Recurrence properties of planar Lorentz process

Abstract

First-return and first-hitting times, local times, and first-intersection times are studied for planar finite-horizon Lorentz processes with a periodic configuration of scatterers. Their asymptotic behavior is analogous to the asymptotic behavior of the same quantities for the two-dimensional simple symmetric random walk (see classical results of Darling and Kac [DK] and Erdős and Taylor [ET]. Moreover, asymptotical distributions for phases in first hittings and in first intersections of Lorentz processes are also proved. The results are also extended to the quasi-one-dimensional model of the linear Lorentz process