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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

ABSTRACT. We study the convergence of the training dynamics of residual neural networks (ResNets) towards their joint infinite depth–width limit. We focus on ResNets with two-layer perceptron blocks, whose shape is determined by the depth L, hidden width M, and embedding dimension D, and we adopt the residual scaling O(√ D/√ (L M) ) recently identified as necessary for local feature learning. We show that after a bounded number of training steps, the error between the finite ResNet and its infinite-size limit is O(1/L + √D/√(L M) + 1/√D), and numerical experiments suggest that this bound is tight in the early training phase. From a probabilistic viewpoint, the D → ∞ limit amounts to a mean-field limit over the coordinates of the embedding, where some interactions scale in 1/√ D contrary to the usual 1/D setting. Our analysis is a rigorous and quantitative instance of the Dynamical Mean Field Theory (DMFT) from statistical physics; it combines propagation of chaos arguments with the cavity method at a functional level.

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

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

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

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

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

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

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

\smallskip

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

ABSTRACT. Random evolutions are stochastic models used to study dynamical systems in random environments; they were originally introduced to seek stochastic solutions to abstract hyperbolic equations. Intuitively, a random evolution can be imagined as a single path taken by a machine with $n$ operating modes, where a control mechanism randomly switches between them. More formally, a random evolution is defined as the random product of semigroup operators on a Banach space. The aim of this talk is to extend random evolution theory to the setting of anomalous dynamics by means of random times (i.e., inverses of subordinators). Furthermore, we study time-fractional abstract hyperbolic differential equations and their solutions in terms of time-changed operators related to random evolutions. In particular, we focus our attention on telegraph-type equations involving non-local operators in time (i.e., convolution-type derivatives), which allow us to obtain an extension of Kac’s solution.

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

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

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

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

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

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

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

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

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

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

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

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

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

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

Joint work with Baptiste Huguet and Pablo López Rivera.

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

ABSTRACT. In many scientific domains, from single-cell genomics to oceanography, we observe populations evolving over time but often only have access to unconnected snapshot samples rather than continuous trajectories. For example, in single-cell sequencing, measuring gene expression destroys the cell, meaning we can access data for any particular cell only at a single time point. Practitioners aim to recover these unobserved trajectories, but standard methods face significant limitations.

While the deep learning community has explored Schrödinger Bridges (SBs) for this task, existing methods typically interpolate between just two time points or require specifying a single, fixed reference dynamic — often Brownian motion. This is restrictive because practitioners can often specify a family of relevant dynamics (e.g., a vortex model for ocean currents) but not the exact parameters. To address this, I introduce a new SB method that learns unobserved trajectories across multiple time points by requiring only a family of reference dynamics. By iteratively refining the reference process against observed data, we bridge the gap between domain knowledge and data-driven inference.

I demonstrate the efficacy of this approach on diverse scientific problems, including modeling ocean debris dispersion in the Gulf of Mexico and inferring gene regulatory networks in the synthetic Repressilator system. By incorporating domain knowledge through parametric families, our method generates trajectories that better respect the underlying physics compared to standard baselines. I will conclude by talking broadly about my research and future work, including alternative approaches to reconstruct trajectories using Maximum Mean Discrepancy.

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

References:

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

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

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

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

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

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