The Vlasov-Fokker-Planck equation in non-convex landscapes: convergence to equilibrium

TL;DR

This paper investigates the long-term behavior of solutions to the Vlasov-Fokker-Planck equation with non-convex potentials, demonstrating convergence to equilibrium using a free-energy approach.

Contribution

It provides the first analysis of convergence to equilibrium for the Vlasov-Fokker-Planck equation in non-convex landscapes.

Findings

Solutions converge to an invariant probability measure.

Convergence is established under suitable assumptions.

The free-energy method is effective for non-convex potentials.

Abstract

In this paper, we study the long-time behaviour of solutions to the Vlasov-Fokker-Planck equation where the confining potential is non-convex. This is a nonlocal nonlinear partial differential equation describing the time evolution of the probability distribution of a particle moving under the influence of a non-convex potential, an interaction potential, a friction force and a stochastic force. Using the free-energy approach, we show that under suitable assumptions solutions of the Vlasov-Fokker-Planck equation converge to an invariant probability.