Anomalous diffusion for multi-dimensional critical kinetic Fokker–Planck equations

We consider a particle moving in $d\geq 2$ 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 $\beta\geq 4+d$, 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.