained via random feature ridge regression. I will derive a dimension-free deterministic equivalent for the risk of the student trained on teacher labels and then, via this deterministic equivalent, I will identify regimes in which the scaling law of the student improves upon that of the teacher. This shows that the improvement can be achieved both in bias-dominated and variance-dominated settings. Strikingly, the student may attain the minimax optimal rate regardless of the scaling law of the teacher -- in fact, when the risk of the teacher does not even decay with the sample size.

ABSTRACT. Branching processes are models used to describe populations that reproduce and die over time. In the classical setting, an individual's reproductive capacity remains constant throughout its lifetime. However, in real-world situations, reproductive capacity typically undergoes ageing - that is, after reaching a peak, it decreases over time. In this work, we study the influence of ageing on the behaviour of the process and how modifying its parameters, along with reproduction rates, affects the destiny of the process. More precisely, we introduce an ageing mechanism through a time-dependent birth rate, focusing on the case of exponential decay governed by a parameter $\alpha$. This modification allows us to capture realistic biological and epidemiological scenarios in which reproduction or transmissibility is strongest at early stages and progressively weakens over time. We analyse how the interplay between the reproduction intensity $\lambda$, the ageing parameter $\alpha$, and the spatial structure of the model determines survival and extinction.

ABSTRACT. Tensor programs \cite{yang1} provide a unified formalism for describing wide neural network architectures and analyzing their infinite-width limits, under appropriate scaling. Classical master theorems establish convergence in distribution of finite-width networks to their infinite-width counterparts, but typically do not provide explicit finite-width error bounds beyond specific settings.

In this work, we prove quantitative master theorems for general tensor programs. Generalizing the main result of \cite{basteri_trevisan}, we establish non-asymptotic bounds in Wasserstein distance between the joint law of the feature variables generated by the finite-width execution and those of the corresponding infinite-width execution. Our results apply under mild assumptions on the activation function and yield explicit convergence rates in terms of the layer widths. As a consequence, we obtain quantitative kernel convergence estimates with matching rates. The proof proceeds by induction over program lines and relies on a detailed analysis of conditional Gaussian updates for matrix multiplication operations, combined with stability estimates the rest of steps in the program.

These results provide a general quantitative refinement of the master theorem in \cite{yang1} and yield explicit finite-width control for a broad class of neural network architectures.

ABSTRACT. We investigate a class of drift-based transformations for multidimensional diffusion processes. The approach is finalized to construct a transformed diffusion whose transition probability density function (p.d.f.) admits a product-form representation with respect to the p.d.f. of the original process. In particular, the ratio between the transformed and original transition densities reduces to a simple expression involving a weight function w. The framework is formulated in terms of stochastic differential equations, from which the weight function w is obtained. Moreover, we establish general conditions under which the transformed p.d.f. remains analytically tractable in the multidimensional setting. Specific choices of the weight function yield mixture representations of the transformed density, revealing structural properties such as bimodality and modified stochastic ordering. The analysis also shows how the product-form relation persists under Poissonian resetting mechanisms, leading in certain cases to explicit stationary distributions and offering insight into diffusions evolving in potential fields. Two fundamental case studies are examined in detail, based on transformations of the Wiener and Ornstein--Uhlenbeck processes. For these models, explicit expressions of the weight function, potential structure, and transition densities are derived. Special attention is devoted to the two-dimensional setting, for which the conditions and behaviors of the transformed processes are analyzed in depth. \par Beyond its theoretical relevance, the representation suggests practical applications in simulation. In particular, it naturally supports rejection sampling schemes, where the original transition density serves as a proposal distribution. Under suitable boundedness conditions, the acceptance probability can be expressed directly in terms of the weight function, resulting in an efficient and implementable algorithm. The results highlight the flexibility of drift-based transformations as a tool for constructing analytically tractable diffusions. While the present work focuses on prototypical Gaussian models, the methodology suggests several possible extensions, including more general diffusion classes, alternative resetting mechanisms, and further analytical and computational developments. This contribution is based on [1].

ABSTRACT. Stochastic growth models and sigmoidal processes are crucial due to their ability to describe phenomena commonly observed in nature. These models are particularly relevant in fields such as medicine and biology, where they are used to represent the spread of diseases, immune responses, and the growth of cellular populations. However, they also have significant applications in finance and physics (see, for example, [2]). This work (cf. [1]) focuses on the lognormal diffusion process subject to random catastrophes, random events which cause jumps and reset the process to a possibly different random state (cf. [3]). The primary contribution of this research is the assumption that the post-catastrophe recovery level follows a binomial distribution. Unlike traditional models where a system might revert to a fixed initial size, our approach allows the population to restart at a random level which reflects a certain survival probability for each element of the population. To demonstrate the usefulness of this framework, we apply it to the population dynamics of wolves subjected to external disturbances. Furthermore, the model effectively captures real-world economic scenarios, such as the trajectories of GDP (Gross Domestic Product) in five European countries impacted by the crises of 2009 and 2020. The findings show that the model can realistically reproduce complex trajectories, displaying periods of gradual growth interspersed with sudden declines triggered by unpredictable external shocks.

ABSTRACT. We describe the critical window for percolation on sparse growing random graphs. In our models, vertices arrive sequentially and connect independently to each earlier vertex with probability proportional to a nonpositive power of its arrival time, continuing until the graph has n vertices. These models include uniformly grown random graphs and inhomogeneous random graphs of preferential attachment type. Whenever the critical percolation threshold is positive, we show that the critical window has width of order (log n)^{-2} and a secondary phase transition at its finite upper boundary. Inside this window the largest component has size of order sqrt(n)/log n, and the susceptibility remains finite and independent of the position in the window. The proofs couple component explorations to branching random walks killed outside an interval of length log n, allowing sharp control of the barely subcritical and critical regimes.

The talk is based on joint work with Joost Jorritsma and Pascal Maillard.

ABSTRACT. Classifier-free guidance is a well-known sampling strategy applied at inference time to manage the trade-off between quality and diversity in diffusion models. Recent work has explored adapting the guidance scale dynamically over time, yielding improved sampling results. However, the theoretical understanding of this approach remains limited. In particular, there is a scarcity of datasets for learning such dynamic guidance schedules, raising concerns about how well these methods generalize. In this paper, we provide statistical guarantees for learning time-dependent guidance scales in conditional diffusion models using classifier-free guidance. We establish concentration bounds for generalization error that do not depend on (i) the dimensionality of the state or prompt spaces, (ii) the specific architectures used for conditional or unconditional score functions, or (iii) hidden constant factors. Furthermore, although the number of timesteps defines the parameter space dimension, we can make the bound independent of it by constraining the sum of the maximal values of the scales weighted by noise schedules. To obtain these results, we develop novel coupling-based PAC-Bayes bounds alongside a coupling framework for diffusion models. A key conceptual contribution is a shift in perspective: we treat fixed diffusion models as prior distributions and guided models as corresponding posteriors within the PAC-Bayes framework.

ABSTRACT. Accurate tuning of hyperparameters is crucial to ensure that models can generalise effectively across different settings. In this talk, we present theoretical guarantees for hyperparameter selection using variational Bayes in the nonparametric regression model. We construct a variational approximation to a hierarchical Bayes procedure, and derive upper bounds for the contraction rate of the variational posterior in an abstract setting. The theory is applied to various Gaussian process priors and variational classes, resulting in minimax optimal rates. Our theoretical results are accompanied with numerical analysis both on synthetic and real world data sets.

ABSTRACT. Originally established in 1931, the Fr\'echet-Shohat Theorem is a fundamental result in the method of moments. It provides sufficient conditions under which the convergence of a sequence of moments $\{m_{k, n}\}_{n=1}^{\infty}$ to a limit sequence $\{m_k\}$ ensures the weak convergence of the associated distribution functions $\{G_n\}$ to a limit $G$. A critical requirement of the classical theorem is the determinacy of the underlying moment problem; that is, $G$ must be the unique distribution characterized by $\{m_k\}$.

This study extends the foundational Fr\'echet-Shohat framework to the setting of indeterminate Hamburger and Stieltjes moment problems, where the classical theorem traditionally fails due to the non-uniqueness of the limiting measure. We demonstrate that by imposing an entropic constraint on the sequence $\{G_n\}$ - specifically, convergence in Shannon entropy - one can recover a unique limit entropy-distinguishable distribution, $G_{hmax}$, from the indeterminate class having the given moments. This result facilitates a unified treatment of the Fr\'echet-Shohat theorem across both determinate and indeterminate frameworks.

The proposed approach provides a rigorous probabilistic foundation for the application of Maximum Entropy methods within statistical inference. Indeed, it strictly adheres to Jaynes' principle of objective inference by ensuring that the derived distribution is uniquely determined by the prescribed moment constraints, thereby precluding the imposition of unjustified or unwarranted assumptions. This methodology operationalizes the MaxEnt desideratum that "{\it \dots we should not assume more than what we know}", maintaining informational parsimony in the presence of indeterminacy.

ABSTRACT. In this talk I will discuss new large deviation results for general stochastic approximation algorithms with state-dependent Markovian noise and decreasing step size.

The theory of stochastic approximations form the theoretical foundation for studying convergence properties of many popular recursive learning algorithms in statistics, machine learning and statistical physics. Large deviation results for stochastic approximations therefore provide asymptotic estimates of the probability that the learning algorithm deviates from its expected path, given by a limit ODE, and the large deviation rate function gives insights to the most likely way that such deviations occur.

The focus of the talk is a new large deviation principle for general stochastic approximations with state-dependent Markovian noise and decreasing step size, obtained using the weak convergence approach. Using this approach, we are able to generalize previous results for stochastic approximations and identify the appropriate scaling sequence for the large deviation principle. We also give a new representation for the rate function, in which the rate function is expressed as an action functional involving the family of Markov transition kernels. Examples of learning algorithms that are covered by the large deviation principle include stochastic gradient descent, persistent contrastive divergence and the Wang-Landau algorithm. Time permitted I will also highlight some connections to weak KAM theory and viscosity solutions to Hamilton-Jacobi equations. In particular regarding the projected Aubry set associated with a stochastic approximation algorithm.

ABSTRACT. Understanding the stability and long-time behavior of generative models is a fundamental problem in modern machine learning. This talk provides quantitative bounds on the sampling error of score-based generative models by leveraging stability and forgetting properties of the Markov chain associated with the reverse-time dynamics. Under weak assumptions, we provide the two structural properties to ensure the propagation of initialization and discretization errors of the backward process: a Lyapunov drift condition and a Doeblin-type minorization condition. A practical consequence is quantitative stability of the sampling procedure, as the reverse diffusion dynamics induces a contraction mechanism along the sampling trajectory. Our results clarify the role of stochastic dynamics in score-based models and provide a principled framework for analyzing propagation of errors in such approaches.

ABSTRACT. We study a general continuous-time random walk (CTRW), by including non-Markovian cases and Lévy flights, under complete stochastic resetting to the initial position with an arbitrary law, which can be power-lawed as well as Poissonian. We provide three linked results. First, we show that the random walk under stochastic resetting is a CTRW with the same jump-size distribution of the non-reset original CTRW but different counting process. Later, we derive the condition for a CTRW with stochastic resetting to be a meaningful displacement process at large elapsed times, i.e., the probability to jump to any site is higher than the probability to be reset to the initial position, and we call this condition the zero-law for stochastic resetting. This law joins with the other two laws for reset random walks concerning the existence and the non-existence of a non-equilibrium stationary state. Finally, we derive master equations for CTRWs when the resetting law is a completely monotone function. The talk is based on the recent paper [1].

