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.
Similar content being viewed by others
References
Babovsky, H.: On Knudsen flows within thin tubes. J. Stat. Phys. 44, 865–878 (1986)
Brézis, H., Rosenkrantz, W., Singer, B.: An extension of Khintchine’s estimate for large deviations to a class of Markov chains converging to a singular diffusion. Commun. Pure Appl. Math. 24, 705–726 (1971)
Cercignani, C.: The Boltzmann Equation and Its Applications. Springer, New York (1988)
Comets, F., Menshikov, M., Popov, S.: Lyapunov functions for random walks and strings in random environment. Ann. Probab. 26, 1433–1445 (1998)
Comets, F., Popov, S., Schütz, G.M., Vachkovskaia, M.: Billiards in a general domain with random reflections. Arch. Ration. Mech. Anal. (2008). Available at arXiv.org as math.PR/0612799. DOI: 10.1007/s00205-008-0120-x
Csáki, E., Földes, A., Révész, P.: Transient NN random walk on the line. Preprint (2007). Available at arXiv.org as math.PR/0707.0734
Dvoretzky, A., Erdős, P.: Some problems on random walk in space. In: Proceedings of the Second Berkeley Symposium on Mathematical Statistics and Probability, 1950, pp. 353–367. University of California Press, Berkeley/Los Angeles (1951)
Esposti, M.D., Magno, G.D., Lenci, M.: An infinite step billiard. Nonlinearity 11, 991–1013 (1998)
Evans, S.N.: Stochastic billiards on general tables. Ann. Appl. Probab. 11, 419–437 (2001)
Fal’, A.M.: Certain limit theorems for an elementary Markov random walk. Ukrainian Math. J. 33, 433–435 (1981). Translated from Ukrain. Mat. Zh. 33, 564–566 (in Russian)
Fayolle, G., Malyshev, V.A., Menshikov, M.V.: Topics in the Constructive Theory of Countable Markov Chains. Cambridge University Press, Cambridge (1995)
Gallardo, L.: Comportement asymptotique des marches aléatoires associées aux polynomes de Gegenbauer et applications. Adv. Appl. Probab. 16, 293–323 (1984) (in French)
Harris, T.E.: First passage and recurrence distributions. Trans. Am. Math. Soc. 73, 471–486 (1952)
Hodges, J.L. Jr., Rosenblatt, M.: Recurrence-time moments in random walks. Pac. J. Math. 3, 127–136 (1953)
Karlin, S., McGregor, J.: Random walks. Ill. J. Math. 3, 66–81 (1959)
Knudsen, M.: Kinetic Theory of Gases: Some Modern Aspects. Methuen’s Monographs on Physical Subjects. Methuen, London (1952)
Lamperti, J.: Criteria for the recurrence and transience of stochastic processes I. J. Math. Anal. Appl. 1, 314–330 (1960)
Lamperti, J.: A new class of probability limit theorems. J. Math. Mech. 11, 749–772 (1962)
Lamperti, J.: Criteria for stochastic processes II: passage-time moments. J. Math. Anal. Appl. 7, 127–145 (1963)
Lenci, M.: Escape orbits for non-compact flat billiards. Chaos 6, 428–431 (1996)
Lenci, M.: Semidispersing billiards with an infinite cusp I. Commun. Math. Phys. 230, 133–180 (2002)
Lenci, M.: Semidispersing billiards with an infinite cusp II. Chaos 13, 105–111 (2003)
Menshikov, M.V., Asymont, I.M., Iasnogorodskii, R.: Markov processes with asymptotically zero drifts. Probl. Inform. Trans. 31, 248–261 (1995). Translated from Problemy Peredachi Informatsii 31 60–75 (in Russian)
Menshikov, M.V., Popov, S.Yu.: Exact power estimates for countable Markov chains. Markov Process. Relat. Fields 1, 57–78 (1995)
Menshikov, M.V., Wade, A.R.: Logarithmic speeds for one-dimensional perturbed random walk in random environment. Stoch. Process. Appl. 118, 389–416 (2008)
Rosenkrantz, W.A.: A local limit theorem for a certain class of random walks. Ann. Math. Stat. 37, 855–859 (1966)
Székely, G.J.: On the asymptotic properties of diffusion processes. Ann. Univ. Sci. Budapest. Eötvös Sect. Math. 17, 69–71 (1974)
Tabachnikov, S.: Billiards. Société Mathématique de France, Paris (1995)
Voit, M.: A law of the iterated logarithm for a class of polynomial hypergroups. Monatsh. Math. 109, 311–326 (1990)
Voit, M.: Strong laws of large numbers for random walks associated with a class of one-dimensional convolution structures. Monatsh. Math. 113, 59–74 (1992)
Voit, M.: A law of the iterated logarithm for Markov chains on ℕ0 associated with orthogonal polynomials. J. Theor. Probab. 6, 653–669 (1993)
Williams, D.: Probability with Martingales. Cambridge University Press, Cambridge (1991)
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Menshikov, M.V., Vachkovskaia, M. & Wade, A.R. Asymptotic Behaviour of Randomly Reflecting Billiards in Unbounded Tubular Domains. J Stat Phys 132, 1097–1133 (2008). https://doi.org/10.1007/s10955-008-9578-z
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10955-008-9578-z