STOCHASTIC BILLIARDS WITH MARKOVIAN REFLECTIONS IN GENERALIZED PARABOLIC DOMAINS

• 成果类型: Article
• 署名作者: Da Costa, Conrado; Menshikov, Mikhail, V; Wade, Andrew r.
• 署名单位: Durham University
• 刊物名称: ANNALS OF APPLIED PROBABILITY
• ISSN/ISSBN: 1050-5164
• DOI: 10.1214/23-AAP1952
• 发表日期: 2023
• 页码: 5459-5496
• 关键词: nonhomogeneous random-walks

摘要

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 transform-ing the stochastic billiards model to a Markov chain on a half-strip R+ x 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.