Asymptotic behaviour of randomly reflecting billiards in unbounded tubular domains

Menshikov, M.V.; Vachkovskaia, M.; Wade, A.R.

doi:10.1007/s10955-008-9578-z

Asymptotic behaviour of randomly reflecting billiards in unbounded tubular domains

Menshikov, M.V.; Vachkovskaia, M.; Wade, A.R.

Authors

Professor Mikhail Menshikov mikhail.menshikov@durham.ac.uk
Professor

M. Vachkovskaia

Professor Andrew Wade andrew.wade@durham.ac.uk
Professor

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.

Citation

Menshikov, M., Vachkovskaia, M., & Wade, A. (2008). Asymptotic behaviour of randomly reflecting billiards in unbounded tubular domains. Journal of Statistical Physics, 132(6), 1097-1133. https://doi.org/10.1007/s10955-008-9578-z

Journal Article Type	Article
Publication Date	Jan 1, 2008
Deposit Date	Mar 1, 2011
Publicly Available Date	Jan 31, 2013
Journal	Journal of Statistical Physics
Print ISSN	0022-4715
Electronic ISSN	1572-9613
Publisher	Springer
Peer Reviewed	Peer Reviewed
Volume	132
Issue	6
Pages	1097-1133
DOI	https://doi.org/10.1007/s10955-008-9578-z
Keywords	Stochastic billiards, Rarefied gas dynamics, Knudsen random walk, Random reflections, Recurrence/transience, Lamperti problem, Almost-sure bounds, Birth-and-death chain.