Ergodicity of some stochastic Fokker-Planck equations with additive common noise

TL;DR

This paper studies stochastic Fokker-Planck equations with common noise, establishing conditions for the uniqueness of invariant measures and analyzing the impact of common noise on system ergodicity.

Contribution

It provides new sufficient conditions ensuring the stochastic Fokker-Planck equation has a unique invariant measure despite multiple measures in the deterministic case.

Findings

Unique invariant measure exists with common noise under certain conditions.

System is attracted to its conditional mean given the common noise.

Small idiosyncratic noise intensity aids in ergodicity.

Abstract

In this paper we consider stochastic Fokker-Planck Partial Differential Equations (PDEs), obtained as the mean-field limit of weakly interacting particle systems subjected to both independent (or idiosyncratic) and common Brownian noises. We provide sufficient conditions under which the deterministic counterpart of the Fokker-Planck equation, which corresponds to particle systems that are just subjected to independent noises, has several invariant measures, but for which the stochastic version admits a unique invariant measure under the presence of the additive common noise. The very difficulty comes from the fact that the common noise is just of finite dimension while the state variable, which should be seen as the conditional marginal law of the system given the common noise, lives in a space of infinite dimension. In this context, our result holds true if, in addition to standard confining properties, the mean field interaction term forces the system to be attracted by its conditional mean given the common noise and the intensity of the idiosyncratic noise is small.