Convergenceof a kinetic equation to a fractional diffusion equation

ABSTRACT. A linear Boltzmann equation is interpreted as the forward equation for the probability density of a Markov process (K(t), Y (t)) on (T × R), where T is the one-dimensional torus. K(t) is a 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 ∫ t 0 v(K(s))ds, where |v | ∼ 1 for small k. We prove that the rescaled process N−2/3Y (Nt) converge in distribution to a symmetric Lévy process, stable with index α = 3/2. 1. INTRODUCTION. The understanding of thermal conductance in both classical and quantum me-chanical systems is one of the fundamental problems of non-equilibrium statistical mechanics. A particular aspect that has attracted much interest is the observation that autonomous translation invariant systems in dimensions one and two exhibi