Long-time behaviors of mean-field interacting particle systems related to McKean-Vlasov equations

TL;DR

This paper establishes uniform-in-time convergence, gradient estimates, and concentration inequalities for mean-field particle systems related to McKean-Vlasov equations, even with complex potentials, using novel coupling and cost functions.

Contribution

It provides explicit, uniform-in-time estimates for convergence and chaos propagation in mean-field systems with complex potentials, avoiding previous technical restrictions.

Findings

Explicit convergence rates in Wasserstein distance

Uniform-in-time propagation of chaos

Sharp concentration inequalities with explicit constants

Abstract

In this paper, we investigate gradient estimate of the Poisson equation and the exponential convergence in the Wasserstein metric $W_{1,d_{l^1}}$, uniform in the number of particles, and uniform-in-time propagation of chaos for the mean-field weakly interacting particle system related to McKean-Vlasov equation. By means of the known approximate componentwise reflection coupling and with the help of some new cost function, we obtain explicit estimates for those three problems, avoiding the technical conditions in the known results. Our results apply when the confinement potential $V$ has many wells, the interaction potential $W$ has bounded second mixed derivative $\nabla^2_{xy}W$ which should be not too big so that there is no phase transition. As an application, we obtain the concentration inequality of the mean-field interacting particle system with explicit and sharp constants, uniform in time. Several examples are provided to illustrate the results.