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One Dimensional Critical Kinetic Fokker–Planck Equations, Bessel and Stable Processes

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Abstract

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.

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Correspondence to Camille Tardif.

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Communicated by C. Mouhot

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We warmly thank Quentin Berger for illuminating discussions. This research was supported by the French ANR-17- CE40-0030 EFI.

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Fournier, N., Tardif, C. One Dimensional Critical Kinetic Fokker–Planck Equations, Bessel and Stable Processes. Commun. Math. Phys. 381, 143–173 (2021). https://doi.org/10.1007/s00220-020-03903-0

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