Diffusion limit for kinetic Fokker-Planck equation with heavy tails equilibria: The critical case

Patrick Cattiaux; Elissar Nasreddine; Marjolaine Puel

doi:10.3934/krm.2019028

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2019, Volume 12, Issue 4: 727-748. Doi: 10.3934/krm.2019028

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Diffusion limit for kinetic Fokker-Planck equation with heavy tails equilibria: The critical case

1.
Institut de Mathématiques de Toulouse. Université de Toulouse. CNRS UMR 5219, 118 route de Narbonne, F-31062 Toulouse cedex 09, France
2.
Mathematics Department, Lebanese University, Faculty of Sciences (I), Hadath, Lebanon
3.
Laboratoire Dieudonné. Université de Nice Sophia Antipolis, Parc Valrose, F-06108 Nice cedex 2, France

^* Corresponding author: Marjolaine Puel
^* Corresponding author: Marjolaine Puel

Received: December 31, 2017

Revised: September 30, 2018

Published: May 2019

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Abstract

This paper is devoted to the diffusion and anomalous diffusion limit of the Fokker-Planck equation of plasma physics, in which the equilibrium function decays towards zero at infinity like a negative power function. We use probabilistic methods to recover and extend the results obtained in [22]. We prove in particular, in the critical case where the classical diffusion coefficient is no more defined, that the small mean free path limit gives rise to a diffusion equation, with an anomalous time scaling and with a variance breaking.

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Mathematics Subject Classification: Primary: 35K10, 35K65, 60F05, 60J55.

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