A criterion for proving entropy chaos on path space

TL;DR

This paper introduces a criterion to establish entropy chaos in interacting diffusion systems, focusing on conservative diffusions with time-reversal symmetry, linking entropy bounds and weak convergence to process-level chaos.

Contribution

It provides a new criterion connecting entropy bounds and weak convergence to strong entropy chaos for a broad class of conservative diffusions.

Findings

Entropy chaos can be proved under bounded relative entropy and weak convergence.

The framework applies to various singular interacting particle systems.

Time-reversal symmetry is key to the analysis.

Abstract

A criterion for proving a strong form of propagation of chaos on the path space, known as entropy chaos, for a general interacting diffusion system is proposed. Our analysis focuses on the class of conservative diffusions introduced by Carlen, which are characterized by infinitesimal characteristic pairs, that is, a time-marginal probability density and a current velocity field. A key property of this broad class is that the processes remain diffusions under time-reversal. We prove that, given a suitable bound on the relative entropy (with respect to the Wiener measure) and the weak convergence of both drifts and fixed-time marginal densities, strong entropy chaos at the process level is achieved in the infinite particle limit, provided the limit drift satisfies a specific regularity condition. This stochastic framework encompasses various singular interacting particle systems and their related asymptotic scenarios.