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 $$\beta \ge 5$$ , 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 rescalings. We recover some results of Nasreddine and Puel (ESAIM Math Model Numer Anal 49:1–17, 2015), Cattiaux et al. (Kinet Relat Models, to appear), Lebeau and Puel (Commun Math Phys, to appear. arXiv:1711.03060 ) and Barkai et al. (Phys Rev X 4:021036, 2014) on the kinetic Fokker–Planck equation, with an alternative approach.