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+|v|).. as |v|.., for some constant .>0. We prove that for large times, after a suitable rescaling, the position process resembles a Brownian motion if ..4+d, a stable process if ..[d,4+d) and an integrated multi-dimensional generalization of a Bessel process if ..(d.2,d). The critical cases .=d, .=1+d and .=4+d require special rescalings.