Large-time behavior in non-symmetric Fokker-Planck equations

TL;DR

This paper studies the long-term behavior of three classes of non-symmetric Fokker-Planck equations, establishing exponential convergence to steady states and analyzing decay rates using a modified entropy method.

Contribution

It introduces a modified entropy method for non-symmetric Fokker-Planck equations, determines optimal decay rates, and analyzes spectral properties under perturbations.

Findings

Exponential convergence to steady states for all three classes.

Explicit decay rate estimates are provided.

Optimal decay rates are achieved and proven.

Abstract

We consider three classes of linear non-symmetric Fokker-Planck equations having a unique steady state and establish exponential convergence of solutions towards the steady state with explicit (estimates of) decay rates. First, "hypocoercive" Fokker-Planck equations are degenerate parabolic equations such that the entropy method to study large-time behavior of solutions has to be modified. We review a recent modified entropy method (for non-symmetric Fokker-Planck equations with drift terms that are linear in the position variable). Second, kinetic Fokker-Planck equations with non-quadratic potentials are another example of non-symmetric Fokker-Planck equations. Their drift term is nonlinear in the position variable. In case of potentials with bounded second-order derivatives, the modified entropy method allows to prove exponential convergence of solutions to the steady state. In this application of the modified entropy method symmetric positive definite matrices solving a matrix inequality are needed. We determine all such matrices achieving the optimal decay rate in the modified entropy method. In this way we prove the optimality of previous results. Third, we discuss the spectral properties of Fokker-Planck operators perturbed with convolution operators. For the corresponding Fokker-Planck equation we show existence and uniqueness of a stationary solution. Then, exponential convergence of all solutions towards the stationary solution is proven with an uniform rate.