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

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. 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. 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. We introduce the dice process, a probabilistic model describing the evolution of a collection of particles moving on a graph according to random local rules. At each time step, all particles occupying the same site use a common, randomly chosen “dice” to determine their next move. This construction gives rise to a rich class of (partially) exchangeable Markov chains.

The first result of the talk establishes that every partially exchangeable col- lection of Markov chains on a finite state space can be represented as a dice

process. As an application, we obtain a natural characterization of multitype Λ-coalescents without restrictions on the migration mechanism (based on joint work with Noemi Kurt, Imanol Nu ̃nes, and Jos ́e Luis P ́erez). We will also briefly discuss a related detour involving the evolutionary rate of

plasmid-bearing bacteria. This part of the talk is based on recent experimen- tal work by Paula Ramiro-Mart ́ınez, Ignacio de Quinto, Laura Jaraba-Soto,

Val F. Lanza, Cristina Herencias-Rodr ́ıguez, Rafael Pe ̃na-Miller, and Jer ́onimo Rodr ́ıguez-Beltr ́an. Finally, we discuss a way to construct functions of sequences of exchangeable random variables that preserve exchangeability. Surprisingly, this leads to a transparent connection between moment duality and de Finetti’s theorem. This last part is based on joint work with Arno Siri-J ́egousse and Ariel Offenstadt.