Diffusion limit for kinetic Fokker-Planck equation with heavy tails equilibria: The critical case

Abstract

This paper is devoted to the diffusion and anomalous diffusion limit of the Fokker-Planck equation of plasma physics, in which the equilibrium function decays towards zero at infinity like a negative power function. We use probabilistic methods to recover and extend the results obtained in [22]. We prove in particular, in the critical case where the classical diffusion coefficient is no more defined, that the small mean free path limit gives rise to a diffusion equation, with an anomalous time scaling and with a variance breaking.