Abstract

We propose a unified method for the large space-time scaling limit of linear collisional kinetic equations in the whole space. The limit is of fractional diffusion type for heavy tail equilibria with slow enough decay, and of diffusive type otherwise. The proof is constructive and the fractional/standard diffusion matrix is obtained. The method combines energy estimates and quantitative spectral methods to construct a “fluid mode”. The method is applied to scattering models (without assuming detailed balance conditions), Fokker–Planck operators and Lévy–Fokker–Planck operators. It proves a series of new results, including the fractional diffusive limit for Fokker–Planck operators in any dimension, for which the formulas for the diffusion coefficient were not known, for Lévy–Fokker–Planck operators with general equilibria, and for scattering operators including some cases of infinite mass equilibria. It also unifies and generalises the results of previous papers with a quantitative method, and our estimates on the fluid approximation error also seem novel.

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