Semi-deterministic processes with applications in random billiards

Abstract

We study the ergodic properties of two classes of random dynamical systems: a type of Markov chain which we call the \textit{alternating random walk} and a certain stochastic billiard system which describes the motion of a free-moving rough disk bouncing between two parallel rough walls. Our main results characterize the types of Markov transition kernels which make each system ergodic -- in the first case, with respect to uniform measure on the state space, and in the second case, with respect to Lambertian measure (a classic measure from geometric optics). In addition, building on results from \cite{rudzis2022}, we give explicit examples of rough microstructures which produce ergodic dynamics in the second system. Both systems have the property that the transition kernel governing the dynamics is singular with respect to uniform measure on the state space. As a result, these systems occupy a kind of mean in the problem space between diffusive processes, where establishing ergodicity is relatively easy, and physically realistic deterministic systems, where questions of ergodicity are far less approachable.