On systems of particles in singular repulsive interaction in dimension one : log and Riesz gas

TL;DR

This paper establishes the first uniform in time propagation of chaos for one-dimensional particle systems with singular repulsive interactions, including Dyson Brownian motion, using novel coupling techniques.

Contribution

It introduces a new approach to prove uniform in time propagation of chaos for Riesz gases, extending previous weak results to strong, uniform ones.

Findings

Proved existence and uniqueness for Riesz gases.

Established uniform in time propagation of chaos.

Developed a coupling method for the analysis.

Abstract

In this article, we prove the first quantitative uniform in time propagation of chaos for a class of systems of particles in singular repulsive interaction in dimension one that contains the Dyson Brownian motion. We start by establishing existence and uniqueness for the Riesz gases, before proving propagation of chaos with an original approach to the problem, namely coupling with a Cauchy sequence type argument. We also give a general argument to turn a result of weak propagation of chaos into a strong and uniform in time result using the long time behavior and some bounds on moments, in particular enabling us to get a uniform in time version of the result of C\'epa-L\'epingle.