[1] Colantoni F, Pagnini G.: Master equations for continuous-time random walks with stochastic resetting. Proc. R. Soc. A 481, 20250641 (2025)

ABSTRACT. We investigate a mean-field game of optimal stopping with common noise, in which a representative agent seeks the optimal stopping time to maximize a reward functional. Both the running and terminal reward functions depend on the mean-field interaction term, which, in equilibrium, corresponds to the conditional law of the optimal stopping time given the common noise. The setting we consider is non-Markovian, as the reward functions are general random functions, and the analysis is performed through a purely probabilistic approach. We seek strong mean-field equilibria in the sense of strong solutions to stochastic differential equations: we fix the probability space and the $\sigma$-algebra representing the common noise and look for adapted solutions. In mean-field games without common noise, strong solutions are usually obtained using fixed point theorems, such as Schauder's or Kakutani's theorems.

Our main contribution is an existence result for strong randomized mean-field equilibria in a setting with continuity assumptions of the reward functions with respect to the interaction terms. We define a strong randomized mean-field equilibrium as a pair, in which the mean-field interaction term is adapted to the common noise, while the stopping time is randomized. In this sense, we allow additional randomization in the stopping times, while maintaining adapted mean-field interaction terms. In order to be able to identify compact subsets within the space of mean-field interactions through tightness arguments, we assume that the common noise is generated by a countable partition of the probability space.

In addition, we study the mean-field game of optimal stopping in a setting with an order structure and monotonicity properties of the reward functions with respect to the mean-field interaction terms. In this framework, the common noise is represented by a general $\sigma$-algebra. We establish the existence of strong mean-field equilibria (with strict optimal stopping times, not randomized) by applying Tarski's fixed point theorem, a result which appears in earlier works. Our contribution lies in a comparative statics analysis of the set of strong mean-field equilibria.

ABSTRACT. Bayesian neural networks, in the overparameterized and infinite-width regime, are now well understood. Under mild assumptions, their prior converges to a Gaussian process (NNGP), and both Bayesian inference and training dynamics can be described by kernel methods. Although, these infinite-width limits provide tractable models and sharp theoretical insights, they also exhibit a fundamental rigidity: the induced feature representation becomes fixed and independent of data. As a result, feature learning disappears in the infinite-width limit, and Bayesian inference reduces to kernel regression with a predetermined kernel.

In this talk, we present a complementary large-deviation perspective on wide Bayesian neural networks. Rather than studying typical Gaussian fluctuations, we analyse exponentially rare, but statistically dominant, configurations that govern posterior concentration as width grows. At this scale, Bayesian inference becomes variational: posterior mass concentrates near minimizers of an explicit functional rate function defined directly on predictors. Our main result shows that, in contrast to the Gaussian-process limit, the posterior large-deviation rate function involves a joint optimization over predictors and internal covariance kernels. This nested variational structure leads to data-dependent kernel selection and provides a mechanism for feature learning that persists even in the infinite-width regime. In particular, we prove that the posterior-optimal kernel generically differs from the NNGP kernel. Joint work with Dario Trevisan.

ABSTRACT. Pearson diffusions form a fundamental class of one-dimensional Markov processes with linear drift and quadratic diffusion coefficients, encompassing the Ornstein–Uhlenbeck, Cox–Ingersoll–Ross, and Jacobi processes. Their generators admit complete spectral decompositions in orthogonal polynomial bases, ensuring analytic tractability and explicit invariant measures, which make them central in stochastic analysis and operator theory.

We develop a non-local framework for Pearson diffusions by replacing the classical first-order time derivative in the Kolmogorov equations with a generalized Caputo-type operator. This extension captures memory effects and yields non-Markovian dynamics, which is obtained via time-change by inverse subordinators. Going beyond standard Caputo fractional derivatives, we introduce a stretched two-parameter non-local operator generating a broader class of memory kernels.

Using the spectral structure of Pearson generators, we establish existence and uniqueness of strong solutions to the associated non-local Cauchy problems. Solutions admit explicit spectral representations in which the classical exponential decay is replaced by Kilbas-Saigo functions, which are generalized Mittag–Leffler–type functions, exhibiting power-law decay. The resulting processes preserve invariant distributions while displaying stretched temporal behavior. These results generalize fractional Pearson diffusions and provide new operator-theoretic insights into anomalous diffusion and long-memory stochastic systems.

ABSTRACT. We consider the discrete Gaussian free field in random environment, where disorder is introduced through random edge conductances on the underlying graph. Such a model describes microscopic fluctuations of a crystal at positive temperature in the presence of inhomogeneities.

We focus on the integer lattice $\mathbb{Z}^d$ for $d\geq 3$, and analyse the maximal fluctuation of the field and its behaviour in the presence of a macroscopic hard wall constraint. First, we derive sharp quenched large deviation asymptotics for the hard wall event. The rate is governed by two key quantities: the homogenized capacity of the associated random conductance model, and the essential supremum of the on-site (random) variances of the field. Secondly, we investigate the law of the field conditioned on the hard wall. We prove that the conditioned field exhibits an entropic push away from the zero height, and identify its expected asymptotic profile. Lastly, we characterize the pathwise behaviour of the conditioned field. This is based on a joint work with Alberto Chiarini.

We conclude by discussing ongoing work with Alberto Chiarini and Alessandra Cipriani, where, still in the supercritical dimension, we replace the lattice $\mathbb{Z}^d$ with different underlying graphs, and study how their structure influences both the decay for the hard wall probability and the asymptotic profile for the expectation of the conditioned field.

ABSTRACT. We consider the spectrum of the adjacency matrix of directed inhomogeneous graphs with independent edges.

Our framework includes, directed stochastic block models and the directed Chung–Lu model. We assume that the expected adjacency matrix has k non-zero eigenvalues of multiplicity 1 and we scale connection probabilities, so that average degrees diverge at least poly-logarithmically in the number of vertices.

In the rank one case, under suitable conditions, we provide a characterization of the asymptotic empirical spectral distribution, in terms of non-dilute Guassian arrays. This is done extending to the inhomogeneous case previous results on the least singular values of complex diagonal shifts for the rescaled matrix. Using tools from free probability, we provide a more precise analysis and an explicit expression of the asymptotic singular value distribution.

Moreover, we establish the existence, with high probability, of k real outliers of the spectrum, whose scale is the square of the bulk's one. We further show that, centering and properly rescaling, the joint law of the k outliers converges in distribution to Gaussian multivariate law with an explicit covariance matrix.

Our analysis complements previous works in the symmetric setting.

Based on an ongoing work with Rajat Hazra.

ABSTRACT. Modeling decision processes under semantic vagueness and contextual uncertainty remains a fundamental challenge in logic and artificial intelligence. Classical probabilistic and symbolic frameworks typically assume that cognitive states are well-defined prior to evaluation and that uncertainty can be fully captured within a Kolmogorovian probability space. However, empirical findings in cognitive science reveal systematic violations of classical probability theory in situations involving ambiguity, order effects, and context dependence [2,5]. In this talk, we propose a quantum-inspired formal framework in which cognitive states are represented as vectors in a Hilbert-like conceptual space, allowing vagueness to be modeled as the structured coexistence of multiple latent and potentially incompatible interpretations [1]. Within this setting, context is formalized as an observable acting on the conceptual state, and decision-making is modeled as a probabilistic collapse governed by the Born rule. This approach naturally accommodates contextuality, non-commutativity, and interference effects, offering a non-Kolmogorovian perspective on uncertainty consistent with broader quantum-like probabilistic frameworks [4]. We complement the theoretical formulation with a computational simulation in which conceptual states evolve via unitary transformations before collapsing under context-defined observables, thereby modeling the transition from indeterminate cognitive potentialities to determinate outcomes. Although illustrated through scenarios of creative cognition [3], the proposed model provides a more general paradigm for reasoning and context-dependent decision dynamics under vagueness. The framework suggests a unified formal perspective for studying uncertainty, semantic ambiguity, and probabilistic contextual selection in AI systems.

ABSTRACT. We study finite-horizon mean field optimal stopping problems in which the state pro- cess is unaffected by the stopping time and is therefore uncontrolled. Such problems arise, for instance, in the pricing of American options when the underlying asset follows McKean–Vlasov dynamics. Due to the intrinsic time inconsistency, we introduce a suitable reformulation on an enlarged state space, referred to as the extended problem, which restores time consistency and admits a dynamic programming principle. In particular, this reformulation allows us to characterize both the value function and the optimal stopping time of the original problem in terms of the extended value function. Building on this theoretical framework, the main focus of the presentation is on recent and ongoing developments concerning the N -player cooperative optimal stopping games associated with our mean field problem. Exploiting the strong connection between the original and extended formulations, we analyze the games corresponding to both settings. In particular, to reflect the cooperative nature of the games, we consider only exchangeable stopping strategies. We first investigate the N -player game linked to the extended problem and rigorously prove the convergence of its value function to that of the reformulated mean field limit as N → ∞. Using this result, together with the established relationship between the extended and original limit value functions, we then derive the analogous convergence result for the original problem. Moreover, studying the extended game yields a key characterization of the optimal stopping strategies: for the original (non-extended) game, restricting our analysis to exchangeable strategies implies that the optimal policy consists of N coinciding stopping times, so that all players stop at the same instant. Finally, relying on the characterization of the limiting optimal stopping time and on the convergence of the value functions, we analyze the asymptotic behavior of the N -player optimal stopping time. To obtain explicit probabilistic bounds on its distance from the limit optimal stopping time, our approach crucially exploits the analytic properties of the mean field free boundary, working in a one-dimensional, time-homogeneous Markovian framework under the assumption of a time-decreasing running gain.

ABSTRACT. We investigate the long-time behavior of one-dimensional models of compressible viscous fluids subject to stochastic forcing. In particular, we focus on the Navier–Stokes–Korteweg equations (NSK), which describe the dynamics of a compressible viscous fluid in regimes where capillarity effects cannot be neglected. In this framework, we establish the existence of invariant measures by adapting the Krylov–Bogoliubov method to the case of a non-Feller Markov semigroup. The analysis of invariant measures and, more generally, of ergodic properties for compressible fluid systems presents several structural obstacles, including the lack of compactness, the possible formation of vacuum regions, and the absence of classical regularity frameworks typically used in incompressible settings. This talk provides an overview of these challenges and discusses recent strategies to overcome them. The proof of the existence of invariant measures for the NSK equations relies on the derivation of suitable a priori estimates providing an appropriate time-growth rate of solutions, despite the presence of high-order nonlinear terms due to capillarity. Building on these estimates, we perform a stochastic compactness argument and introduce a class of functions that is invariant under the Markov semigroup while remaining compatible with the available convergence result. Overall, the present result highlights specific properties of Korteweg fluids that remain unknown in models where capillarity effects are neglected.

ABSTRACT. A long-standing open problem of L. E. Dubins seeks to determine the maximal expected range of Walsh's spider process on $n$ edges per root of the expected stopping time. The solution is known for $n=1$ (1988) and $n=2$ (2009). In this paper we present the solution for $n \ge 3$.

ABSTRACT. Diffusion models for continuous state spaces based on Gaussian noising processes are now relatively well understood from both practical and theoretical perspectives. In contrast, results for diffusion models on discrete state spaces remain far less explored and pose significant challenges, particularly due to their combinatorial structure and their more recent introduction in generative modelling. In this work, we establish new and sharp convergence guarantees for three popular discrete diffusion models (DDMs). Two of these models are designed for finite state spaces and are based respectively on the random walk and the masking process. The third DDM we consider is defined on the countably infinite space $\mathbb{N}^d$ and uses a drifted random walk as its forward process. For each of these models, the backward process can be characterized by a discrete score function that can, in principle, be estimated. However, even with perfect access to these scores, simulating the exact backward process is infeasible, and one must rely on time discretization. In this work, we study Euler-type approximations and establish convergence bounds in both Kullback–Leibler divergence and total variation distance for the resulting models, under minimal assumptions on the data distribution. To the best of our knowledge, this study provides the \emph{optimal non-asymptotic} convergence guarantees for these noising processes that do not rely on boundedness assumptions on the estimated score. In particular, the computational complexity of each method scales only \emph{linearly in the dimension, up to logarithmic factors}.

