Abstract
This paper is devoted to the diffusion approximation for the 1-d Fokker Planck equation with a heavy tail equilibria of the form \({(1+v^2)^{-\beta/2}}\), in the range \({\beta\in ]1,5[}\). We prove that the limit diffusion equation involves a fractional Laplacian \({\kappa\vert \Delta\vert^{\frac{\beta+1}{6}}}\), and we compute the value of the diffusion coefficient \({\kappa}\). This extends previous results of Nasreddine and Puel (ESAIM: M2AN 49(1):1, 2015) in the case \({\beta > 5}\), and of Cattiaux et al. (Diffusion limit for kinetic Fokker–Planck equation with heavy tails equilibria: the critical case, preprint, 2016) in the case \({\beta=5}\).
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Communicated by C. Mouhot
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Gilles Lebeau was supported by the European Research Council, ERC-2012-ADG, Project Number 320845: Semi Classical Analysis of Partial Differential Equations.
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Lebeau, G., Puel, M. Diffusion Approximation for Fokker Planck with Heavy Tail Equilibria: A Spectral Method in Dimension 1. Commun. Math. Phys. 366, 709–735 (2019). https://doi.org/10.1007/s00220-019-03315-9
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DOI: https://doi.org/10.1007/s00220-019-03315-9