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

Abstract

In this paper, we extend the spectral method developed (Dechicha and Puel (2023)) to any dimension $\mathit{d}⩾1$ , in order to construct an eigen-solution for the Fokker–Planck operator with heavy tail equilibria, of the form ${\mathrm{(}1+|\mathit{v}{|}^2\mathrm{)}}^{-\frac{\mathit{\beta }}{2}}$ , in the range $\mathit{\beta }\in ]\mathit{d}\mathrm{,}\mathit{d}+4[$ . 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 $\mathit{\rho }:={\int }_{{\mathbb{R}}^{\mathit{d}}}\mathit{f}\mathrm{d}\mathit{v}$ , with a fractional Laplacian $\mathit{\kappa }{\mathrm{(}-\mathrm{\Delta }\mathrm{)}}^{\frac{\mathit{\beta }-\mathit{d}+2}{6}}$ and a positive diffusion coefficient κ.