Convergence to equilibrium in Wasserstein distance for Fokker-Planck equations

TL;DR

This paper establishes conditions under which solutions to certain Fokker-Planck equations converge exponentially to equilibrium in Wasserstein distance, highlighting the diffusion term's role in the convergence rate.

Contribution

It introduces practical criteria linking functional inequalities to convergence rates, emphasizing the diffusion term’s contribution, a novel aspect in the analysis of Fokker-Planck equations.

Findings

Provides conditions for exponential convergence in Wasserstein distance.

Links functional inequalities to spectral gaps and convergence rates.

Quantifies the diffusion term's impact on convergence speed.

Abstract

We describe conditions on non-gradient drift diffusion Fokker-Planck equations for its solutions to converge to equilibrium with a uniform exponential rate in Wasserstein distance. This asymptotic behaviour is related to a functional inequality, which links the distance with its dissipation and ensures a spectral gap in Wasserstein distance. We give practical criteria for this inequality and compare it to classical ones. The key point is to quantify the contribution of the diffusion term to the rate of convergence, which to our knowledge is a novelty.