Anomalous diffusion for multi-dimensional critical Kinetic Fokker-Planck equations

2018/12/17 by Fournier, Nicolas, Tardif, Camille
#FOS: Mathematics #Probability (math.PR)

paper · doi:10.48550/arxiv.1812.06806

Abstract

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

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