Fractional diffusion for Fokker–Planck equation with heavy tail equilibrium: An à la Koch spectral method in any dimension
Résumé
In this paper, we extend the spectral method developed (Dechicha and Puel (2023)) to any dimension $d \geqslant 1$, in order to construct an eigen-solution for the Fokker–Planck operator with heavy tail equilibria, of the form $(1+|v|^2)^{−\beta/2}$ , in the range $\beta \in ] d , d+4 [$. 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 $\rho : = \int_{\mathbb{R}^d} f \mathrm{d}v$, with a fractional Laplacian $\kappa (-\Delta)^{(\beta−d+2)/6}$ and a positive diffusion coefficient $\kappa$.