Stochastic Perturbations of Convex Billiards

TL;DR

This paper studies how random perturbations in the reflection angles of a convex billiard table affect the system's long-term behavior, proving conditions for ergodicity in the resulting stochastic process.

Contribution

It introduces a model of stochastic billiards with boundary perturbations and establishes uniform ergodicity for a broad class of these perturbations.

Findings

Markov chain is uniformly ergodic under certain perturbations

Ergodicity does not hold universally for all perturbation types

Provides conditions under which stochastic billiards mix rapidly

Abstract

We consider a strictly convex billiard table with $C^2$ boundary, with the dynamics subjected to random perturbations. Each time the billiard ball hits the boundary its reflection angle has a random perturbation. The perturbation distribution corresponds to the physical situation where either the scale of the surface irregularities is smaller than but comparable to the diameter of the reflected object, or the billiard ball is not perfectly rigid. We prove that for a large class of such perturbations the resulting Markov chain is uniformly ergodic, although this is not true in general.