Title / Abstract

SPDEs driven by symmetric α-stable Lévy  noise 

In this talk, we prove the existence of solution of SPDEs driven by a symmetric α-stable Lévy noise. The solution is a random field, in the sense of Walsh (1986). As examples, we consider the heat and wave equations, also called the parabolic/hyperbolic Anderson models. For these models, in the case of the Gaussian noise, the solution has an explicit Wiener chaos expansion, and is studied using tools from Malliavin calculus.

These tools cannot be used for an infinite-variance Lévy noise. Nevertheless, we will show that a solution can be constructed using a "stable'' chaos expansion. For this, we will use the multiple stable integrals, which were developed by Samorodnitsky and Taqqu (1991), and originate from the LePage series representation of the noise. To give a meaning to the stochastic integral which appears in the definition of solution, we embed the noise into a Lévy basis, and use the Bichteler-Jacod (2002) stochastic integration theory with respect to this object, similarly to other studies of SPDEs with heavy-tailed noise, e.g. Chong-Dalang-Humeau (2019). This talk is based on joint work with Juan Jimenez (University of Ottawa).

The Smoluchowski-Kramers approximation for stationary solutions of damped stochastic wave equations with variable friction

We investigate the convergence, in the small mass limit, of the stationary solutions of a class of stochastic damped wave equations, where the friction coefficient depends on the state and the noisy perturbation if of multiplicative type. We show that the Smoluchowski-Kramers approximation which has been previously shown to be true in any fixed time interval, is still valid in the long time regime. Namely, we prove that the first marginals of any sequence of stationary solutions for the damped wave equation converge to the unique invariant measure of the limiting stochastic quasilinear parabolic equation. The convergence is proved with respect to the Wasserstein distance associated with the H−1 norm.

Interpolating Stochastic Heat and Wave Equations Through Fractional SPDEs

This talk explores Stochastic Partial Differential Equations (SPDEs) with fractional differential operators, which give a parametric family that bridges the stochastic heat equation (SHE) and the stochastic wave equation (SWE). We will showcase recent developments and findings pertinent to this interpolation.

A critical stochastic heat equation with long-range noise 

I will describe a critical nonlinear stochastic heat equation in dimension d≥3 with noise that is white in time with spatial covariance that looks like the Riesz kernel at large scales. If the noise is attenuated by a logarithmic factor, then the limiting pointwise statistics of this equation can be described in terms of a forward-backward SDE (FBSDE). The FBSDE is similar to but not the same as a similar FBSDE obtained for analogous limits of the 2D nonlinear stochastic heat equation with spatially uncorrelated noise. This work also complements 2004 work by Mueller and Tribe in the linear case, where there is no logarithmic attenuation and a measure-valued process is obtained. Joint work in progress with Martin Hairer and Xue-Mei Li.

Interface and its fluctuation in interacting particle systems

We first discuss the separation of particles into sparse and dense phases in Glauber-Kawasaki dynamics of non-gradient type and show that the phase-separating interface evolves according to the anisotropic curvature flow. Then, we study the fluctuation of the interface in a simple situation and derive a linear SPDE via the Boltzmann-Gibbs principle. We also discuss heuristically the derivation of nonlinear SPDEs. The talk is partly based on joint work with Chenlin Gu, Han Wang (Tsinghua University), Hyunjoon Park (Meiji University), Claudio Landim (IMPA), Sunder Sethuraman (University of Arizona). The results for the first part can be found in arXiv:2404.18364, arXiv:2404.12234, arXiv:2403.01732. 

Support properties of solutions to nonlinear stochastic partial differential equations

In this talk, we will investigate the support properties of solutions to nonlinear second-order stochastic partial differential equations driven by random noise. The noise can be space-time white noise or spatially homogeneous colored noise that meets the reinforced Dalang's condition. We will illustrate conditions on the diffusion coefficient that ensure the solution has compact support in the spatial direction. Additionally, we will introduce conditions on the diffusion coefficient that guarantee the solution is strictly positive. Furthermore, we will suggest possible extensions of these conditions. This research is a collaboration with Kunwoo Kim and Jaeyun Yi. 

A random string among random obstacles

We consider a random string on R^d modelled by a vector valued stochastic heat equation driven by additive space time white noise. The obstacles are given by balls of a fixed radius centered around points of a Poisson process. The string dies when any part of the string touches any of the obstacles. We give asymptotic bounds on the probability of survival of the string up to a large time T. The talk is based on joint work with Siva Athreya and Carl Mueller.

An existence and uniqueness theory to stochastic partial equations involving  pseudo-differential operators driven by space-time white noises

In this talk, we introduce a new weak formulation to guarantee existence and uniqueness of a solution to stochastic partial differential equations with pseudo-differential operators whose symbols are allowed to be sign-changing.

Markov processes with jump kernels decaying at the boundary

In this talk, we discuss pure-jump Markov processes on smooth open sets whose jumping kernels vanishing at the boundary and part processes obtained by killing at the boundary or (and) by killing via the killing potential. The killing potential  may be subcritical or critical.

This work can be viewed as  developing a general theory for non-local singular operators whose kernel vanishing at the boundary. Due to the possible degeneracy at the boundary, such operators are, in a certain sense, not uniformly elliptic. These operators cover the restricted, censored and spectral Laplacians in smooth open sets and much more.

The main results are the boundary Harnack principle and its possible failure, and sharp two-sided Green function estimates.

