Seminar

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We study a nonlocal approximation of the standard and fractional Fokker-Planck equation in which we can estimate the speed of convergence to equilibrium in a way which does not degenerate in the scaling limit of the equation. This uniform estimate cannot be easily obtained with standard inequalities or entropy methods, but can be obtained through the use of Harris’s theorem, finding interesting links to quantitative versions of the central limit theorem in probability. As a consequence one can prove that solutions of this nonlocal approximation converge to those of the standard (fractional) Fokker-Planck equation uniformly in time. The associated equilibrium has some interesting properties, which we also study. In the last part of the talk we discuss some extensions to the non-autonomous setting, motivated by a self-similar type change of variable for nonlocal diffusion and growth fragmentation equations.