Invariant Measures for a Stochastic Fokker-Planck Equation

TL;DR

This paper investigates a stochastic kinetic Fokker-Planck equation, establishing existence, uniqueness, and exponential mixing of invariant measures under small noise conditions using hypocoercivity techniques.

Contribution

It provides the first rigorous analysis of invariant measures for a stochastic Fokker-Planck equation with a Vlasov force, including explicit conditions on noise intensity.

Findings

Existence of solutions for small noise levels

Unique invariant measure with exponential mixing

Explicit example demonstrating noise restriction necessity

Abstract

We study the kinetic Fokker-Planck equation perturbed by a stochastic Vlasov force term. When the noise intensity is not too large, we solve the Cauchy Problem in a class of well-localized (in velocity) functions. We also show that, when the noise intensity is sufficiently small, the system with prescribed mass admits a unique invariant measure which is exponentially mixing. The proof uses hypocoercive decay estimates and hypoelliptic gains of regularity. At last we also exhibit an explicit example showing that some restriction on the noise intensity is indeed required.