Anomalous diffusion for multi-dimensional critical Kinetic Fokker-Planck equations

Abstract

We consider a particle moving in $d\ge 2$ dimensions, its velocity being a reversible diffusion process, with identity diffusion coefficient, of which the invariant measure behaves, roughly, like $(1+|v|{)}^{-\beta }$ as $|v|\to \infty$, for some constant $\beta >0$. We prove that for large times, after a suitable rescaling, the position process resembles a Brownian motion if $\beta \ge 4+d$, a stable process if $\beta \in [d,4+d)$ and an integrated multi-dimensional generalization of a Bessel process if $\beta \in (d-2,d)$. The critical cases $\beta =d$, $\beta =1+d$ and $\beta =4+d$ require special rescalings.

40 pages