Exponential convergence to equilibrium for kinetic Fokker-Planck equations

TL;DR

This paper proves exponential convergence to equilibrium for a class of kinetic Fokker-Planck equations with geometric assumptions, extending results to the relativistic case at low temperature.

Contribution

It introduces a geometric framework to analyze convergence rates and applies the method to relativistic Fokker-Planck equations, previously lacking such results.

Findings

Global solutions converge exponentially to equilibrium.

The approach uses a modified entropy functional and geometric assumptions.

Results include the relativistic Fokker-Planck equation at low temperature.

Abstract

A class of linear kinetic Fokker-Planck equations with a non-trivial diffusion matrix and with periodic boundary conditions in the spatial variable is considered. After formulating the problem in a geometric setting, the question of the rate of convergence to equilibrium is studied within the formalism of differential calculus on Riemannian manifolds. Under explicit geometric assumptions on the velocity field, the energy function and the diffusion matrix, it is shown that global regular solutions converge in time to equilibrium with exponential rate. The result is proved by estimating the time derivative of a modified entropy functional, as recently proposed by Villani. For spatially homogeneous solutions the assumptions of the main theorem reduce to the curvature bound condition for the validity of logarithmic Sobolev inequalities discovered by Bakry and Emery. The result applies to the relativistic Fokker-Planck equation in the low temperature regime, for which exponential trend to equilibrium was previously unknown.