Diffusion approximation for Fokker Planck with heavy tail equilibria : a\n spectral method in dimension 1

2017/11/08 by Gilles Lebeau, Lebeau, Gilles, Marjolaine Puel +1
Chemistry · Mathematics · #Analysis of PDEs (math.AP) #Diffusion Coefficients in Liquids #FOS: Mathematics #Fractional Differential Equations Solutions #Nonlinear Partial Differential Equations

paper · pdf · doi:10.48550/arxiv.1711.03060

openalex publication_date 2017/11/08 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

This paper is devoted to the diffusion approximation for the 1-d Fokker\nPlanck equation with a heavy tail equilibria of the form (1+v²)^-\β/2, in\nthe range beta\∈ ]1,5[. We prove that the limit diffusion equation involves a\nfractional Laplacian kappa|\Δ|^\(\β+1)/(6), and we compute the\nvalue of the diffusion coefficient kappa. This extends previous results of E.\nNasreddine and M. Puel in the case beta>5, and of P. Cattiaux, E. Nasreddine\nand M. Puel in the case beta=5.\n

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