Fractional diffusion for Fokker–Planck equation with heavy tail equilibrium: An à la Koch spectral method in any dimension

Article type: Research Article

Authors: Dechicha, Dahmane^{a; *} | Puel, Marjolaine^b

Affiliations: [a] Laboratoire J.-A. Dieudonné, Université Côte d’Azur, UMR 7351, Parc Valrose, 06108 Nice Cedex 02, France | [b] Laboratoire de recherche AGM, CY Cergy Paris Université, UMR CNRS 8088, 2 Avenue Adolphe Chauvin, 95302 Cergy-Pontoise Cedex, France

Correspondence: [*] Corresponding author. E-mail: [email protected].

Abstract: In this paper, we extend the spectral method developed (Dechicha and Puel (2023)) to any dimension d⩾1, in order to construct an eigen-solution for the Fokker–Planck operator with heavy tail equilibria, of the form (1+|v|2)−β2, in the range β∈]d,d+4[. The method developed in dimension 1 was inspired by the work of H. Koch on nonlinear KdV equation (Nonlinearity 28 (2015) 545). The strategy in this paper is the same as in dimension 1 but the tools are different, since dimension 1 was based on ODE methods. As a direct consequence of our construction, we obtain the fractional diffusion limit for the kinetic Fokker–Planck equation, for the correct density ρ:=∫Rdfdv, with a fractional Laplacian κ(−Δ)β−d+26 and a positive diffusion coefficient κ.

Keywords: Kinetic Fokker–Planck equation, Fokker–Planck operator, heavy-tailed equilibrium, anomalous diffusion, fractional diffusion, spectral theory, eigen-solutions

DOI: 10.3233/ASY-231870

Journal: Asymptotic Analysis, vol. 136, no. 2, pp. 79-132, 2024