Fractional diffusion as the limit of a short range potential Rayleigh gas

The fractional diffusion equation is rigorously derived as a scaling limit from a deterministic Rayleigh gas, where particles interact via short range potentials with support of size ε and the background is distributed in space R³ according to a Poisson process with intensity N and in velocity according some fat-tailed distribution. As an intermediate step a linear Boltzmann equation is obtained in the Boltzmann-Grad limit as ε tends to zero and N tends to infinity with N ε² =c. The convergence of the empiric particle dynamics to the Boltzmann-type dynamics is shown using semigroup methods to describe probability measures on collision trees associated to physical trajectories in the case of a Rayleigh gas. The fractional diffusion equation is a hydrodynamic limit for times t ∈ [0,T], where T and inverse mean free path c can both be chosen as some negative rational power ε⁻ᵏ.