Long Time Behavior of Fokker-Planck Equations

Abstract

We consider the large-time behavior of Fokker-Planck type equations (advection-diffusion equations). These equations describe the evolution of a cloud of particles undergoing diffusion. There is a unique equilibrium state for these equations, and solutions are supposed to converge to it as time goes to infinity. When these equations are non-degenerate (i.e., the diffusion matrix is positive definite), their large-time behavior is comprehensively studied, for example, one can use the well-known Bakry-Emery theory. However, degenerate Fokker-Planck equations are challenging and there are many open problems. We develop a method for proving the convergence of solutions to the equilibrium state for degenerate Fokker-Planck equations, and, in particular, we apply this method to the kinetic Fokker-Planck equation and the Vlasov-Poisson-Fokker-Planck system. Our method lets us obtain explicit and constructive estimates on the rate of convergence and it is based on construction of appropriate Lyapunov functionals.