ABSTRACT. We consider a tick-by-tick model of price formation, in which buy and sell orders are modeled as self-exciting point processes (Hawkes process). We adopt an agent based approach by studying the aggregation of a large number of these point processes, mutually interacting in a mean-field sense.

The financial interpretation is that of an asset on which several labeled agents place buy and sell orders following these point processes, influencing the price. The mean-field interaction introduces positive correlations between order volumes coming from different agents that reflect features of real markets such as herd behavior and contagion. When the large scale limit of the aggregated asset price is computed, if parameters are set to a critical value, a singular phenomenon occurs: the aggregated model converges to a stochastic volatility model with leverage effect and faster-than-linear mean reversion of the volatility process.

The faster-than-linear mean reversion of the volatility process is supported by econometric evidence, and we have linked it in a previous work to the observed multifractal behavior of assets prices and market indices. This seems connected to the Statistical Physics perspective that expects anomalous scaling properties to arise in the critical regime.

The presentation is based on a joint work with Paolo Dai Pra.

ABSTRACT. In this talk, we present a nonparametric estimator for the diffusive interaction function in particle systems, constructed from $Nn$ discrete observations of the trajectories. We comment its statistical performance and provide theoretical guarantees on the estimation error within suitable function classes and norms. We also discuss the main challenges arising in this setting and comment on optimality properties of the estimator.

ABSTRACT. We study the geometric structure of the space of random measures $\mathcal{P}_p (\mathcal{P}_p(X))$, endowed with the Wasserstein-on-Wasserstein metric, where $(X, d)$ is a complete separable metric space. In this setting, we prove a metric superposition principle, that will allow us to recover important geometric features of the space.

When $X$ is $\mathbb{R}^d$, we study the differential structure of \(\mathcal{P}_p(\mathcal{P}_p(\mathbb{R}^d))\) in analogy with the simpler Wasserstein space $\mathcal{P}_p(\mathbb{R}^d)$. We show that continuity equations for laws of random measures involving the abstract concept of derivation acting on cylinder functions can be more conveniently described by suitable non-local vector fields $b:[0,T]\times \mathbb{R}^d \times \mathcal{P}_p(\mathbb{R}^d) \to \mathbb{R}^d$. In this way, we can: characterize the absolutely continuous curves on the Wasserstein-on-Wasserstein space; define and characterize its tangent bundle; prove a Benamou-Brenier-like formula; prove a superposition principle for the solutions to the standard non-local continuity equation in terms of solutions of interacting particle systems.

ABSTRACT. The regression importance index of a coherent system evaluates a component’s importance based on the system’s conditional mean lifetime when the component’s failure time is known (see Arriaza et al. [1]). We aim to introduce the ``regression importance signature”, a tool designed to identify, for a given number of components, the subgroup that should be prioritized in reliability analysis and failure localization. To achieve this, the concept of importance index is generalized for subgroups of components, taking into account the occurrence of failures within the subgroup. General results for systems with dependent components are provided, with a particular focus on system modules, as well as sufficient conditions for comparing the importance of individual components and subgroups. This analysis highlights how a component’s relevance depends not only on its reliability but also on its structural role within the system. As an application, we consider the ship control system already discussed in [1], extending the original analysis to the computation of the full signature. This allows us to identify the most influential subgroups of components and to explore how the dependence modeled by the FGM copula and the variation of its parameters affect the importance of different subsets of components.

References:

[1] Arriaza, A., Navarro, J., Sordo, M.A., Suárez-Llorens, A. A variance-based importance index for systems with dependent components. Fuzzy Sets and Systems 467, 108482 (2023). https://doi.org/10.1016/j.fss.2023.02.003

[2] Di Crescenzo, A., Pisano, G., Suárez-Llorens, A. Analysis of systems with dependent components through a variance-based index and regression importance signature. Reliability Engineering & System Safety, 273, 112357 (2026). https://doi.org/10.1016/j.ress.2026.112357

ABSTRACT. Real-world data frequently violate the assumptions underlying classical parametric inference, including normality, homoscedasticity, and balanced experimental designs. These challenges are particularly relevant in applied contexts characterized by heterogeneous samples, mixed outcome types, and small data sizes. This work investigates permutation tests and the Nonpara metric Combination (NPC) methodology as a unified framework for robust statistical inference under minimal assumptions. Permutation tests reconstruct the null distribution of test statistics through resampling based on exchangeability, yielding exact or Monte Carlo–exact inference without distributional constraints. The NPC methodology ex tends this approach to multivariate settings by combining partial permu tation tests using suitable combining functions, such as Fisher statistics, enabling global inference while preserving dependence structures through synchronized permutations and controlling the family-wise error rate. From an applied perspective, this framework naturally accommodates mixed mea surement scales, correlated endpoints, and unbalanced designs. The empirical contribution is illustrated through three interdisciplinary applications. In education, the methodology evaluates teaching effectiveness using a randomized classroom experiment with heterogeneous outcomes, in cluding continuous improvement measures and ordinal satisfaction indica tors, providing strong global evidence of learning gains. In clinical research, the NPC framework is applied to a dataset of patients affected by necrotiz ing fasciitis, combining continuous biomarkers and binary survival outcomes within a small heterogeneous sample. The analysis identifies key predictors and demonstrates the stability of permutation-based inference in noisy medi cal data. Finally, in the social security domain, permutation tests and NPC methodology are employed to assess regional pension disparities in Italy using administrative microdata, revealing statistically significant territorial inequality and highlighting the ability of the framework to integrate corre lated socio-economic indicators into a single coherent inferential measure. Overall, permutation-based NPC methods emerge as robust, interpretable, and distribution-free tools for interdisciplinary statistical analysis, offering a practically relevant alternative to classical parametric approaches. MSC 2020: 62G09; 62H15; 62P10; 62P20; 62P25 References 1. Pesarin, F. (2001). Multivariate Permutation Tests with Applications in Biostatistics. Wiley. 2. Pesarin, F., Salmaso, L. (2010). Permutation Tests for Complex Data: Theory, Applications and Software. Wiley. 3. Giacalone, M., Piscopo, G., Bandaru, S.T. (2025). Permutation-based analysis of clinical variables using NPC methodology. Mathematics

ABSTRACT. We study non-linear additive functionals of Gaussian fields over anisotropically growing domains on $\mathbb R^d$ — for instance, spatiotemporal ones — and show that Gaussian or Rosenblatt-type limits arise under non-separable covariance structures, depending on precise long-range dependence conditions, thereby extending existing spatiotemporal limit theorems beyond the separable and short-memory frameworks. In particular, we prove that 2-domain Rosenblatt distributions emerge as scaling limits for Gaussian fields with Gneiting-type covariance functions, widely used in spatiotemporal applications. The talk is based on a joint work with N. Leonenko, I. Nourdin, and L. Maini.

ABSTRACT. Quantum Markov Semigroups (QMSs) have been used in the literature to model the reduced evolution of a quantum system coupled to the environment. Gaussian QMSs are a specific type of such semigroups, acting on bounded operators on the Boson Fock space, characterized by the fact that their predual map gaussian states into other gaussian states. They are not only relevant in the applications but also from a mathematical standpoint, since they constitute an amenable class of not uniformly continuous semigroups, where many computations can still be performed, and are closely related to the classical Ornstein- Uhlenbeck semigroups. In this talk I will present a summary of results from a double perspective. On the one hand, they provide algebraic conditions on the parameters that define the semi- group to characterize relevant properties of the semigroup itself, such as the spectral gap, the decoherence-free subalgebra, the existence of an invariant state and the symmetry of the semigroup with respect to it. On the other hand, these properties shape the semigroup, forcing it to have a specific, simpler structure.

ABSTRACT. We model complexity by introducing a complexity order that ranks lotteries by their Wasserstein distances from degenerate lotteries, which carry no risk. The resulting relation is a continuous incomplete preorder whose properties reflect the geometry of the outcome space. We relate it to the convex order, showing that they coincide for univariate monetary lotteries, while this equivalence fails in higher dimensions.

To address incompleteness, we introduce a complexity measure defined by how well a lottery can be approximated by a degenerate one. This measure provides a natural completion of the complexity order and inherits many of its properties. It enables comparative statics for mixtures of lotteries and yields explicit maximally complex lotteries in several cases.

Finally, we apply these notions to choice under risk. Combining the complexity order with first-order stochastic dominance yields a choice criterion that, for monetary lotteries, is equivalent to second-order stochastic dominance. Using our complexity measure, we define Complexity-Sensitive Expected Utility (CSEU) preferences. For this class of preferences, we analyze how complexity aversion interacts with risk aversion and, in particular, prove that complexity aversion is a component of risk aversion.

ABSTRACT. I will begin by introducing the paper Flandoli-Gubinelli-Priola IM 2010 regarding a singular stochastic transport equation. This appears to have been the first paper concerning a PDE of interest in fluid dynamics that becomes well-posed under the influence of a (multiplicative) Wiener noise. Next, I will discuss several developments and research directions stemming from this work. The first direction involves papers on regularization by transport noise of Wiener type. This field has grown significantly, leading to several important ramifications. I will primarily focus on works published up to 2015. A second direction concerns singular stochastic evolution equations in infinite dimensions. In this regard, the mentioned paper and the theory of Kolmogorov equations in infinite dimensions inspired the seminal paper Da Prato-Flandoli JFA 2010 on strong uniqueness for SPDEs with H\"older continuous and bounded coefficients. I will also mention some recent developments in this area. Finally, I will explore regularization by Lévy noise.

ABSTRACT. We present a Feller-type test for explosions of one-dimensional continuous stochastic Volterra processes of convolution type. We focus on dynamics driven by nonsingular kernels, which preserve the semimartingale property of the processes while incorporating memory effects through a path-dependent drift. For the Volterra square-root diffusion, also known as the Volterra CIR process, we provide a detailed discussion of the approximation of the singular fractional kernel by a sum of exponentials, a technique commonly used in the mathematical finance literature.

ABSTRACT. I will first introduce a general class of mean-field-like spin systems with random couplings that comprises both the Ising model on inhomogeneous dense random graphs and the randomly diluted Hopfield model. I will then present quantitative estimates of metastability in large volumes at fixed temperatures when these systems evolve according to a Glauber dynamics, i.e. where spins flip with Metropolis rates at inverse temperature $\beta$. The main result identifies conditions ensuring that with high probability the system behaves like the corresponding system where the random couplings are replaced by their averages. More precisely, we prove that the metastability of the former system is implied with high probability by the metastability of the latter. Moreover, we consider relevant metastable hitting times of the two systems and find the asymptotic tail behaviour and the moments of their ratio. Based on a joint work in collaboration with Anton Bovier, Frank den Hollander, Saeda Marello and Martin Slowik. 

ABSTRACT. Bilevel optimization problems consist of minimizing a value function whose evaluation depends on the solution of an inner optimization problem. These problems are typically tackled using first-order methods that require computing the gradient of the value function ({\it the hypergradient}). However, in several practical settings, first-order information is unavailable ({\it zeroth-order setting}), rendering these methods inapplicable. Finite-difference methods provide an alternative by approximating hypergradients using function evaluations along a set of directions. Nevertheless, such surrogates are notoriously expensive, and existing finite-difference bilevel methods rely on two-loop algorithms that are poorly parallelizable. To tackle these limitations, we propose ZOBA, the first finite-difference single-loop algorithm for bilevel optimization. Our method leverages finite-difference hypergradient approximations based on delayed information to eliminate the need for nested loops. We analyze the proposed algorithm and establish convergence rates in the non-convex setting, achieving a complexity of $\mathcal{O}(p(d + p)^2\varepsilon^{-2})$, where $p$ and $d$ denote the dimension of inner and outer spaces respectively and $\varepsilon \in (0,1)$ an accuracy parameter, which is better than prior approaches based on Hessian approximation. We further introduce and analyze HF-ZOBA, a Hessian-free variant that yields additional complexity improvements. Finally, we corroborate our findings with numerical experiments on synthetic functions and a real-world black-box task in adversarial machine learning. Our results show that our methods achieve accuracy comparable to state-of-the-art techniques while requiring less computation time.

