Skip to main content

Research Repository

Advanced Search

Asymptotic behaviour of randomly reflecting billiards in unbounded tubular domains

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

Asymptotic behaviour of randomly reflecting billiards in unbounded tubular domains Thumbnail


Authors

M. Vachkovskaia



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.

Files

Accepted Journal Article (382 Kb)
PDF

Copyright Statement
The original publication is available at www.springerlink.com





You might also like



Downloadable Citations