Purdue Probability Seminar, Fall 2025

In this talk, I will describe some recent progress on singular stochastic partial differential equations in the setting of non-commutative probability theory - examples will include the stochastic quantization of Fermionic quantum field theories and the setting of free probability. This is based on joint work with Martin Hairer and Martin Peev.

This talk will discuss various notions of boundaries at infinity of random walks in random potentials. Recent results on existence and uniqueness will be presented for a class of models that generalizes first- and last-passage percolation, random walks in random environments, and directed polymers. The resulting boundary structures are related to jointly stationary distributions, geodesic rays, Busemann functions, harmonic functions and the associated Martin boundary, and extremal Gibbs-DLR measures. Based on joint works with Sean Groathouse, Sergazy Nurbavliyev, Firas Rassoul-Agha, and Timo Seppäläinen.

Reinforcement learning (RL) is one of the three main paradigms of machine learning. Traditionally, it has been studied in discrete time and space via Markov decision processes. In 2020, Wang, Zariphopoulou and Zhou [WZZ] formulated a continuous version of RL under the machinery of stochastic control theory and proved results under this formulation. Non-Markovian dynamics in mind, Chakraborty Honnappa and Tindel [CHT] recasted this formulation and introduced rough paths as the "random" driver, instead of Brownian motion. In this talk we will review the [WZZ] construction and interpretation of continuous time RL. Moreover, we will mention the results by [CHT] and present our new results in this direction. This is based on a joint work with Prakash Chakraborty, Harsha Honnappa and Samy Tindel.

This talk will discuss the area process and its asymptotics on partial flag manifolds with blocks of equal size. A Brownian motion on these manifolds can be represented as a matrix valued diffusion obtained from a unitary Brownian motion in block form. To obtain an explicit expression for the characteristic function of the area processes the matrix Jacobi operators on the simplex are introduced and studied. These polynomials simultaneously generalise the Heckman-Opdam polynomials of type BC and the Jacobi polynomials on the simplex. This work generalises the results for the full flag manifold, obtained by Fabrice Baudoin, Nizar Demni, Jing Wang and myself, to a more general class of partial flag manifolds.

A Ray-Knight theorem is a description of the local time profile of a stochastic process when stopped at some inverse local time. Since a Ray-Knight theorem contains a lot of information about the underlying process, and since a number of results have been obtained for self-interacting random walk models by proving Ray-Knight theorems for the walk, one naturally wonders if a Ray-Knight theorem can be used directly to deduce the scaling limit of the walk. Somewhat surprisingly, a recent result of myself with Kosygina and Mountford shows that this is not the case.

In this talk, I will show that while Ray-Knight theorems are not sufficient for proving scaling limits, one can obtain a functional limit for the walk through what we call joint Ray-Knight theorems. As an application of our main result we prove the convergence of polynomially self-repelling random walks to a limiting process which appears to be a new stochastic process. This is based on joint work with Elena Kosygina, Laure Mareche, and Tom Mountford.

The challenge of estimating an unobserved signal from noisy measurements known as the filtering problem, is fundamental in control theory and signal processing. We consider a system whose signal-observation dynamics are modeled by a stochastic differential equation (SDE). The objective is to compute the optimal estimate, given by the conditional expectation, of the signal trajectory based on the observation path which is called the filter. I will specifically address the robust filtering problem, where we want to show that there is a version of the filter which is continuous in the observation paths in a suitable topology.

In my talk, I will first formally define the robust filtering problem and summarize the methods that have been used to demonstrate the filter's path continuity. The latter part of the talk will introduce the theory of rough paths which readily give us continuity estimates for differential equations driven by paths of finite p-variation (like Brownian motion). I will then present results showing how the framework of rough paths provides allows us to construct a robust filter under the rough topology.

In a series of recent developments, the scaling limits of stationary measures of open ASEP have been computed with various choices of parameters, and for some choices the limits are stationary measures of the open KPZ equation and the conjectured open KPZ fixed point. An overview of these developments will be provided. The talk is based on several joint works with Wlodek Bryc, Zoe Himwich, Alexey Kuznetsov, Joseph Najnudel, Jacek Wesolowski, and Zongrui Yang.

Stochastic partial differential equations (PDEs) model diverse phenomena across physics, biology, and materials science, where randomness plays a central role. In this talk, we explore the fractal geometry of extremes in such equations, focusing on the (1+1)-dimensional KPZ equation and the Parabolic Anderson Model (PAM)-two canonical systems exhibiting rich intermittent behavior and multifractal structure. We show that the spatial and spatio-temporal peaks of their solutions attain infinitely many macroscopic Hausdorff dimensions, characterizing their multifractality in a precise quantitative framework. We also present new results for the (2+1)-dimensional critical stochastic heat flow, the only known critical SPDE where certain fractal dimensions have been computed. Our techniques draw on a broad array of tools, including integrable probability, Gibbsian line ensembles, the machinery of regularity structures and paracontrolled calculus using newly found sequential coarse graining technique.

These findings are part of an emerging program aimed at unraveling the universal fractal geometry behind singular SPDEs, with several open directions to be discussed.

We identify conditions for which a Dirichlet space(a metric measure space with diffusion) admitting a sub-Gaussian heat kernel would be in the Da Prato-Debussche regime of the $Phi^{n+1}$ equation. For this purpose, we use heat kernel based Besov spaces, where regularity of Schwartz-type distributions is measured using the small time behavior of the heat kernel. In the process, we show how many nontrivial parts of the solution theory such as construction of paraproducts and energy estimates are made more difficult by the roughness of the underlying geometry. These difficulties in fact produce a more restrictive regime than one may first expect by typical scaling heuristics.

We are interested in the question of the long time accuracy of solutions of nonlinear stochastic partial differential equations (SPDEs) under various approximations. We will present a powerful and quite general framework which addresses such questions. It centers on 1) proving a (parameter uniform) contraction of the Markovian dynamic in an appropriate Wasserstein metric 2) establishing the accuracy of approximations over a finite time window. Our main paradigmatic application for this approach concerns the numerical approximation of the stochastically driven Navier-stokes equations where the question of long time accuracy was open.

First-passage percolation on the square lattice is a random growth model in which each edge of Z^2 is assigned an i.i.d. nonnegative weight. The passage time between two points is the smallest total weight of a nearest-neighbor path connecting them, and a path achieving this minimum is called a geodesic. Typically, the number of edges in a geodesic is comparable to the Euclidean distance between its endpoints. However, when the edge-weights take the value 0 with probability exactly 1/2, a strikingly different behavior occurs: geodesics travel primarily on critical clusters of zero-weight edges, whose internal graph distance scales superlinearly with Euclidean distance. Determining the precise degree of this superlinear scaling is a challenging and ongoing endeavor. I will discuss recent progress on this front (joint with David Harper, Xiao Shen, and Evan Sorensen), along with complementary results on a dual problem, where we restrict path lengths and analyze passage times (joint with Jack Hanson and Daniel Slonim).