Sharp uniform-in-time propagation of chaos

TL;DR

This paper establishes the optimal rate at which interacting diffusions converge to their mean-field limit uniformly over time, using entropy methods and log-Sobolev inequalities, for systems with convex interactions and on tori.

Contribution

It provides the first proof of uniform-in-time propagation of chaos at the optimal rate for certain interacting diffusion models, extending previous work to the time-uniform setting.

Findings

Convergence rate of $O((k/n)^2)$ for $k$-particle marginals

Uniform-in-time bounds on the distance between particle systems and limits

Application of entropy and log-Sobolev techniques to time-uniform propagation of chaos

Abstract

We prove the optimal rate of quantitative propagation of chaos, uniformly in time, for interacting diffusions. Our main examples are interactions governed by convex potentials and models on the torus with small interactions. We show that the distance between the $k$-particle marginal of the $n$-particle system and its limiting product measure is $O((k/n)^2)$, uniformly in time, with distance measured either by relative entropy, squared quadratic Wasserstein metric, or squared total variation. Our proof is based on an analysis of relative entropy through the BBGKY hierarchy, adapting prior work of the first author to the time-uniform case by means of log-Sobolev inequalities.