From a kinetic equation to a diffusion under an anomalous scaling

A linear Boltzmann equation is interpreted as the forward equation for the probability density of a Markov process (K(t ), i(t ),Y (t)) on (T2 ×{1, 2} × ℝ2 ), where T2 is the two-dimensional torus. Here (K(t ), i(t)) is an autonomous reversible jump process, with waiting times between two jumps with finite expectation value but infinite variance. Y(t) is an additive functional of K , defined as (Formula presented). ds , where |v| ∼ 1 for small k . We prove that the rescaled process (N lnN) -1/2 Y(Nt) converges in distribution to a two-dimensional Brownian motion. As a consequence, the appropriately rescaled solution of the Boltzmann equation converges to the solution of a diffusion equation.