This is a joint work with Soobin Cho (University of Illinois, USA), Renming Song (University of Illinois, USA) and Zoran Vondra\v{c}ek (University of Zagreb, Croatia)

Scaling Limits of Functional Stochastic Games on Graphs

We consider a general class of finite-player stochastic games on graphs with nearest neighbor interaction, in which the linear-quadratic cost functional includes linear operators acting on controls in L^2. We propose a novel approach for deriving the Nash equilibrium of the game semi-explicitly in terms of operator resolvents, by reducing the associated first order conditions to a system of stochastic Fredholm equations of the second kind and deriving their solution in semi-explicit form. Furthermore, by proving stability results for the system of stochastic Fredholm equations, we derive the convergence of the equilibrium of the N-player games on dense graphs to the corresponding infinite-dimensional graphon game.

Finally, we apply our general framework to solve various examples, such as stochastic Volterra linear-quadratic games, models of systemic risk and advertising with delay, and optimal liquidation games with transient price impact.

This talks is based on joint projects with Eduardo Abi-Jaber, Moritz Voss and Sturmius Tuschmann. 

Global solutions for superlinear SPDEs 

The classical existence and uniqueness theorems for SPDEs prove that, under appropriate assumptions, SPDEs with Lipschitz continuous forcing terms have unique global solutions. This talk outlines recent results about SPDEs exposed to superlinearly growing deterministic and stochastic forcing terms. I describe sufficient conditions that guarantee that, despite the superlinear growth, the SPDEs have unique global solutions. 

Gaussian fluctuation of Euclidean Φ⁴ QFT

I will discuss the asymptotic expansions of the Euclidean Φ⁴-measure in the low-temperature regime. Consequently, we derive limit theorems, specifically the law of large numbers and the central limit theorem for the Φ⁴-measure in the low-temperature limit. In the second part of the talk, I will focus on the infinite volume limit of the focusing Φ⁴-measure. Specifically, with appropriate scaling, the focusing Φ⁴-measure exhibits Gaussian fluctuations around a scaled solitary wave, that is, the central limit theorem.

This talk is based on joint works with Benjamin Gess, Pavlos Tsatsoulis, and Philippe Sosoe. 

Stochastic quantization of Yang-Mills

We will discuss the stochastic Yang-Mills flow, which is the deterministic Yang-Mills flow driven by a (very singular) space-time white noise. We will start with a gentle introduction to the Yang-Mills model, review the fundamental challenges, and then discuss our construction over 2 and 3 dimensional tori, including the meaning of "gauge equivalence” and "gauge invariant observables" in the singular setting. 

Some rough paths techniques in reinforcement learning

In this talk I will start by reviewing some classical results relating machine learning problems with control theory. I will mainly discuss some very basic notions of supervised learning as well as reinforcement learning. Then I will show how noisy environments lead to very natural equations involving rough paths. This will include a couple of motivating examples. In a second part of the talk I will try to explain the techniques used to solve reinforcement learning problems with a minimal amount of technicality. In particular, I will focus on rough HJB type equations and their respective viscosity solutions. If time allows it, I will give an overview of our current research program in this direction. This talk is based on a joint work with Prakash Chakraborty (Penn State) and Harsha Honnappa (Purdue, Industrial Engineering).

Solving marginals of the LDP for the directed landscape

Recently, Das, Dauvergne, and Virág proved the upper-tail Large Deviation Principle (LDP) for the directed landscape, with a rate function written in terms of (directed) metrics. In this talk, we explain how one can solve the corresponding metric-level variational problem to obtain some marginal LDPs. Specifically, we prove the upper-tail LDP for the parabolic Airy process and characterize the limit shape of the directed landscape under the upper-tail conditioning. Our method is PDE-based and uses geometric arguments, connecting the variational problem to the weak solutions of Burgers' equation.

Joint work with Sayan Das.

Comments on an anisotropic percolation system on Z²

This talk will be mostly based on a paper that was published a few years back, joint with Thomas Mountford and Hao Xue [1]. It deals with a question that arose in the frame of ferromagnetic Ising models on the plane with a layered interaction, and which is still unanswered.  I plan to motivate a bit the original model and then shift to the simplest case of percolation, where the interplay with SPDEs has been particularly helpful.   We consider the following anisotropic graph on  the two dimensional lattice Z²: horizontal edges are set for each pair of vertices within a  given Euclidean distance N and vertical edges connect only nearest neighbor vertices. On this graph one considers a Bernoulli percolation model, so that horizontal edges are open with probability 1/(2N), corresponding to the critical mean field value. The goal is the determination of a critical exponent  for the opening probability of the vertical bonds  which would allow us to observe a phase transition for all large N. This brings naturally to an analysis of the scaling limit of the growth process restricted to each horizontal layer. Our arguments for this analysis are inspired by the work of Mueller and Tribe on the long range contact process.  A renormalization scheme is used for the percolative regime.

Reference:

[1] T.S. Mountford, M. E. Vares, H. Xue. Critical scaling for an anisotropic percolation system on Z^2,Electronic Jr. Probab.  25, p. 129, 44pp. 2020

Ill-posedness of stochastic PDEs by the technique of convex integration 

I will describe recent developments on the well-posedness and ill-posedness of stochastic PDEs. I will especially focus on the technique of convex integration that has its roots in geometry but has recently proven to be a new effective tool to prove various types of non-uniqueness results for stochastic PDEs in fluid mechanics such as the Navier-Stokes equations, magnetohydrodynamics system, and surface quasi-geostrophic equations.

Lower bounds on the Lyapunov exponents of SPDEs from fluid dynamics

We discuss the Lyapunov exponent of stochastic PDEs, especially from fluid dynamics. In particular, we introduce strategies to obtain lower bounds on the Lyapunov exponents, focusing on the passive scalar advection-diffusion model with stochastic velocity. Our approach relies on the introduction of a novel Lyapunov functional for projective processes, based on the dynamics of the energy median. This ongoing work is joint with Martin Hairer, Samuel Punshon-Smith, and Tommasi Rosati.