Purdue Probability Seminar, Fall 2026

In work by Hairer, rough path theory is used to construct solutions to the stochastic Burgers type equation and to establish their regularity. In later work by Hairer, Maas, and Weber, the authors prove that spatial discretizations of the equation converge with rate up to 1/6, which is suboptimal. In the talk, I will discuss how these properties are affected when the classical derivative is replaced by fractional derivatives.

We show that the supremum of a canonical process under a given marginal index distribution can be characterized by an integral of the rate-distortion function of the index distribution across various scales. This gives a sharp two-sided version of Dudley's integral by replacing the log covering number by the rate-distortion function from information theory. As applications, we provide new streamlined proofs of the majorizing measure theorem and the Bednorz-Latala theorem (formerly the Bernoulli conjecture). Furthermore, the value of entropic optimal transport can be sharply characterized by a truncated rate-distortion integral. (Based on arXiv:2508.18682, 2604.14061, and 2608.11031).

It is a striking and recurrent theme in probability and physics that seemingly different microscopic mechanisms can produce the same universal statistical laws. Even more surprising is that mathematical structures invented for one purpose often resurface, in unexpected ways, in seemingly distant fields. In this talk, I will give a glimpse of these two manifestations of universality through the study of random interfaces. I will use the KPZ universality class as means of demonstrating integrability in probability and the Critical 2d Stochastic Heat Flow to demonstrate emerging structures at criticality. I will try to give emphasis on discussing how these objects connect such seemingly diverse mathematical fields as stochastic analysis, stochastic PDEs, statistical mechanics, integrable systems, representation theory, algebraic combinatorics and number theoretic objects.

Two dimensional abelian Yang–Mills–Higgs is perhaps the simplest gauge theory that does not enjoy exact solvability. The first construction of this model was carried out by Brydges, Fröhlich, and Seiler using phase space expansion techniques. Recently, Bringmann and Cao proved global-in-time well-posedness for the corresponding Langevin dynamic; this result yields the finite volume measure via tightness bounds. I will describe how to construct the finite volume measure directly using a stochastic control representation called the Boué–Dupuis formula. This is joint work with Nikolay Barashkov, Ilya Chevyrev, Andreas Koller, and Abdulwahab Mohamed.

Cookie random walks are a type of self-interacting random walk. In the most basic model, an $(M,p)$-cookie random walk, M cookies of strength p are placed at each site on Z. When a random walker reaches a site with a cookie, he steps right with probability $p>1/2$ and eats the cookie there. At sites with no cookies, the walker steps left and right with equal probability. Pinsky proposed a variation in which only one cookie is placed at each site, but the random walker only eats it after stepping left from that site. He was able to compute the speed of this walk for small values of p and sufficiently large values of p using martingale arguments. To close the gap he left open, we study a certain branching process. We calculate the mean of this process's stationary distribution, obtaining formula for the speed of both Pinsky's random walk and other related walks as a consequence. For a further application of our methods, we obtain an explicit (but not particularly attractive) formula for the speed of $(3,p)$-cookie random walk. We discuss how beautifying this formula is equivalent to establishing an asymptotic expansion of several orders for certain generalized Hypergeometric-like functions.