Billiards in a General Domain with Random Reflections

Abstract

We study stochastic billiards on general tables: a particle moves according to its constant velocity inside some domain $\fancyscript{D}\subset {\mathbb{R}}^d$ until it hits the boundary and bounces randomly inside, according to some reflection law. We assume that the boundary of the domain is locally Lipschitz and almost everywhere continuously differentiable. The angle of the outgoing velocity with the inner normal vector has a specified, absolutely continuous density. We construct the discrete time and the continuous time processes recording the sequence of hitting points on the boundary and the pair location/velocity. We mainly focus on the case of bounded domains. Then, we prove exponential ergodicity of these two Markov processes, we study their invariant distribution and their normal (Gaussian) fluctuations. Of particular interest is the case of the cosine reflection law: the stationary distributions for the two processes are uniform in this case, the discrete time chain is reversible though the continuous time process is quasi-reversible. Also in this case, we give a natural construction of a chord “picked at random” in $\fancyscript{D}$ , and we study the angle of intersection of the process with a (d − 1)-dimensional manifold contained in $\fancyscript{D}$ .