ANOMALOUS DIFFUSION FOR MULTI-DIMENSIONAL CRITICAL KINETIC FOKKER-PLANCK EQUATIONS

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