Periodicity and longtime diffusion for mean field systems in $\mathbb{R}^d$

TL;DR

This paper investigates the long-term dynamics of large finite populations modeled by stochastic differential equations, demonstrating their empirical measures remain near stable periodic solutions with a stochastic dephasing described by a Brownian motion.

Contribution

It establishes the convergence of empirical measures to periodic solutions and characterizes the stochastic dephasing in large populations.

Findings

Empirical measures stay close to stable periodic solutions.

Dephasing converges to a Brownian motion with drift.

Provides insights into long-time behavior of mean field systems.

Abstract

We study in this paper the longtime behavior of some large but finite populations of interacting stochastic differential equations whose (infinite population) limit Fokker-Planck PDE admits a stable periodic solution. We show that the empirical measure for the population of size $N$ stays close to the periodic solution, but with a random dephasing at the timescale $Nt$ that converges weakly to a Brownian motion with constant drift.