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

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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.


Original languageEnglish
Article number19
JournalElectronic Communications in Probability
Publication statusPublished - 15 Mar 2018


  • Kinetic equation, Vlasov-Fokker-Planck equation, Free-energy, Asymptotic behaviour, Granular media equation, Stochastic processes