Convergence to equilibrium for the kinetic Fokker-Planck equation on the torus

TL;DR

This paper proves exponential convergence to equilibrium for the kinetic Fokker-Planck equation on the torus using stochastic differential equations and Wasserstein distance, and explores coupling methods affecting convergence bounds.

Contribution

It establishes exponential convergence in Wasserstein distance for the kinetic Fokker-Planck equation on the torus and analyzes the limitations of co-adapted couplings.

Findings

Exponential convergence in Wasserstein $$ distance.

Coupling bounds depend on the square root of initial distance.

Limitations of co-adapted couplings for this convergence.

Abstract

We study convergence to equilibrium for the kinetic Fokker-Planck equation on the torus. Solving the stochastic differential equation, we show exponential convergence in the Monge-Kantorovich-Wasserstein $\mathcal{W}_2$ distance. Finally, we investigate if such a coupling can be obtained by a co-adapted coupling, and show that then the bound must depend on the square root of the initial distance.