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. 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. 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. 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 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. 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. 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. 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. 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. 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. 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. 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. 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. 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 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 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. 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. 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. 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. 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 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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.

We discuss pathwise (strong) existence and uniqueness under assumptions that allow generalized drift components, including local time terms at the interface. As a motivating example, we consider a threshold Cox-Ingersoll-Ross model, where the drift and/or 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 based on joint works with Julia Budzinski, Madalina Deaconu, Benoît Nieto, and Paolo Pigato.

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. 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. 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. 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. 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. 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. 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. 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. 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. 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 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 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 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. 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 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. 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. 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 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. 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. 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. 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. 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. 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. 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 recent years, the mathematical analysis of overparameterized learning systems has increasingly relied on tools from probability theory, optimal transport, and interacting par- ticle systems. In this talk, I present a mean-field limit result for Mixture of Experts (MoE) models trained via gradient flow, and discuss how the same framework naturally applies to quantum neural networks. We consider a supervised learning problem with quadratic loss, where the predictor is given by the uniform average of N identical experts. Each ex- pert is parametrized by a vector in a compact parameter space, and training is performed through gradient flow. This reformulation allows us to interpret learning as the evolution of a deterministic interacting particle system in parameter space: each parameter follows a drift that depends on the empirical distribution of all the others. The main result es- tablishes propagation of chaos as the number of experts tends to infinity. More precisely, the empirical measure of the parameters converges to a deterministic probability measure solving a nonlinear continuity equation. We provide a quantitative convergence rate in Wasserstein distance between the empirical measure and the mean-field limit, thereby giv- ing explicit control on finite-width effects. In the second part of the talk, we apply the theory to quantum neural networks. Each expert is taken to be a parametrized quantum circuit whose output is the expectation value of a fixed observable. This provides a rigorous interacting-particle description of quantum models built as mixtures of quantum experts. Importantly, this regime differs from previously studied infinite-width limits leading to Gaussian process behavior. Here, the limit concerns the number of experts rather than the number of qubits, and the dynamics is genuinely nonlinear: the model operates beyond the lazy-training regime. Overall, we provide a unified probabilistic framework for classical and quantum mixture architectures, clarifying the geometric structure of their training dynamics and the deterministic mean-field evolution.

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. 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 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. 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. 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 propose 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. 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. 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. 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. 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. 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. 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.