Stochastic billiards with Markovian reflections in generalized parabolic domains

da Costa, Conrado; Menshikov, Mikhail V.; Wade, Andrew R.

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 sub-linear 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 R+ × 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 R+ component are of generalized Lamperti type, and the S component is asymptotically Markov; this extends earlier work that dealt with finite S

Citation

da Costa, C., Menshikov, M. V., & Wade, A. R. (2023). Stochastic billiards with Markovian reflections in generalized parabolic domains. Annals of Applied Probability, 33(6B), 5459-5496. https://doi.org/10.1214/23-AAP1952

• Journal Article Type: Article
• Acceptance Date: Mar 11, 2023
• Online Publication Date: Dec 13, 2023
• Publication Date: 2023-12
• Deposit Date: Aug 9, 2021
• Publicly Available Date: Dec 13, 2023
• Journal: Annals of Applied Probability
• Print ISSN: 1050-5164
• Publisher: Institute of Mathematical Statistics
• Peer Reviewed: Peer Reviewed
• Volume: 33
• Issue: 6B
• Pages: 5459-5496
• DOI: https://doi.org/10.1214/23-AAP1952
• Public URL: https://durham-repository.worktribe.com/output/1237971
• Related Public URLs: https://doi.org/10.48550/arXiv.2107.13976