Convergence of a Kinetic Equation to a Fractional Diffusion Equation

#### G. Basile, A. Bovier

2010, v.16, Issue 1, 15-44

ABSTRACT

A linear Boltzmann equation is interpreted as the forward equation for the probability density of a Markov process $(K(t)$, $Y(t))$ on $(\t\times\r)$, 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 $\int_0^t v(K(s)) ds$, where $|v|\sim 1$ for small $k$. We prove that the rescaled process $N^{-2/3}Y(Nt)$ converges in distribution to a symmetric Levy process, stable with index $\alpha=3/2$.

Keywords: anomalous diffusion,Levy process,Boltzmann equation,coupledoscillators,kinetic limit,heat conductance