Schedule and abstracts

Many systems of interest in the applied sciences share the common feature of possessing multiple scales, either in time or in space, or both. Multiscale modelling approaches are built on the ambition of treating both scales at the same time, with the aim of deriving (rather than empirically obtaining) efficient coarse grained models which incorporate the effects of the smaller/faster scales.

In this talk we will consider SDE systems that are multiscale in time, which are typically studied via (stochastic) averaging and homogenization. When applying such methods a key assumption is that the dynamics for the fast scale is ergodic, i.e. that it has a unique equilibrium (invariant measure). If the fast scale has multiple invariant measures (which is a rather common occurrence in many scenarios, for example for McKean-Vlasov SDEs) then truly very little is known. In particular both averaging and homogenization theory are currently not equipped to help tackle this problem. In this talk we will present situations in which this scenario occurs, with particular reference to multiscale interacting particle systems,  summarise recent progress and point out gaps in current theory.

Based on work with K.Painter and I. Souttar.

In this talk I will introduce a class of continuity equations with noisy perturbations. I will briefly introduce the theory of rough paths and show how this perspective can give meaning to the noisy part of the equation. In the linear setting, we shall see how to obtain well-posedness of the continuity equations under DiPerna-Lions regularity conditions on the drift. If time permits, we shall also see how these results can be extended to a non-linear drift and how this leads to so-called Yudovich theory of the Euler equation with rough perturbations when d=2. 

We study the propagation of chaos phenomenon for controlled systems of N particles whose dynamics solving Markovian optimal control problems. In the large-N limit, these systems are expected to converge to deterministic optimal control problems of McKeanVlasov (mean-field) type. We focus on the cases where the limit problem is a mean-field planning problem or a mean-field Schrödinger problem, which generalize classical optimal transport by prescribing both the initial and terminal distributions. The proof combines compactness arguments for flows of probability measures in Wasserstein spaces with a generalized de Finetti theorem adapted to the associated transport equation. This talk is based on joint work with Chiara Rigoni. 

In a toy model of interacting particles system with common noise, we make a decomposition of the first particle and of the mean-field limit yielding both the law of the process, its long-time convergence and in the specific case of the small-noise limit, we will obtain a Large Deviations Principle and the first time that the processes touches -in some sense - a given point in the space. Finally, we will study the influence of the parameter tuning the common noise with respect to the idiosyncratic one. 

I will be presenting a deep dive into the proof of Sanov type LDP results for for the law of the empirical measure associate to an ensemble of interacting equations converging to the law of a McKean-Vlasov equation. 

Given we study convergence of the law, we only need work with weak solutions. Therefore, I will introduce some new results for weak existence, uniqueness and PoC before developing and extending the techniques first used by [Budhiraja, Dupuis, Fischer 2012] to prove this LDP. Critically, their method rely on the existence of strong solutions AND martingale techniques which we bypass efficiently. 

This is joint work with Jiawei Li and Goncalo dos Reis.

It is by now well understood that SDEs driven by additive (fractional) Brownian motion can be well-posed in the presence of irregular, possibly even distributional drifts, in a regularization by noise fashion. Scaling arguments and explicit counterexamples also establish that positive results can only be expected in so-called subcritical regularity regimes. However, under additional structural assumptions on the drift, like it being divergence free, it was recently understood in the Brownian case that the regularity conditions on the drift can be weakened, by means of PDE arguments and martingale problems. In this talk I will show how similar results can be established for fBm-driven SDEs; the proofs are based on rather different methods, based on a Lagrangian approach, combined with Girsanov transform and relative entropy bounds. In particular, for a class of supercritical drifts of bounded divergence, the associated SDE can be shown to admit a unique-in-law solution for Lebesgue a.e. initial condition, and the resulting solution kernel is stable in the total variation norm. Leveraging on this, wellposedness results for McKean-Vlasov equations are also derived.

Based on a joint work with Zimo Hao (Bielefeld).

In this talk we present some results on a moderately interacting particle system with drift in negative Besov spaces (hence the drift is a distribution), and its limiting McKean SDE. We show well-posedness of the system as well as well-posedness of the limiting equation. Propagation of chaos is also investigated, and some partial results are presented. As a by-product of this analysis, we have also well-posedness of two classes of related PDEs with coefficient in Besov spaces, namely a Kolmogorov-type PDE and a Fokker-Plank PDE.

This is a joint work-in-progress with Luca Bondi (UniTo) and Francesco Russo (ENSTA Paris).

