The distance between the Kac process and the Wiener process with applications to generalized telegraph equations

Abstract

In this note we obtain a local central limit theorem and an expansion of length two for the Kac processYα(t) that describes the position of a particle at timet after collisions. In particular, we obtain a rate of convergence for the distance of total variation for the distributions oft−1/2Yα(t) and the Wiener process at timet. The results apply to the probabilistic solutions of abstract telegraph and heat equations which heavily rely on the Kac and Wiener processes. Under very mild assumptions we establish a rate of convergence for a singular perturbation problem of an abstract heat equation.