ABSTRACT. We argue for the use of separate exchangeability as a modeling principle in Bayesian nonparametric (BNP) inference. Separate exchangeability is de facto widely applied in the Bayesian parametric case, e.g., it naturally arises in simple mixed models. However, while in some areas, such as random graphs, separate and (closely related) joint exchangeable models are widely used, they are curiously underused for several other applications in BNP. We briefly review the definition of separate exchangeability, focusing on the implications of such a definition in Bayesian modeling. We then discuss two tractable classes of models that implement separate exchangeability, which are the natural counterparts of familiar partially exchangeable BNP models.

The first is nested random partitions for a data matrix, defining a partition of columns and nested partitions of rows, nested within column clusters. Many recent models for nested partitions implement partially exchangeable models related to variations of the well-known nested Dirichlet process. We argue that inference under such models in some cases ignores important features of the experimental setup. We obtain the separately exchangeable counterpart of such partially exchangeable partition structures.

The second class is about setting up separately exchangeable priors for a nonparametric regression model when multiple sets of experimental units are involved. We highlight how a Dirichlet process mixture of linear models, known as ANOVA DDP, can naturally implement separate exchangeability in such regression problems. Finally, we illustrate how to perform inference under such models in two real data examples.

This presentation is based on a joint work with Qiaohui Lin and Peter Müller.

ABSTRACT. We consider the strategic interaction of traders in a continuous-time financial market with Epstein-Zin-type recursive intertemporal preferences and performance concerns. We derive explicitly a Nash equilibrium for the finite player game and a mean-field equilibrium for the mean-field version of the game, based on a study of geometric backward stochastic differential equations of Bernoulli type that describe the best replies of traders. Our results show that Epstein-Zin preferences can lead to substantially different equilibrium behavior.

ABSTRACT. We develop a probabilistic model for free-floating car-sharing systems. In these systems, shared vehicles occupy the same parking spaces as private cars. The availability of parking spaces across different zones of a city depends on private cars, which are far more numerous than free-floating vehicles. The system dynamics are described by a closed network of queues in which private and free-floating cars move among the same nodes, representing city zones. Nodes have finite capacity, and saturation is handled through a blocking and rerouting policy. We show that these dynamics preserve the product-form structure of the invariant distribution. The model accounts for spatial heterogeneity in both user demand and availability of parking spaces. We identify, in this setting, phase transitions between an overloaded regime where all nodes are saturated and underloaded regimes with saturated and non-saturated nodes. Scaling limits and stochastic averaging methods are used to analyze the behavior of the system when the capacity of the nodes is large. The analysis is performed when the average number of private cars per zone increases linearly with capacity, while the number of free-floating cars remains of smaller order. Our goal is to characterize the macroscopic behavior of the system and provide insights for optimizing vehicle distribution.

ABSTRACT. We present a construction of the measure of the fractional $\Phi^4$ Euclidean quantum field theory on $\mathbb{R}^3$ in the full subcritical regime via parabolic stochastic quantisation. Our approach is based on the use of a truncated flow equation for the effective description of the model at sufficiently small scales and on coercive estimates for the nonlinear stochastic partial differential equation describing the interacting field. The constructed measure is invariant under translations, reflection positive and has quartic exponential tails.

ABSTRACT. Homophily is the tendency of people to interact more with others who are similar to them (for instance, by age, gender, or opinion). It is often summarized by a single assortativity number for the entire network, but this can hide important structural features. From a probabilistic viewpoint, this is a loss of information: two networks may share the same global homophily yet have very different local organization and large-scale behavior.

A key observation is that social interactions naturally occur at different “scales”. Some interactions are one-to-one, while others occur within small groups (for example, a work team, a classroom, or a recurring social circle). These scales need not display the same mixing pattern: a society may have many cross-group acquaintances but mostly same-group close circles, or vice versa. Standard network statistics typically blend these effects together.

In this talk, we present a modeling framework that makes this multi-scale structure explicit while remaining analytically tractable. The idea is to represent a sparse network as the superposition of group interactions of different sizes. For each size we control, we estimate how strongly interactions tend to occur within versus across groups. The model is built using a maximum-entropy principle: among all random networks consistent with basic constraints (such as the overall group proportions and the observed amount of same-group interaction at each scale), we select the least biased distribution. This provides a clean probabilistic baseline, a transparent interpretation of parameters, and a practical route to inference from data.

Empirically, fitting the model to social networks reveals that homophily can be strongly scale-dependent. Networks that appear similar when judged by a single assortativity score can differ markedly once we separate direct contacts from small-group structure. This yields an informative “homophily profile” across scales and helps explain which parts of the network architecture are responsible for perceived segregation or integration.

We also discuss why these distinctions matter for probabilistic questions about connectivity and spreading. In many applications, transmission (of information, behaviors, or infections) occurs through a combination of reinforcement inside groups and occasional bridges between groups. Changing where homophily sits, mostly in direct links versus mostly inside groups, can change the onset and size of large connected components and alter the effectiveness of interventions. Overall, the framework offers a principled way to connect interpretable social mechanisms to the behavior of random graph models at scale.

ABSTRACT. Once upon a time, the problem started with a question: can the addition of (some kind of) noise improve structural properties of infinite-dimensional systems, such as existence, uniqueness, regularity, stability, etc.? First, we will discuss some finite-dimensional examples in which the answer is positive, as well as some infinite dimensional examples, to convince our readers that this question is worth asking. We will then tell about some of the early attempts, made by a few groups, to obtain additional properties of solutions (mainly Markovianity, relaxation to equilibrium) by means of the underlying probabilistic structure.

ABSTRACT. We will describe a model for interacting vertex-reinforced random walks, each taking values on a complete subgraph of a locally finite undirected graph. The transition probability of a walk to a given vertex depends on the cumulative proportion of visits by all walks that have access to that vertex. Proportions are modified by multiplication by a real valued interaction parameter and the addition of a parameter representing the intrinsic preference of the walk for the vertex. This model covers a wide range of interactions, including the cooperation or competition of several walks at single vertices. Under mild regularity conditions, the proportion of visits to each vertex by all walks converges almost surely towards the set of fixed points of the transition probabilities. Convergence to a single fixed point is in fact the generic behaviour as this is shown to hold for almost all parameters. Far beyond convergence, the model allows for a detailed description of the asymptotic behaviour depending on the interaction parameters and subgraph geometries. This will be illustrated by few examples including competing walks on complete graphs and complete subgraphs of stars and cycles.

We will also consider the case where each walk $i$ makes transitions at independent random times $t_1^i, t_2^i, t_3^i, \ldots$ with geometrically distributed inter-transition times with parameter $p^i \in (0,1]$. Independently of the value of $p^i$, we prove that the vertex occupation measure converges almost surely to the same set of fixed points as the synchronous version of the process--remarkably, these accumulation points do not depend on the parameters $p^i$. However, an interesting open question remains: although the accumulation points are invariant to $p^i$, the probability distribution over these limit points may be substantially affected by these parameters. In competitive dynamics, walks with larger $p^i$ transition more frequently and may occupy more attractive vertices with higher probability, potentially excluding slower opponents more effectively.

If time allows, we will mention few open problems and discuss possible directions for future research. The results to be presented are based in part on those described in arxiv.org/abs/2508.15992.

ABSTRACT. We consider the problem of learning ergodic dynamical systems from a finite trajectory. We derive learning guarantees for a basic least squares estimator and contrast them with classical supervised learning results for independent and identically distributed data. We further provide extensions to higher-order systems, systems with finite state spaces, and learning Koopman operators. Our analysis integrates tools from statistical learning theory and Markov processes, together with suitable concentration results for non-i.i.d.\ Hilbert space--valued random variables.

ABSTRACT. We provide an analytic proof for celebrated relative density formulas of the open KPZ equation with respect to white noise. The proof relies on a Girsanov transform, a time reversal and a subtle use of the theory of regularity structures to reconstruct the force of the solution to the KPZ equation at the boundary of the domain. This is joint work with A. Dunlap and Y. Gu.

ABSTRACT. We propose a new algorithm generating discrete stochastic processes with tailored long-range auto-correlation. The algorithm is based on a Markov chain characterized by equispaced real eigenvalues ranging from zero (excluded) to one (included). The stochastic matrix of the Markov chain can be efficiently and accurately obtained by using Soules matrices. Once the N x N Markov chain is produced, by setting a specific projection of the visited states of the chain we generate a stochastic process with a tailored autocorrelation function associated with the selected projection. In this talk, we consider stochastic processes with power-law or logarithmic autocorrelation functions. Among these stochastic processes, we are able to generate an approximate 1/f noise with spectral density power covering in frequency a few orders of magnitude.

ABSTRACT. In this talk, we explore central and non-central limit theorems for functionals of stationary Gaussian random fields, together with their regularity properties, through the lens of their decomposition into Wiener chaoses. We will emphasize three main settings: the case in which the functional is ``concentrated on a single chaos''; the Breuer–Major setting, where all chaotic components contribute equally to the limit; and the challenging scenario in which the dominant contribution to the variance arises from the tail of the chaotic expansion. Throughout the presentation, we will illustrate these phenomena with key examples and discuss some open questions.

This talk is mainly based on joint works with L. Maini, N. Turchi and G. Zheng.

ABSTRACT. After recalling the role played by the tail algebra of a sequence of random variables in Classical Probability, I will discuss the equivalence of two natural definitions of tail algebra in the framework of quantum processes based on twisted tensor products. I will then move on to explain why a canonical conditional expectation onto the tail algebra always exists in this setting as well. This will put me in a position to provide a statement of de Finetti's theorem for spreadable quantum processes on twisted tensor products in terms of (orderly/full) conditional independence with respect to the tail algebra: spreadability is the same as orderly/full independece w.r.t. the tail algebra. Time permitting, I would also like to highlight a striking difference of the quantum case as opposed the classical case: the bilateral tail algebra of a spreadable bilateral sequence may well fail to coincide with the unilateral tail algebra.

ABSTRACT. In this paper we introduce a stochastic volatility model with correlated jumps, incorporating a self-exciting effect in the intensity dynamics. First we derive a pricing formula based on the compound CARMA(p, q)-Hawkes framework, where the stochastic volatility is influenced by the quadratic variation of the counting process in the log-price dynamics. Additionally, we construct a simulation algorithm for the jump term founded on the thinning algorithm. This algorithm is rooted in the existence of a Hawkes intensity with exponential kernel, which serves as an upper bound for the CARMA(p, q)-Hawkes intensity. Finally, we present numerical and empirical analyses.

