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 (T-2 x {1, 2} x R-2), where T-2 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 integral(t)(0) v(K (s)) ds, where |v| similar to 1 for small k. We prove that the rescaled process (N In N)Y-1/2 (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.