Invariant measures for a stochastic Fokker-Planck equation

Sylvain De Moor,,Irmar, Umr Cnrs 6625, Éns Rennes, Bruz, France , ,Luis Miguel Rodrigues,Julien Vovelle

doi:10.3934/krm.2018017

Abstract

We study a kinetic Vlasov/Fokker-Planck equation perturbed by a stochastic forcing term. When the noise intensity is not too large, we solve the corresponding Cauchy problem in a space of functions ensuring good localization in the velocity variable. Then we show under similar conditions that the generated dynamics, with prescribed total mass, admits a unique invariant measure which is exponentially mixing. The proof relies on hypocoercive estimates and hypoelliptic regularity. At last we provide an explicit example showing that our analytic framework does require some smallness condition on the noise intensity.

Highlights

We study a kinetic Vlasov/Fokker-Planck equation perturbed by a stochastic forcing term
We focus on the case where the singularity is of a very specific type, namely when it is given by a time-white noise
Our goal is to study the large-time dynamics induced by (1.1) and we stress that even in a purely deterministic setting (λ = 0) where the large-time dynamics is trivial — in the sense that it leads at exponential rate to the convergence towards a unique stationary solution — the quantitative analysis of simplest relevant models is a very recent achievement, see [8] and [7]

Summary

Introduction

We are interested in the large-time dynamics generated by the following stochastic FokkerPlanck equation (1.1). From a modeling point of view it would be more satisfactory to consider a force that instead of being purely singular would be the sum of a time-white noise and of a part that would be given as a smooth deterministic function of the density of the law This would result in a nonlinear stochastic partial differential equation. Though interesting from a modeling point of view, a space-time white noise seems too singular to be handled in (1.4) by currently available techniques in the analysis of stochastic partial differential equations. As we already pointed out, even for the Fokker-Planck operator acting on functions constant in space — an elliptic operator — some localization is needed to obtain global-in-time estimates. In Appendix A we gather some basic background material that may be skipped by the expert reader but may be useful to the reader unfamiliar with some of the crucial underlying concepts

Existence and uniqueness of solutions

E Πmgin 2 2 φ

Regularization and decay

L02 ds