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September 2020 Anomalous diffusion for multi-dimensional critical kinetic Fokker–Planck equations
Nicolas Fournier, Camille Tardif
Ann. Probab. 48(5): 2359-2403 (September 2020). DOI: 10.1214/20-AOP1426


We consider a particle moving in $d\geq2$ 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 $\beta\geq4+d$, 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.


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Nicolas Fournier. Camille Tardif. "Anomalous diffusion for multi-dimensional critical kinetic Fokker–Planck equations." Ann. Probab. 48 (5) 2359 - 2403, September 2020.


Received: 1 March 2019; Published: September 2020
First available in Project Euclid: 23 September 2020

MathSciNet: MR4152646
Digital Object Identifier: 10.1214/20-AOP1426

Primary: 35Q84 , 60F05 , 60J60

Keywords: anomalous diffusion phenomena , Bessel processes , central limit theorem , heavy-tailed equilibrium , Homogenization‎ , Kinetic diffusion process , kinetic Fokker–Planck equation , Local times , Stable processes

Rights: Copyright © 2020 Institute of Mathematical Statistics

Vol.48 • No. 5 • September 2020
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