Convergence rates for the Vlasov-Fokker-Planck equation and uniform in time propagation of chaos in non convex cases

TL;DR

This paper establishes convergence rates for the Vlasov-Fokker-Planck equation in Wasserstein distance without requiring convexity, and demonstrates uniform in time propagation of chaos in non-convex scenarios using coupling methods.

Contribution

It introduces a novel approach to prove convergence and propagation of chaos for kinetic equations under non-convex potentials without convexity assumptions.

Findings

Proves contraction rate in Wasserstein distance for Vlasov-Fokker-Planck.

Achieves uniform in time propagation of chaos in non-convex settings.

Utilizes coupling methods adapted to the kinetic framework.

Abstract

We prove the existence of a contraction rate for Vlasov-Fokker-Planck equation in Wasserstein distance, provided the interaction potential is (locally) Lipschitz continuous and the confining potential is both Lipschitz continuous and greater than a quadratic function, thus requiring no convexity conditions. Our strategy relies on coupling methods suggested by A. Eberle adapted to the kinetic setting enabling also to obtain uniform in time propagation of chaos in a non convex setting.