Uniform-in-time estimates on corrections to mean field for interacting Brownian particles

TL;DR

This paper establishes uniform-in-time estimates for corrections to mean field in interacting Brownian particles, covering both Langevin and overdamped dynamics, using Lions expansions and new diagrammatic tools.

Contribution

It introduces novel diagrammatic methods and ergodic estimates for the linearized mean-field equations, advancing the understanding of propagation of chaos.

Findings

Uniform-in-time estimates on many-particle correlation functions

A new ergodic estimate for the linearized Vlasov-Fokker-Planck equation

A uniform-in-time quantitative central limit theorem and concentration results

Abstract

We consider a system of classical Brownian particles interacting via a smooth long-range potential in the mean-field regime, and we analyze the propagation of chaos in form of sharp, uniform-in-time estimates on many-particle correlation functions. Our results cover both the kinetic Langevin setting and the corresponding overdamped Brownian dynamics. The approach is mainly based on so-called Lions expansions, which we combine with new diagrammatic tools to capture many-particle cancellations, as well as with fine ergodic estimates on the linearized mean-field equation, and with discrete stochastic calculus with respect to initial data. In the process, we derive some new ergodic estimates for the linearized Vlasov-Fokker-Planck kinetic equation that are of independent interest. Our analysis also leads to a uniform-in-time quantitative central limit theorem and to uniform-in-time concentration estimates for the empirical measure associated with the particle dynamics.