One dimensional critical Kinetic Fokker-Planck equations, Bessel and stable processes

We consider a particle moving in one dimension, its velocity being a reversible diffusion process, with constant diffusion coefficient, of which the invariant measure behaves like $(1+|v|)^{-\beta}$ for some $\beta>0$. We prove that, under a suitable rescaling, the position process resembles a Brownian motion if $\beta\geq 5$, a stable process if $\beta\in [1,5)$ and an integrated symmetric Bessel process if $\beta\in (0,1)$. The critical cases $\beta=1$ and $\beta=5$ require special rescaling. We recover some results of G.Lebeau and M.Puel [LP17], P.Cattiaux, E.Nasreddine and M. Puel [CNP16], and E.Barkai, E.Aghion and D. A. Kesslerwith [BAK14] with an alternative approach.