Brownian particle systems with singular interactions

Abstract: We present a class of Brownian interacting particle systems known as \textit{rank-based diffusions} and their alter ego, \textit{systems of competing Brownian particles}. The former originally appeared as a model in stochastic portfolio theory, while the latter model—obtained by considering the order statistics of the former—is related to skew-reflected Brownian motion. This talk will be mainly expository, describing for a broad audience the fundamental properties of these processes, including their associated stationary distributions. We will also discuss the infinite-particle versions of these models, where the stationarity structure is richer. As a representative calculation, we will show that the distribution of the lowest particle in equilibrium is often Gumbel or related. Finally, we will describe some of our results on the equilibrium fluctuations of a certain space-time random field associated with the infinite Atlas model (a prototypical model in the class of rank-based diffusions). These fluctuations have a scaling limit given by a two-parameter Gaussian process with explicit covariance structure, equivalently described as the solution to a certain stochastic partial differential equation (SPDE). As a result, tagged particles exhibit fluctuations that locally behave as fractional Brownian motion with Hurst parameter 1/4. This work is joint with Sayan Banerjee and Amarjit Budhiraja (UNC Chapel Hill).

machine learningprobabilitystatistics theory

Audience: researchers in the discipline

( paper )

Gothenburg statistics seminar

Series comments: Gothenburg statistics seminar is open to the interested public, everybody is welcome. It usually takes place in MVL14 (http://maps.chalmers.se/#05137ad7-4d34-45e2-9d14-7f970517e2b60, see specific talk). Speakers are asked to prepare material for 35 minutes excluding questions from the audience.