ABSTRACT. We prove the existence and pathwise uniqueness of a strong solution for a system of $N$ interacting stochastic particles driven by independent Brownian motions. Particles are subject to two types of interactions: a pairwise force generated by a strongly singular drift and a nonlocal interaction with an underlying field $c$. Each particle evolves up to a random reaction time. The coupled process $(X,H,c)$, describing respectively the particles positions, their activity state and the underlying field, satisfies the following system for $t\in(0,T]$ and $ i \in N^*=\{1,...,N\}$: \begin{equation*} \begin{split} dX_t^i &= \left[ - \frac{1}{N}\sum_{j\ne i}^N \nabla V (X_t^i-X_t^j) + F(c(t,\cdot))(X_t^i) \right]\,dt + \sigma\, \,dW_t^i, \hspace{2.2cm} t <\tau_i;\\ H_t^i &= H_0^i+ \int_{(0,t]\times N^*\times \mathbb R_+} \mathbbm 1_{\{i\}}(j)\,\mathbbm 1_{\{0\}}(H_{s^-}^i)\,\ \mathbbm 1_{\left\{z\le {\lambda}\,c(s,X_s^i)\right\}} \,M(ds,dj,dz), \hspace{0.8cm} t>0;\\ \partial_t c(t,x) &= -\lambda\,c(t,x)\,\frac{1}{ N}\sum_{j=1}^N K(x-X_t^j), \hspace{4.5cm} (t,x) \in (0,T]\times \mathbb R^d. \end{split} \end{equation*} The pairwise interaction is governed by means of the strongly singular Lennard-Jones potential $$ V(x) := \frac{A}{|x|^\alpha} - \frac{B}{|x|^\beta}, \qquad A,B>0,\quad \alpha>\beta>0, $$ which induces a repulsive-attractive force with singular behavior at the origin. The underlying field $c$ influences the particle dynamics from two different perspectives: on the one hand, it biases the motion of active particles through the non-local drift $F(c(t,\cdot))$; on the other hand, it affects the switching rate of particles from active ($ H_i(t)=0$) to inactive ($H_i(t)=1$) through reaction random times $\tau_i$ driven by a Poisson random measure $M$ on $(0,T] \times N^* \times \mathbb R_+$ with intensity modulated by $c$. In turn, the field $c$ is random itself: it evolves according to the instantaneous particle configuration, yielding a fully coupled stochastic system.

\smallskip

The proof of well-posedness proceeds in two steps. We first establish existence and pathwise uniqueness for the system without reactions. We then treat the full coupled dynamics using an interlacing argument to incorporate the jump mechanism.

\smallskip The model is motivated by microscopic stochastic descriptions of sulphation processes in cultural-heritage materials.

ABSTRACT. The Boltzmann equation describes the dynamics of a density in position and velocity of a rarified gas expanding in vacuum. Ludwig Eduard Boltzmann (1844 -1906) derived the Boltzmann equation, by assuming any gas molecule of a rarified gas to travel straight in vacuum until an elastic collision occurs with another molecule of the same gas. In the Boltzmann equation, only binary centered collisions are considered. In this talk we present the “Boltzmann process” [1], i.e the process whose density evolves according to the Boltzmann equation. Using the Ito formula, we prove that this is a solution of a stochastic differential equation of McKean Vlasov type, for which we prove the existence [2].

[1] Albeverio, S., Rüdiger B., Sundar P.: On the construction and identifcation of Boltzmann processes. In: Brasesco S., Buttà P., Cassandro M., Picco P., Vares M. E. (eds.) , ENSAIOS MATHEMATICOS, vol.38, pp. 1-22, .Papers in honor of Errico Presutti, Sociedade Brasileira de Mathematica (2023). \doi{10.21711/217504322023/em381}

[2] Rüdiger B., Sundar P.: .: Identification and existence of Boltzmann processes, arxiv 2301.08662v2 (2025). \doi{10.48550/arXiv.2301.08662}

ABSTRACT. We study a broad class of inhomogeneous spatial random graphs, including long-range and scale-free percolation and preferential attachment-like models. Vertices are placed on the discrete d-dimensional torus and are equipped with heavy tailed random weights. The probability of linking any pair of vertices decays in their distance but increases as a function of the weights. We focus on the adjacency matrix of such graphs in the dense regime and prove that, as the size of the torus goes to infinity, the empirical spectral distribution converges. The corresponding limiting measure is given by an operator-valued semicircle law that we show to be absolutely continuous and to have finite second moment, even when the weights have infinite variance. We also characterize its Stieltjes transform by a fixed point equation in an appropriate Banach space.

ABSTRACT. The transition function of the Wright--Fisher diffusion with selection is central to understanding non-neutral evolution but, unlike in the neutral case, is not available in a form that is straightforward to evaluate: duality-based approaches typically lead to dual processes with intractable rates, while spectral methods rely on truncation whose computational burden grows rapidly with the strength of selection and model complexity.

We develop a \emph{gamma duality} framework for a multi-allelic Wright--Fisher diffusion with parent-independent mutation and genic selection, based on an exponential augmentation of polynomial duality. We show that the resulting birth-and-death dual process has tractable infinitesimal rates, identify its stationary distribution, and describe its small-time behavior. This dual yields an explicit representation of the transition function as a mixture of standard Dirichlet components, with mixing weights characterized by the dual started from an entrance boundary.

The representation supports computation for arbitrary numbers of alleles and selection coefficients, including regimes where existing approaches are unavailable or impractical. We illustrate that our algorithms deliver substantially improved runtimes over specialized methods, when these do apply.

ABSTRACT. The increasing complexity of low-voltage distribution assets calls for scalable, non-intrusive monitoring solutions that are sustainable in terms of energy consumption and maintenance. This contribution presents Sentinel-GRID, a distributed, beacon-based approach for multi-parameter monitoring of secondary substations and distribution transformers, built around ultra-low-power IoT sensor nodes operating in an event-driven "fit-and-forget" paradigm. The nodes integrate heterogeneous sensing (electromagnetic signatures, temperature, vibration and acoustic indicators) and remain mostly in deep sleep, waking only when relevant conditions occur, thus drastically reducing energy draw and enabling multi-year autonomy, potentially supported by micro energy-harvesting. Data collection relies on a hybrid communication architecture: BLE beacons [2] for opportunistic acquisition through mobile gateways (e.g., smartphones carried by maintenance crews interacting in harsh environments via a PWA and Data over Audio links [1]) and longrange connectivity (e.g., LoRaWAN) for periodic reporting and resilient backhaul. In critical situations, the system enables priority alerting over high-availability channels. Alongside physical sensors, the solution also includes low-power computer-vision nodes for visual inspection: by monitoring retroreflective strips applied near tightening points, it is possible to estimate micro-displacements and movements of transformer bolts [3], providing a direct indicator of mechanical loosening and supporting multimodal diagnostics. On the backend, a cloud platform integrates a Digital Twin and ML/AI analytics tailored to sparse and asynchronous data streams, offering anomaly detection, explainable diagnostics, and remaining useful life (RUL) estimation. A representative use case is the early identification of loosened bolted connections, a typical precursor to overheating and reliability degradation. The proposed approach aims to reduce OPEX, increase safety, and accelerate predictive maintenance adoption in distribution networks.

ABSTRACT. We address the nonparametric empirical Bayes problem of denoising observations of latent variables distributed according to an unknown distribution on a compact Riemannian manifold. We demonstrate that the gradient of the log-marginal density defines a score-based field that captures both geometric and probabilistic structures. This field induces a denoiser that achieves near-Bayes risk while bypassing the computationally intensive posterior Fréchet mean; instead, it utilizes intrinsic, locally defined updates driven by the score field. We develop a fully data-driven approximation of this oracle denoiser via a novel approximate Tweedie–Eddington formula for Riemannian Gaussian mixture models, and establish a near-parametric rate of convergence.

ABSTRACT. Measure-valued Pólya sequences (MVPS) are stochastic processes whose dynamics are governed by generalized Pólya urn schemes with infinitely many colors. Assuming a general reinforcement rule, MVPSs can be viewed as extensions of Blackwell and MacQueen's Pólya sequence, which characterizes an exchangeable sequence with a Dirichlet process (DP) prior distribution. In this talk, we give a complete account of the class of exchangeable MVPSs. We show that under exchangeability, an MVPS is necessarily balanced and its reinforcement kernel is, after normalization, a proper regular conditional distribution. As a result, its prior distribution is that of a DP mixture with respect to a latent parameter, which is associated with the conditioning $\sigma$-algebra. Furthermore, we examine the effects of relaxing exchangeability to conditional identity in distribution (c.i.d.) and find that the two are equivalent for balanced MVPSs. In the unbalanced case, it is still possible to have c.i.d. MVPSs that are not exchangeable, but this necessitates a particular form of the reinforcement kernel.

This is joint work with Yoana R. Chorbadzhiyska and Mladen Savov.

ABSTRACT. We establish a solution theory (global weak existence, local strong existence and weak-strong uniqueness) for the incompressible Navier-Stokes-Fourier system with thermal noise, posed on the three-dimensional torus. While in the incompressible deterministic setting the equation for the velocity can be solved independently of the temperature, the inclusion of the effects of thermal fluctuations by means of the GENERIC framework leads to a nonlinear gradient noise term, which couples the dynamics of both variables. Therefore, the analysis of the stochastic system poses new challenges, which are absent in deterministic Navier-Sokes-Fourier equations. This talk is based on joint work with Benjamin Gess and Zhengyan Wu.

ABSTRACT. We introduce a gradient-based swarm optimization method built on a Softmin Energy interaction function $J_\beta(\mathbf{x})$, which provides a smooth approximation of the minimum value among particles. By defining a stochastic gradient flow with Brownian exploration and an annealing-like control parameter $\beta$, our approach retains gradient efficiency while promoting global exploration. The main theoretical result shows that our dynamics reduce effective potential barriers compared to Simulated Annealing, leading to faster transitions between local minima and improved exploration of the energy landscape. Analytical estimates of hitting times and experiments on benchmark functions, such as double-well and Ackley landscapes, confirm accelerated convergence and better global search performance.

ABSTRACT. Normalizing flows provide a flexible class of invertible transformations for learning probability distributions and can be interpreted as flows on spaces of measures. In this talk, we present a theoretical framework in which normalizing flows are viewed as approximations of optimal transport maps, constructed via neural ordinary differential equations with linear control structure. Within this setting, we establish approximation results showing that suitably constrained neural ODEs can approximate optimal transport maps between absolutely continuous measures. In order to formulate a tractable finite-dimensional optimization problem, the transport is approximated using discrete empirical measures; consistency as the number of atoms tends to infinity is guaranteed by a $\Gamma$-convergence result \cite{article1}. The optimal transport plans associated to the discrete approximating measures naturally encode information only in an $L^2$-type topology. This creates a mismatch with the underlying approximation results for diffeomorphisms, which are stated in a stronger topology, namely uniform convergence on compact sets, and prevents a direct exploitation of these results. We discuss ongoing work aimed at bridging this gap by incorporating risk measures into the optimization problem, thereby providing a principled way to interpolate between $L^2$ and $C^0$ topologies. Finally, we outline future directions toward quantitative estimates, with the goal of expressing the approximation error of the optimal transport map in terms of the Wasserstein distance between discrete empirical measures and their continuous counterparts.

ABSTRACT. I summarise results of three papers [1,2,3] on a fractional Hawkes process with kernel proportional to the probability density function of Mittag-Leffler random variables. This is joint work with Jane A. Aduda, Maggie Chen, the late Alan G. Hawkes, Cassien Habyarimana, and Federico Polito. The code used to generate simulations and figures is freely available from https://github.com/habyarimanacassien/Fractional-Hawkes.

1. Chen, J., Hawkes, A.G., Scalas, E.: A Fractional Hawkes Process, In: Beghin, L., Mainardi, F., Garrappa, R. (eds.) Nonlocal and Fractional Operators. SEMA SIMAI Springer Series, vol 26. Springer, (2021). 2. Habyarimana C., Aduda J.A., Scalas, E., Chen, J., Hawkes A.G., Polito F.: A fractional Hawkes process II: Further characterization of the process. Physica A 615, 128596, (2023). 3. Habyarimana C., Aduda J.A., Scalas E.: Parameter estimation for the fractional Hawkes process. Journal of Agricultural, Biological and Environmental Statistics, (2024). https://doi.org/10.1007/s13253-024-00663-5.

