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.
Keywords
transport process, kinetic theory, anomalous diffusion,
scattering operator, Fokker–Planck operator,
Lévy–Fokker–Planck operator, spectral theory