Uniform-in-Time Estimates on the Size of Chaos for Interacting Particle Systems

TL;DR

This paper provides uniform-in-time bounds on the correlation functions of weakly interacting particle systems with bounded kernels, highlighting the role of diffusion in controlling chaos and enabling fluctuation analysis.

Contribution

It introduces new uniform-in-time estimates for correlation functions in particle systems, improving understanding of chaos propagation under varying kernel conditions.

Findings

Uniform bounds on correlation functions for large diffusion coefficients

Removal of initial data dependence under restrictive kernel conditions

Facilitation of fluctuation analysis around mean-field limits

Abstract

For any weakly interacting particle system with bounded kernel, we give uniform-in-time estimates of the $L^2$ norm of correlation functions, provided that the diffusion coefficient is large enough. When the condition on the kernels is more restrictive, we can remove the dependence of the lower bound for diffusion coefficient on the initial data and estimate the size of chaos in a weaker sense. Based on these estimates, we may study fluctuation around the mean-field limit.