Asymptotic behaviour of randomly reflecting billiards in unbounded tubular domains

TL;DR

This paper analyzes the long-term behavior of stochastic billiards in unbounded tubular domains, revealing conditions for recurrence or transience and establishing super-diffusive escape rates, with new bounds for related stochastic processes.

Contribution

It introduces a classification of stochastic billiards in unbounded domains and provides almost-sure asymptotic results, including super-diffusive escape rates, connecting to zero-drift stochastic processes.

Findings

Classified billiard processes into recurrent and transient cases.

Proved super-diffusive escape rates in the transient case.

Derived new bounds for one-dimensional stochastic processes with zero drift.

Abstract

We study stochastic billiards in infinite planar domains with curvilinear boundaries: that is, piecewise deterministic motion with randomness introduced via random reflections at the domain boundary. Physical motivation for the process originates with ideal gas models in the Knudsen regime, with particles reflecting off microscopically rough surfaces. We classify the process into recurrent and transient cases. We also give almost-sure results on the long-term behaviour of the location of the particle, including a super-diffusive rate of escape in the transient case. A key step in obtaining our results is to relate our process to an instance of a one-dimensional stochastic process with asymptotically zero drift, for which we prove some new almost-sure bounds of independent interest. We obtain some of these bounds via an application of general semimartingale criteria, also of some independent interest.