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

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 + vertical bar v vertical bar(2))(-beta/2), in the range beta is an element of] 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 := integral(Rd) f dv, with a fractional Laplacian kappa(-Delta)(beta-d+/6) and a positive diffusion coefficient kappa.