ABSTRACT. We establish strong Feller property and irreducibility for the transition semigroup associated to a class of nonlinear stochastic partial differential equations with multiplicative degenerate noise. As a by-product, we prove uniqueness of the invariant measure under very mild assumptions. The drift of the equation diverges exactly where the noise coefficient vanishes, resulting in a competition between the dissipative effects and the degeneracy of the noise. The main idea is to introduce a mathematical method to measure the accumulation of the solution towards the potential barriers, allowing to give rigorous meaning to the inverse of the noise operator even in the degenerate case. If the singularity of the drift and the degeneracy of the noise are suitably balanced, the dynamics are shown to stabilise for large times. From the mathematical point of view, the results provide a first generalisation of the classical work by Peszat & Zabczyk [1] to the case of degenerate multiplicative diffusions. From the application perspective, the models cover interesting scenarios in physics, in the context of evolution of relative concentrations of mixtures, under the influence of thermodynamically-relevant potentials of Flory-Huggins type.

[1] Peszat, S., Zabczyk, J.: Strong Feller property and irreducibility for diffusions on Hilbert spaces. The Annals of Probability, 157–172, (1995).

ABSTRACT. We build a class of additive inhomogeneous processes by subordination of a multiparameter Brownian motion. The subordinator is chosen to be a Sato process (see e.g. \cite{eberlein2009sato}) and it is constructed to incorporate both a time transform common to all assets and an idiosyncratic one. The resulting process is a generalization of multivariate L\'evy processes with good fit properties on financial data, see \cite{LuciSem1}. We specify the model to have unit time normal inverse Gaussian distribution, introduced in \cite{barndorff1995normal} to model asset returns, and we discuss the ability of the model to fit time inhomogeneous correlations on real data.

ABSTRACT. We study two-type competing first-passage percolation on random graphs generated by the configuration model with a power-law degree distribution with exponent \tau in (1,2), corresponding to the infinite-mean regime. In the classical nearest-neighbor setting, the competition is dominated by giant-degree hubs: the type that first reaches a hub rapidly infects the entire network, leading to a "winner takes it all but one" phenomenon. We extend this model by introducing long-range infections: each infected vertex infects a uniformly chosen vertex at rate \gamma>0, independently of the edge-based dynamics. This global transmission mechanism competes with the local spread and fundamentally changes the phase diagram.

In cybersecurity terms, this models malware or information campaigns that spread both through local network connections and via global mechanisms, such as phishing, mass email, or broadcast exploits, which can reach arbitrary devices. This provides a natural framework for studying attacks on heterogeneous networks, many of which have heavy-tailed degree distributions with \tau in (1,2) or \tau in (2,3).

We identify a sharp threshold for coexistence as a function of $\gamma$. In the subcritical regime, the "winner takes it all" phenomenon arises, with the losing type infecting either finitely many vertices or even infinitely many but a vanishing proportion of the graph. In the supercritical regime, long-range transmission enables macroscopic coexistence, including an extreme case in which the final proportions of the two types converge to a random limit characterized by a Pólya urn.

ABSTRACT. We introduce the Random Quadratic Form (RQF): a stochastic differential equation which formally corresponds to the gradient flow of a random quadratic functional on a sphere. While the one-point motion of the system is a Wiener process and thus has no preferred direction, the two-point motion exhibits nontrivial synchronizing behaviour. In this work we study synchronization of the RQF, namely we give both distributional and path-wise characterizations of the solutions by studying invariant measures and random attractors of the system.

The RQF model is motivated by the study of the role of linear layers in transformers and illustrates the synchronization by common noise phenomena arising in the simplified models of transformers. In particular, we provide an alternative (independent of self-attention) explanation of the clustering behaviour in deep transformers and show that tokens cluster even in the absence of the self-attention mechanism.

ABSTRACT. I will present some recent results obtained for the facilitated exclusion process in one dimension. This stochastic lattice gas model is subject to strong kinetic constraints that create a continuous phase transition to an absorbing state at a critical particle density value. If the microscopic dynamics is symmetric, its macroscopic behavior (with periodic boundary conditions and in the diffusive time scale) is governed by a nonlinear PDE belonging to free boundary problems (or Stefan problems). One of the major ingredients is to show that the system reaches the “ergodic” component in a subdiffusive time. When the particle system is put in contact with reservoirs (which can either destroy or inject particles at both boundaries), it leads to a Dirichlet boundary-value problem. Starting from a suitable initial condition, the weakly asymmetric case gives rise to a new KPZ-type equation on the half line. All these results rely, to various extent, on mapping arguments (towards auxiliary processes), which completely fail in dimension higher than 1. I will finally discuss some open problems and questions, especially in dimension 2. Based on several joint works with G. Barraquand, O. Blondel, H. Da Cunha, C. Erignoux, M. Sasada and L. Zhao.

ABSTRACT. The theory of stochastic partial differential equations has seen rapid progress over the past decade, spurred by the introduction of the theories of regularity structures [7] and paracon- trolled calculus [6]. Despite the close connections of singular SPDEs to physical phenomena— for instance through statistical mechanics where heterogeneous environments arise naturally (for instance defects in Φ^4, a toy model for ferromagnets) or through quantum field theory, where models are naturally geometric (for example, Yang–Mills theory, canonically formu- lated on principal bundles)—the theory of singular SPDEs was until recently focused mainly on homogeneous settings. In this overview talk, I will motivate and present recent developments that make it pos- sible to treat singular SPDEs in genuinely inhomogeneous environments, including parabolic equations with heterogeneous operators [2, 9, 4] and bundle-valued equations on Riemannian manifolds [1, 5, 3, 8].

ABSTRACT. We investigate a fundamental object in operations research: the stability region of a randomly modulated scheduling problem. Specifically, we consider a queueing system comprising multiple queues and a single server, where the scheduling decisions are influenced by a dynamic, autonomous, random, and stationary environment that modulates the queue capacities. In the setting where the modulation space is finite, we identify this problem with two structures arising, respectively, in combinatorial optimisation and convex geometry: a generalised network flow -- or network flow with gains -- and a cephoid -- the Minkowski sum of deGua simplices. These novel identifications yield strongly polynomial algorithms for feasibility, new characterisations of the stability region -- explicit in some cases -- and algorithms for computing its minimal descriptions. In the ON/OFF setting, where the first identification reduces to a classical network flow, we present a unified framework tied together by the max-flow/min-cut theorem.

ABSTRACT. We consider the control of a McKean-Vlasov stochastic differential equation (SDE) and present a novel approach to the proof of Peng's maximum principle. The main step is the introduction of a third adjoint equation, a conditional McKean-Vlasov backward SDE: Peng's maximum principle is derived from a second-order Taylor expansion of the cost functional, which in the McKean-Vlasov case, due to the structure of the Lions derivative, introduces quadratic terms that contain independent copies of the variational processes. To accommodate the dualization of these terms, we introduce this third adjoint equation. We only treat SDEs in $\mathbb{R}^d$ but the dependence on the distribution already makes these equations inherently infinite dimensional. Our approach will also be useful in further extensions to the common noise setting in mean-field control and the control of Hilbert space valued McKean-Vlasov SDEs.

ABSTRACT. Stochastic processes provide a fundamental framework for describing systems that evolve randomly over time. A key aspect in the analysis of stochastic dynamics is the study of the first-passage time (FPT), defined as the time required for a trajectory to reach a prescribed region for the first time. FPT problems are essential for characterizing threshold phenomena, rare events, and barrier-crossing mechanisms, and they offer deep insights into the statistical structure of the underlying processes.

FPT problems for diffusion processes in discs, spheres, and general closed domains are often investigated with the aim of deriving closed-form solutions. However, the inherent mathematical complexity of these problems typically limits such analysis, leading researchers to focus on approximate results or on low-order moments, such as the mean or variance, rather than on the distribution of the FPT. For regular elliptic domains, the problem has been addressed in a recent paper for the Wiener and Ornstein–Uhlenbeck two-dimensional processes, considering both interior and exterior initial states. In this setting, the Laplace transform of the FPT density and the corresponding moments are derived and analyzed, and numerical inversion techniques are employed to obtain approximate probability density functions. Particular attention is devoted to the asymptotic behavior of FET moments, highlighting the differences between the dynamics of the two processes.

The FPT problem through closed boundaries has been considered also for perturbated closed curves. For example, some authors estimate the mean FPT for irregular domains obtained by perturbing the boundary of a disk or an ellipse, with applications to geographical settings where islands are modeled as perturbed elliptic shapes. Further investigations on mean FPT in elongated planar domains, including elliptic geometries, are reported in other papers.

The present contribution aims to investigate FPT problems for two-dimensional diffusion processes through perturbed elliptic boundaries. In particular, we focus on two-dimensional Wiener and Ornstein–Uhlenbeck processes, which play a central role in probability theory and have applications ranging from physics and biology to economics and information theory. The analysis combines analytical, probabilistic, and computational approaches, with the goal of providing a comprehensive characterization of FPT distributions in geometrically nontrivial settings.

Since explicit analytical solutions are not available for perturbed elliptic domains, in addition to a simulation analysis, a comparison-based approach is adopted. In particular, we focus on the perturbed region enclosed between two regular ellipses, allowing the corresponding FPT distributions to be suitably bounded. We resort to stochastic ordering theory, with special reference to the usual stochastic order and the Laplace transform ratio order. Both such orders lead to robust inequalities for the FPT distribution and rigorous estimates of first-passage statistics for complex geometries.

ABSTRACT. This paper introduces the signed subjective expected utility (SSEU) model in which an individual’s willingness-to-bet (WTB) on an event reflects not only the event's subjective likelihood but also its ``valence''---a measure of intrinsic attractiveness or aversiveness of the event. As a result, an event's WTB may be greater than $1$ or less than $0$. Our model directly extends the subjective expected utility (SEU) model by weakening the Monotonicity axiom. We show that SSEU accounts for behavioral phenomena such as hedging aversion, the conjunction fallacy, coexistence of insurance and gambling, the choice of dominated actions in strategy-proof mechanisms, and the home equity bias puzzle. Finally, we show how to extend SSEU to allow for a stake-dependent (and non-additive) WTB. This extension accommodates recent experimental evidence showing that subjects jointly violate monotonicity and independence.

ABSTRACT. We investigate the random variable defined by the volume of the zero set of a smooth Gaussian field, on a general Riemannian manifold possibly with boundary. We prove a new explicit formula for its Wiener-Itô chaos decomposition that is notably simpler than existing alternatives and which holds in greater generality, without requiring the field to be compatible with the geometry of the manifold. A key advantage of our formulation is a significant reduction of the complexity of the computations of the variance of the nodal volume. Unlike the standard Hermite expansion, which requires evaluating the expectation of products of 2+2n Hermite polynomials, our approach reduces this task - in any dimension n - to computing the expectation of a product of just four Hermite polynomials. As a consequence, we establish a new exact formula for the variance.

ABSTRACT. Berry’s random wave is obtained by superimposing plane waves whose directions are chosen uniformly at random on the unit sphere. It serves as a canonical uninversal local model for high-energy Laplace eigenfunctions on chaotic manifolds. Simulations on large domains reveal persistent filamentary patterns, often called scars, that are absent from other natural Gaussian fields. We provide the first rigorous description of this phenomenon, showing that the fluctuations of Berry’s field become asymptotically indistinguishable from those of a Poisson line process, and converge to a fractional Gaussian field of index 1/2. Complementary to such concentration along lines is the question of oscillation along them. We prove that, along each scar, the field oscillates according to a universal frequency profile that depends on the dimension. In dimension two only, it reduces to a single frequency: the field oscillates like a pure sinusoid along every scar line.

