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 $d\geqslant 1$, in order to construct an eigen-solution for the Fokker-Planck operator with heavy tail equilibria, of the form $(1+|v|^2)^{-\frac{\beta}{2}}$, 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 [Koch, Nonlinearity, 2015]. 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 $\kappa(-\Delta_x)^{\frac{\beta-d+2}{6}}$ and a positive diffusion coefficient $\kappa$.