Large-time behavior of degenerate Fokker-Planck type equations

par Gayrat Toshpulatov

mercredi 31 janv. 2024, 14:00 → 15:00 Europe/Paris

n this talk 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 (which is degenerate).  Our method lets us obtain  explicit and constructive  estimates on the rate of convergence and it is based on  construction of appropriate Lyapunov functionals.