ABSTRACT. We develop a finite-network growth model in which capital accumulates at each loca- tion and productivity follows a continuous-time regime-switching process. A social planner chooses location-specific consumption to maximize discounted utility, under linear AK dy- namics with mobility across nodes and state constraints. After introducing the model and the associated control problem, we prove existence and uniqueness of an optimal control and establish regularity properties of the value function that support a feedback charac- terization of the optimal policy. The resulting Hamilton–Jacobi–Bellman system is solved numerically delivering computed optimal paths for capital and consumption. We illustrate the framework with a numerical application for a two-location economy with symmet- ric links, specialized to a three-regime specification, and show how risk aversion and the intensity of regime switches shape the value function and the resulting trajectories.

ABSTRACT. In this work, we propose, analyze and empirically validate a lazy-update approach to maintain accurate approximations of the $2$-hop neighborhoods of dynamic graphs resulting from sequences of edge insertions.

We first show that under random input sequences, our algorithm exhibits an optimal trade-off between accuracy and insertion cost: it only performs $O(\frac{1}{\varepsilon})$ (amortized) updates per edge insertion, while the estimated size of any vertex's $2$-hop neighborhood is at most a factor $\varepsilon$ away from its true value in most cases, \emph{regardless} of the underlying graph topology and for any $\varepsilon > 0$.

As a further theoretical contribution, we explore adversarial scenarios that can force our approach into a worst-case behavior at any given time $t$ of interest. We show that while worst-case input sequences do exist, a necessary condition for them to occur is that the \textit{girth} of the graph released up to time $t$ be at most $4$.

Finally, we conduct extensive experiments on a collection of real, incremental social networks of different sizes, which typically have low girth. Empirical results are consistent with and typically better than our theoretical analysis anticipates. This further supports the robustness of our theoretical findings: forcing our algorithm into a worst-case behavior not only requires topologies characterized by a low girth, but also carefully crafted input sequences that are unlikely to occur in practice.

Combined with standard sketching techniques, our lazy approach proves an effective and efficient tool to support key neighborhood queries on large, incremental graphs, including neighborhood size, Jaccard similarity between neighborhoods and, in general, functions of the union and/or intersection of $2$-hop neighborhoods.

ABSTRACT. We develop a Bayesian nonparametric framework for inference in multivariate spatio-temporal Hawkes processes, extending existing theoretical results beyond the purely temporal setting. Our framework encompasses modelling both the background and triggering components of the Hawkes process through Gaussian process priors. Under appropriate smoothness and regularity assumptions on the true parameter and the nonparametric prior family, we derive posterior contraction rates for the intensity function and the parameter, in the asymptotic regime of repeatedly observed sequences. These results provide, to our knowledge, the first theoretical guarantees for Bayesian nonparametric methods in spatio-temporal point data. We also show that we can numerically approximate the posterior via variational inference and demonstrate the benefit of nonparametric modelling in the context of spatio-temporal events.

ABSTRACT. We investigate a Mean-Field Game (MFG) posed in an infinite-dimensional Hilbert space and driven by degenerate noise. The associated MFG system consists of a Hamilton–Jacobi–Bellman (HJB) equation coupled with a nonlinear Fokker–Planck (FP) equation, both governed by a degenerate Kolmogorov operator.

The degeneracy of the noise introduces significant analytical challenges. In particular, the HJB equation is treated in the viscosity sense, while the FP equation is interpreted in a suitable weak formulation. A major difficulty stems from the degeneracy of the Kolmogorov operator, which makes the uniqueness of solutions to the FP equation particularly delicate.

Under appropriate structural assumptions, we establish well-posedness of the MFG system. As an application, we consider Mean-Field Games arising from stochastic delay differential equations, highlighting how delay effects naturally lead to infinite-dimensional and degenerate dynamics.

This talk is based on joint work with Andrzej \'{S}wi\k{e}ch.

ABSTRACT. It is recognized that the addition of polymers is very efficient in reducing the friction drag in turbulent regimes. My talk is about the effects of small-scale turbulence on polymers distribution by using a stochastic scaling and singular limits. Many works have been done in recent years using the scaling limit in both scalar and vector cases. The second one is characterized by the presence of stretching, which adds complications over the scalar case.

In \cite{Art1}, we investigate the stretching mechanism of stochastic models of turbulence acting on a simple model of polymer. Namely, we investigate a scaling limit problem, under suitable intensity assumption. The polymer density equation, initially an SPDE converges (in the first step) weakly to a limit deterministic equation with a new degenerate term with some singular parameter. Recently, in \cite{Art2} we investigate the singular limit in the spirit of the hydrodynamic limit techniques. One consequence is that the limiting density shows a power-law decay in the polymer length, which is consistent with physical predictions.

The activities mentioned herein were performed in the framework of the project: EU-HORIZON EUROPE ERC-2021-ADG “Noise in Fluids” (NoisyFluid), no. 101053472.

\bibitem{Art1}Flandoli, F. and Tahraoui Y.: Stretching of polymers and turbulence: Fokker Planck equation, special stochastic scaling limit and stationary law. Journal of Differential Equations 452 : 113789 (2026)

\bibitem{Art2}Tahraoui, Y.: Small-scale turbulence limit of Fokker-Planck equation for polymers in turbulent flow. arXiv preprint arXiv:2503.18143 (2025)

ABSTRACT. We introduce a framework for constructing orthogonal polynomials on path space. Beginning with an introduction to signatures which play the role of polynomials, and we orthogonalise these features to obtain \(L^2\)-convergent series for square-integrable path functionals. Under an infinite radius of convergence assumption, we prove linear functionals on the signature are dense in \(L^p\).

Identifying the shuffle algebra with polynomials over the free Lie algebra, we generalise orthogonal polynomial theory: establishing recurrence relations, a Favard-type theorem, and connections to spectral measures. For Brownian motion, a natural (dimension-independent) orthogonal basis exists only with time-augmentation, yielding explicit Itô-orthogonal polynomials.

In ongoing work with Markus Reiß and Christian Bayer, we apply these methods to classify Ornstein-Uhlenbeck processes, obtaining closed-form expected signatures and optimal discriminative features for hypothesis testing

ABSTRACT. In this presentation we deal with mild solutions to semilinear stochastic partial differential equations (SPDEs) of jump-diffusion type, driven by a trace class Wiener process and a Poisson random measure. The state space of the SPDE is a separable Hilbert space, and the linearity is the generator of a strongly continuous semigroup on the Hilbert space. Consider a closed convex cone in the state space. We say that the cone is invariant for the SPDE if for each starting point the corresponding solution process stays in the cone.

The goal of this talk is to characterize stochastic invariance of the closed convex cone by means of the coefficients of the SPDE. Moreover, we will present applications of our findings to SPDEs arising in mathematical finance.

ABSTRACT. Poisson point processes are the simplest, yet fundamental, models for the analysis of spatial and spatio–temporal point patterns. They can be used to describe the locations of events or objects of interest and to estimate the intensity of these point patterns within a de- fined region. Beyond Poisson models, many widely used spatial and spatio–temporal point process models are built on a Poisson–type intensity. For instance, log–Gaussian Cox pro- cesses [1] and self–exciting or Hawkes–type processes [2] typically relies on a well–specified first–order intensity, and on a Poisson (or Poisson–like) likelihood to estimate it. Conse- quently, obtaining consistent, low–bias first–order intensity estimates is critical not only for Poisson models but also for the robustness of more complex models built upon them. Likelihood-based inference for three-dimensional Poisson point process models requires approximating the integral term in the log-likelihood over a 3D observation window. In practice this is done via a quadrature scheme [3], which replaces the integral with a weighted sum over observed points and a set of dummy points. Despite its widespread use, the accu- racy of the approximation strongly depends on a few tuning parameters: whether dummy points are placed on a regular grid or randomly, the dummy-to-data multiplier q, so that nd = q n dummy points are generated (where n represents the number of observed points in the pattern), and the window partition resolution nc, providing n3 c voxels and cubature weights. Clear guidance on how to select these parameters to obtain reliable inference is missing, especially in three dimensions, while concerns about quadrature accuracy have long been noted [4]. We formalise the cubature scheme and its replicated version for multi-type/categorical marks, and develop data-driven guidelines through an extensive simulation study. We simu- late three-dimensional inhomogeneous Poisson processes across different scenarios in which the intensity depends on coordinates, external covariates and categorical marks, with mul- tiple sample sizes, fitting models over a wide grid of (q, nc) values under both dummy-point layouts. Performance is assessed via the mean squared error of parameter estimates and by second-order diagnostics such inhomogeneous K-function weighted by the fitted intensity. Across scenarios, we identify well-delimited regions of (q, nc) that provides stable likelihood approximations and accurate intensity recovery. A validation study confirms the robustness of these recommendations. Finally, an application to 2008 Greece background seismicity shows that baseline cubature choices can fail diagnostics, whereas guideline-based settings provide coherent parameter estimates and pass global envelope tests.

ABSTRACT. We address two probabilistic approaches for associating a specific stochastic dynamics with a McKean-type partial differential equation featuring a reaction term and path-dependent coefficients.

The non-conservative nature of the macroscopic dynamics leads to two possible interpretations of both the sub-probability measure and the associated SDE equation at the microscale: on the one hand, as a measure-valued solution of a Feynman–Kac-type equation; on the other hand, as the sub-probability associated with an SDE with memory defined up to a survival time with a reaction-dependent rate. These different interpretations give rise to two different microscopic stochastic models and therefore to two different techniques of probabilistic analysis. Then, by considering the interacting particle systems associated with each of the microscopic models, we discuss how their empirical densities provide two different kernel estimators for the PDE solution.

Finally, we discuss how the convergence to a standard advection-diffusion-reaction McKean-type PDE is achieved by rescaling the interaction kernel at an intermediate scale and using a semigroup approach.

The PDE model under consideration arises in applications in materials science: it describes the sulphation phenomenon, a degradation process affecting marble surfaces exposed to atmospheric pollutants.

ABSTRACT. Expectiles provide a smooth and naturally tail-sensitive alternative to quantiles, and have recently emerged as powerful tools for describing dispersion and asymmetry. In this talk, we develop a framework in which expectiles serve as the basis for measuring inequality, leading to a new class of expectile-based inequality indices that offer a natural geometric counterpart to classical Lorenz-Gini methodology.

The key observation is that comparisons in convex stochastic order can be expressed in terms of inclusions between suitably defined expectile regions. This allows distributional spread to be described geometrically through a nested family of regions capturing tail behavior. Building on this idea, we introduce law-invariant functionals obtained by integrating expectiles or inter-expectile ranges across asymmetry levels. These constructions give rise not only to generalized deviation and inequality measures, but also to expectile-based risk measures, while remaining fully consistent with convex-order comparisons and preserving a clear probabilistic interpretation.

A central aspect of the approach is a geometric representation of expectiles via a star-shaped set in the plane, whose boundary is traced by scaled expectiles. The area of this set naturally defines an inequality index, playing a role analogous to that of the Gini index, but arising from expectile geometry rather than from quantile-based Lorenz curves.

We also extend the construction to multivariate settings by defining expectile regions through directional projections, thereby obtaining inequality measures capable of capturing genuinely multidimensional heterogeneity. Finally, we discuss empirical implementation and computational aspects, showing that the proposed functionals can be evaluated efficiently in practice.

ABSTRACT. In this talk we will briefly introduce several stochastic models of anomalous diffusion. In particular, we will focus on random processes subject to trapping effects, diffusion processes subject to reflecting barriers and kinetic processes subject to random obstacles. For these different models, we will discuss the anomalous diffusive behavior, the (non-) Markov property, the (non-local) PDE connections and, in particular and simulation methods. Particular attention will be dedicated to their connection with scaling limits of Continuous Time Random Walks / L\'evy walks.

