Peter Rudzis: Brownian particle systems with singular interactions

Abstract: The focus of this talk is a class of Brownian interacting particle systems on the real line known as \textit{systems of competing Brownian particles}. The interactions in this model are singular, determined by the collision local times and a certain `asymmetry' parameter $p \in [0,1]$. The symmetric ($p = 1/2$) version of this model describes the order statistics of rank-based diffusions, a type of particle system first appearing in stochastic portfolio theory to model the market capitalizations of stocks. Asymmetric versions, on the other hand, are related to last passage percolation and to certain eigenvalue processes in random matrix theory.

The first part of this talk will be mainly expository, describing for a broad audience the fundamental properties of these particle systems, including associated stationary distributions and connections to skew-reflected Brownian motion, and also discussing the infinite-particle versions, where the inter-particle gaps exhibit infinite families of product-form stationary distributions. Second, we will describe the equilibrium fluctuations of the associated counting processes of the particle systems. In the symmetric setting ($p = 1/2$), the fluctuations have a scaling limit given by a two-parameter Gaussian process with explicit covariance structure, equivalently described as the solution to a certain SPDE. As a result, tagged particles exhibit fluctuations that locally behave as fractional Brownian motion with Hurst parameter 1/4. In the weakly asymmetric setting ($p = 1/2 - \epsilon$), we establish well-posedness of the infinite system of SDEs describing the model and show that when the particles are started from a homogeneous stationary distribution, the fluctuations converge to a solution to the KPZ equation. This work is joint with Sayan Banerjee and Amarjit Budhiraja (UNC Chapel Hill).