Stochastic billiards with Markovian reflections in generalized parabolic domains

TL;DR

This paper analyzes the recurrence and transience of a stochastic billiard model with Markovian boundary reflections in unbounded, generalized parabolic domains, extending previous work to more complex boundary conditions.

Contribution

It introduces a new classification of recurrence for stochastic billiards with Markovian reflections in domains with unbounded directions, generalizing earlier finite-state results.

Findings

Recurrence characterized by reflection kernel and domain growth rate.

Developed recurrence classification for near-critical regimes with Lamperti-type drifts.

Extended previous models to include asymptotically Markov boundary components.

Abstract

We study recurrence and transience for a particle that moves at constant velocity in the interior of an unbounded planar domain, with random reflections at the boundary governed by a Markov kernel producing outgoing angles from incoming angles. Our domains have a single unbounded direction and sublinear growth. We characterize recurrence in terms of the reflection kernel and growth rate of the domain. The results are obtained by transforming the stochastic billiards model to a Markov chain on a half-strip $\mathbb{R}_+ \!\times S$ where $S$ is a compact set. We develop the recurrence classification for such processes in the near-critical regime in which drifts of the $\mathbb{R}_+$ component are of generalized Lamperti type, and the $S$ component is asymptotically Markov; this extends earlier work that dealt with finite $S$.