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.
"The Vlasov-Fokker-Planck equation in non-convex landscapes: convergence to equilibrium." Electron. Commun. Probab. 23 1 - 10, 2018. https://doi.org/10.1214/18-ECP116