ABSTRACT. A Thorin process is a stochastic process with independent and stationary increments whose laws are weak limits of finite convolutions of gamma distributions. Many popular L\'evy processes fall under this class. The Thorin class can be characterized by a representing triplet that conveys more information on the process compared to the L\'evy triplet. We provide a full account of the theory of multivariate Thorin processes, starting from the Thorin--Bondesson representation for the characteristic exponent, and highlight the roles of the Thorin measure in the analysis of density functions, moments, path variation and subordination. Various old and new examples are discussed. It is illustrated how univariate Brownian subordination with respect to Thorin subordinators produces Thorin processes whose representing measure is given by a pushforward with respect to a hyperbolic function, leading to easier formulae compared to the Bochner integral for the L\'evy measure. We further detail a treatment of subordination of gamma processes with respect to negative binomial subordinators which is made possible by the Thorin--Bondesson representation, and show some examples of applications in finance (from a joint work with D. Madan).

ABSTRACT. We study a slow-fast system of coupled two- and three-dimensional Navier-Stokes equations in which the fast component is perturbed by an additive fractional Brownian noise with Hurst parameter $\mathcal{H}>\frac{1}{3}$. The system is analyzed using rough path theory, and the limiting behaviour strongly depends on the value of $\mathcal{H}$. We prove convergence in law of the slow component to a Navier–Stokes system with an additional It\^o-Stokes drift when $\mathcal{H}<\frac{1}{2}$. In contrast, for $\mathcal{H}\in (\frac{1}{2},1)$, the limit equation features only a transport noise driven by a rough path.

ABSTRACT. Consider a viral agent spreading through a network with prescribed infection and healing rates. The multiple-node optimization problem requires identifying a set of $k$ nodes to immunize to minimize the spread of the infection. This problem is computationally hard, and various heuristic methods have been considered to address it. We propose an algorithm that chooses the nodes to immunize as the complement of the set of roots of a suitably sampled random forest. We provide a theoretical description of the algorithm's features and offer numerical evidence that it improves the results of the reference deterministic heuristic while maintaining the same asymptotic computational cost.

ABSTRACT. In the framework of urn models, we introduce a probability distribution designed to quantify the concentration of attributes among members of small groups. This new distribution addresses a specific occupancy problem, focusing on how particular marbles are allocated to urns. We fully characterize this distribution, referred to as the Reverse Hypergeometric distribution, and propose a statistical test based on it. The model enables testing for excess intra-group similarity against a null hypothesis of random co-occurrence of marbles with the same attribute in the urns. We compare it with established models, including the Multinomial and the Multivariate Hypergeometric distributions. We also provide an asymptotic approximation of the Reverse Hypergeometric distribution by gauging a Multinomial distribution and demonstrate how the model results from urn exchangeability. We illustrate its use through three real-world applications in the domains of network science, social science, and text analysis: investigating the presence of homophily in relationship networks, assessing the excess of same-sex children within households, and analyzing the concentration of sentiment-polarized sentences in the abstracts of scientific papers. Finally, we present a generalization of the model that accommodates groups of varying sizes, enhancing its versatility for different domains and data structures. Session number: CS191 Session name: Selected Topics in Probability and Statistics First organizer: Andrea Simonetti

ABSTRACT. We introduce a model of epidemics among moving particles on any locally finite graph. At any time, each vertex either is empty, occupied by a healthy particle, or occupied by an infected particle. Infected particles recover at rate 1 and transmit the infection to healthy particles at neighboring vertices at rate $\lambda$. In addition, particles perform an interchange process with rate $\mathsf v$, that is, the states of adjacent vertices are swapped independently at rate $\mathsf v$, allowing the infection to spread also through the movement of infected particles. On the $d$-dimensional Euclidean lattice, we start with a single infected particle at the origin and with all the other vertices independently occupied by a healthy particle with probability $p$ or empty with probability $1-p$. We define $\lambda_c(\mathsf v,p)$ as the threshold value for $\lambda$ above which the infection persists with positive probability and analyze its asymptotic behavior as $\mathsf v \to \infty$ for fixed $p$.

ABSTRACT. The analogue of de Finetti’s theorem for random graphs is the Aldous–Hoover theorem, which provides a representation of exchangeable random graphs as a mixture of graphon models.

We study exchangeable random graphs subject to structural constraints, such as bipartiteness, transitivity, or the presence or absence of specific subgraphs. A notable example is the transitive case, where exchangeable random graphs reduce to exchangeable partitions.

We prove de Finetti–style representation theorems for constrained exchangeable graphs, showing that they correspond to mixtures over restricted classes of graphons satisfying explicit algebraic or functional constraints. This perspective unifies, in a single framework, several previously studied objects, including exchangeable partitions and exchangeable posets. We also establish finite exchangeability results for these constrained graph models.

Our results provide a principled foundation for Bayesian modeling of networks with structural constraints.

ABSTRACT. Functional principal component analysis (FPCA) is a fundamental tool for exploring variation in samples of random curves or surfaces. We propose a new approach to FPCA for functional data observed irregularly and sparsely over their domains, based on smoothing directly at the level of the eigenfunctions. Our formulation leads to an efficient optimization-based procedure whose computational and storage costs are comparable to those of standard multivariate PCA for regularly observed data. The method is flexible with respect to domain geometry and model class, accommodates structural constraints and penalties, and facilitates uncertainty quantification via resampling and asymptotic theory.

ABSTRACT. I will talk about a model of two dimensional random growth (namely, polynuclear growth) where we can find nice exact expressions for the distributions of key statistics, via the RSK correspondence. By analysing this model in half-space with external sources, we can show the appearance of a universal interface fluctuations associated with stationary random growth, previously studied by Beta, Ferrari and Occelli, and then Barraquand, Le Doussal and Krajenbrink. We also find a distribution which interpolates between the half-space stationary one and different Tracy—Widom distributions (in other words, a half-space analogue of a distribution of Baik and Rains). Our approach uses connections between enumeration of Young tableaux, symmetric functions, matrix integrals, and Hankel determinants, plus a Riemann—Hilbert problem. I’ll discuss how we can extend this approach to an inhomogeneous version of TASEP.

ABSTRACT. This talk is devoted to the stochastic optimal control problem of infinite-dimensional differential systems allowing for both path-dependence and measurable randomness. As opposed to the deterministic path-dependent cases studied by Bayraktar and Keller [J. Funct. Anal. 275 (2018) 2096-2161], the value function turns out to be a random field on the path space and it is characterized by a stochastic path-dependent Hamilton-Jacobi (SPHJ) equation. A notion of viscosity solution is proposed and the value function is proved to be the unique viscosity solution to the associated SPHJ equation.

ABSTRACT. We consider the problem of nonparametric estimation of the conformational variability in a population of related structures, based on low-dose tomography of a random sample of representative individuals. In this context, each individual represents a random perturbation of a common template and is imaged noisily and discretely at but a few projection angles. Such problems arise in the cryo Electron Microscopy of structurally heterogeneous biological macromolecules. We model the population as a random field, whose mean captures the typical structure, and whose covariance reflects the heterogeneity. We show that consistent estimation is achievable with as few as two projections per individual, and derive uniform convergence rates reflecting how the various parameters of the problem affect statistical efficiency, and their trade-offs. Our analysis formulates the domain of the forward operator to be a reproducing kernel Hilbert space, where we establish representer and Mercer theorems tailored to question at hand. This allows us to exploit pooling estimation strategies central to functional data analysis, illustrating their versatility in a novel context. We provide an efficient computational implementation using tensorized Krylov methods and demonstrate the performance of our methodology by way of simulation.

ABSTRACT. We consider ergodic McKean stochastic differential equations with a unique stationary state and study the linearized (in the sense of McKean) diffusion process obtained by replacing the law of the nonlinear process with its unique invariant measure. We prove that the law of the nonlinear McKean process and its linearized counterpart are exponentially close in time, both in relative entropy and in Wasserstein distance. The analysis, based on entropy estimates and logarithmic Sobolev inequalities, is carried out on both the whole space and the torus. We then show how the resulting linearized diffusion can be used to replace the original nonlinear process for tasks depending on the long-time behavior of the dynamics, with a particular focus on parameter estimation from a single observed long trajectory.

ABSTRACT. Gibbs measures are often thought of as the equilibrium measures of thermodynamical systems, but this statement is not universally proven. Indeed, while many detailed results on the dynamics of lattice systems exist, very little is known about the dynamical aspects of Gibbs point processes, in particular regarding out-of-equilibrium dynamics and convergence. In this talk, I will present a model for a continuous-time birth-and-death dynamics in d-dimensional Euclidean space. For such a model, thanks to a de Bruijn-type identity relating the time evolution of the specific relative entropy along trajectories to the Fisher information, we were able to confirm the intuition that the Gibbs measures are the long-time (weak) limit points of the dynamics.

This talk is based on joint works with B. Jahnel, J. Köppl, and Y. Steenbeck, available at arXiv:2508.21196 and arXiv:2602.13474.

ABSTRACT. We explore the geometry of the Bures-Wasserstein space for potentially degenerate Gaussian measures on a separable Hilbert space. In this general setting, the optimal transport map is formally the subgradient of a convex function that is infinite almost everywhere, rendering conventional duality-based variational methods ineffective. We overcome this analytical barrier by exploiting a constructive operator-theoretic approach. Our central result proves that the Kantorovich problem for any pair of Gaussian measures reduces to a Monge problem; that is, an optimal transport map exists in at least one direction between two measures. This reduction allows for a complete characterization with explicit formulas for all optimal (potentially unbounded) Monge transport map and Kantorovich couplings, as well as establishing their uniqueness. Furthermore, we provide a full description of the convex set of geodesics between degenerate measures, revealing a rich geometric structure where the classical McCann interpolants arise as the extreme points. We apply these findings to construct transport maps for Gaussian processes and introduce a novel framework for Wasserstein barycenters based on random Green's operators.

ABSTRACT. We consider a stochastic epidemic model with dynamics driven by mixing events gatherings of two or more individuals), and the corresponding early-phase branching process approximation. We introduce into the model a special type of contact tracing, sideward contact tracing, which aims at tracing individuals who were infected at the same event as a diagnosed individual. In order to deal with the dependencies caused by the tracing, we define a new branching process where sibling groups of the original branching process correspond to multi-type macro-individuals on an enriched state space, free of dependencies. This then allows us to analyse the impact of sideward contact tracing on the epidemic dynamics.

ABSTRACT. Discrete curvatures are quantities associated to the nodes and edges of a graph that reflect the local geometry around them. These curvatures have a rich mathematical theory and they have recently found success as a tool to analyze networks across a wide range of domains. We consider the problem of constructing graphs with a prescribed set of discrete edge curvatures, and explore the space of such graphs. In particular, we solve the exact reconstruction problem for the specific case of Forman–Ricci curvature. By leveraging the algebraic theory of Markov bases, we obtain a finite set of rewiring moves that connects the space of all graphs with a fixed discrete curvature. These moves allow us to define a Markov chain to sample from the space of graphs with a given curvature, providing a foundation for generating curvature-constrained null models. Based on joint work with Michelle Roost, Karel Devriendt and Jürgen Jost and ongoing work with Jane Ivy Coons.

ABSTRACT. Graph limit theory studies the convergence of sequences of graphs as the number of vertices grows, providing an effective framework for representing large networks. In this talk, I will give a brief introduction to graph limits and report on recent extensions to weighted graphs, colored graphs and multiplex networks (probability graphons and P-variables). As an application of this theory I will present a large deviation principle (LDP) for random weighted graphs that generalizes the LDP for Erdős-Rényi random graphs by Chatterjee and Varadhan (2011), based on joint work with Pierfrancesco Dionigi.