We study SDEs driven by fractional Brownian motion with singular drift, of distributional type in the Catellier—Gubinelli regime and a confining potential of linear growth. Building on Hairer’s theory of stochastic dynamical systems (Hairer ’05 & Hairer—Ohashi ’07) we leverage stochastic averaging estimates to show existence and uniqueness of invariant measures. The central innovation is to combine stochastic sewing (an intrinsically local tool) with large time estimates to obtain tightness, an analogue of the strong Feller property, topological full support and quasi-Markovianity. The latter three are combined to prove an analogue of the Doob—Khashminskii theorem for stochastic dynamical systems (Hairer—Ohashi ’07).

Joint work ongoing with Ł. Mądry.

This talk presents tools for studying geodesic convexity of various functionals on submanifolds of Wasserstein spaces with their induced geometry. We obtain short new proofs of several known results, such as the strong convexity of entropy on sphere-like submanifolds due to Carlen-Gangbo, as well as new ones, such as the semiconvexity of entropy on the space of couplings. The arguments revolve around a simple but versatile principle, which crucially requires no knowledge of the structure or regularity of geodesics in the submanifold. In these setting, we derive strengthened forms of Talagrand and HWI inequalities on submanifolds, which we show to be related to large deviation bounds for conditioned empirical measures. This work is collaboration with Prof. Daniel Lacker. 

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 R^d and are connected by certain conductances. On top of this resistor network, particles move according to random walks with the exclusion rule.

Under soft assumptions on the simple point process measure and conductances, which includes ergodicity, stationarity and certain moment conditions, it is known that the empirical density of particles homogenises converging for almost all realisation of the environment to the solution of an heat equation with a certain effective diffusivity.

In this talk, we examine its equilibrium fluctuations. For d≥3, 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.

TBA

The aim of this talk is to present some regularizing effects of an infinite-dimensional common noise on mean-field control models. It is known that the usual finite-dimensional common noise is often insufficient to enhance well-posedness in mean-field systems with non-regular data. In contrast, an infinite-dimensional common noise is expected to enforce the uniqueness of solutions. However, constructing such a forcing requires the introduction of a diffusion process that takes values in the space of probability measures. In this work, we examine the impact of a Dirichlet-Ferguson type noise (see [1]). We first discuss a system of non-controlled interacting particles and the associated backward Kolmogorov equation on the space of probability measures. We then move on to a mean-field control problem and analyze the corresponding second-order Hamilton-Jacobi-Bellman equation.

This presentation is based on joint work with François Delarue and Giacomo Sodini.

[1] L. Dello Schiavo. The Dirichlet–Ferguson diffusion on the space of probability measures over a closed Riemannian manifold, The Annals of Probability, 50(2), 591-648, 2022.

The entropic optimal transport problem, a.k.a. Schrödinger problem, is a regularised version of the classical optimal transport problem which consists in minimising relative entropy against a reference distribution among all couplings of two given marginals. In this talk, we study the stability of optimisers and exponential convergence of Sinkhorn’s algorithm, which is at the heart of the success of entropic regularisation in machine learning and engineering applications.  In the first part of the talk, we will illustrate how semiconcavity of dual optimal variables, known as entropic potentials, plays a key role in establishing tight stability bounds and the convergence of Sinkhorn iterates with sharp rates. In the second part of the talk we discuss how to establish semiconcavity bounds in examples of interest such as  (weakly) log-concave marginals or marginals with bounded support. 

Transformers are a central architecture in modern deep learning, forming the backbone of large language models such as ChatGPT. In this talk, I will present a mathematical framework for studying how information—represented as "tokens"—evolves through the layers of such neural networks. Specifically, we consider a family of partial differential equations that describe how the distribution of tokens—modeled as particles interacting in a mean-field way—changes with depth.

Numerical experiments reveal that, under certain conditions, these dynamics exhibit a metastable clustering phenomenon, where tokens group into well-separated clusters that evolve slowly over time. A rigorous analysis of this behavior uncovers a range of open questions and unexpected connections to analysis and geometry.

In this talk, I will discuss a system of kinetic stochastic differential equations in which each particle is additively perturbed by a Brownian motion. In the deterministic setting, no assumptions are required beyond the convergence of the empirical measure. By contrast, the stochastic framework typically requires either strong exchangeability of the initial particle configuration or certain technical moment assumptions to study the convergence of the empirical measure. Using a family of anisotropic Sobolev spaces, a “simple” SPDE satisfied by the empirical measure, and the Garsia–Rodemich–Rumsey lemma, I will show how one can recover the deterministic framework and establish a classical law of large numbers for kinetic diffusions.

This is based on joint work with Fabio Coppini (Utrecht University).