##
**Billiards in a general domain with random reflections.**
*(English)*
Zbl 1186.37049

Arch. Ration. Mech. Anal. 191, No. 3, 497-537 (2009); erratum 193, No. 3, 737-738 (2009).

Stochastic billiards on general tables are studied in this paper and a rigorous mathematical foundation is established for them. A particle is assumed to move with constant velocity inside a bounded d-dimensional domain and the bounces of the particle with the boundary occur according to some postulated reflection law. The boundary of the domain is locally Lipschitz and corresponds to an almost everywhere continuously differentiable function. With these assumptions, the authors then construct probabilistic models for the motion of the particle. In particular, discrete and continuous time processes recording the bounces are investigated. The authors further show that these are Markov processes with exponential ergodicity. Their invariant distributions and discrepancy from the Gaussian normal distribution are also obtained. Specific attention is given to the case of the cosine reflection law and the corresponding distributions are determined to be uniform in this case. The motion of the particle is viewed as a random walk and a variety of probabilistic, measure theoretic (such as Hausdorff measures) and geometric methods (such as random chords) are used as key ingredients in this paper.

Reviewer: Emre Alkan (Istanbul)

### MSC:

37D50 | Hyperbolic systems with singularities (billiards, etc.) (MSC2010) |

60D05 | Geometric probability and stochastic geometry |

### Keywords:

stochastic billiards; reflection laws; locally Lipschitz domains, Markov processes; random walks; random chords; Hausdorff measures; exponential ergodicity
PDF
BibTeX
XML
Cite

\textit{F. Comets} et al., Arch. Ration. Mech. Anal. 191, No. 3, 497--537 (2009; Zbl 1186.37049)

### References:

[1] | Bardos, C.; Golse, F.; Colonna, J.-F., Diffusion approximation and hyperbolic automorphisms of the torus, Phys. D, 104, 32-60, (1997) · Zbl 0902.35088 |

[2] | Billingsley P. (1995) Probability and Measure. Wiley, New York · Zbl 0822.60002 |

[3] | Boatto, S.; Golse, F., Diffusion approximation of a Knudsen gas model: dependence of the diffusion constant upon the boundary condition, Asymptot. Anal., 31, 93-111, (2002) · Zbl 1216.76063 |

[4] | Borovkov, K. A., On a new variant of the Monte Carlo method, Theory Probab. Appl., 36, 355-360, (1991) · Zbl 0776.65067 |

[5] | Borovkov, K. A., On simulation of random vectors with given densities in regions and on their boundaries, J. Appl. Probab., 31, 205-220, (1994) · Zbl 0804.65004 |

[6] | Brzank, A.; Schütz, G. M., Amplification of molecular traffic control in catalytic grains with novel channel topology design, J. Chem. Phys., 124, 214701, (2006) |

[7] | Caprino, S.; Pulvirenti, M., The Boltzmann-Grad limit for a one-dimensional Boltzmann equation in a stationary state, Commun. Math. Phys., 177, 63-81, (1996) · Zbl 0852.76080 |

[8] | Cercignani C. (1988) The Boltzmann Equation and its Applications. Springer, New York · Zbl 0646.76001 |

[9] | Coppens, M.-O.; Dammers, A. J., Effects of heterogeneity on diffusion in nanopores, From inorganic materials to protein crystals and ion channels. Fluid Phase Equilib., 241, 308-316, (2006) |

[10] | Coppens, M.-O.; Malek, K., Dynamic Monte-Carlo simulations of diffusion limited reactions in rough nanopores, Chem. Eng. Sci., 58, 4787-4795, (2003) |

[11] | Evans, S. N., Stochastic billiards on general tables, Ann. Appl. Probab., 11, 419-437, (2001) · Zbl 1015.60058 |

[12] | Federer H. (1969) Geometric Measure Theory. Springer, New York · Zbl 0176.00801 |

[13] | Feres, R.; Yablonsky, G., Knudsen’s cosine law and random billiards, Chem. Eng. Sci., 59, 1541-1556, (2004) |

[14] | Feres, R.: Random Walks Derived from Billiards. Preprint, 2006 · Zbl 1145.37009 |

[15] | Goldstein, S.; Kipnis, C.; Ianiro, N., Stationary states for a mechanical system with stochastic boundary conditions, J. Statist. Phys., 41, 915-939, (1985) · Zbl 0642.60059 |

[16] | Jaynes, E. T., The well-posed problem, Found. Phys., 3, 477-493, (1973) |

[17] | Keil, F. J.; Krishna, R.; Coppens, M.-O., Modeling of diffusion in zeolites, Rev. Chem. Eng., 16, 71, (2000) |

[18] | Knudsen M. (1952) Kinetic Theory of Gases—Some Modern Aspects. Methuen’s Monographs on Physical Subjects, London · JFM 60.1420.10 |

[19] | Lalley, S.; Robbins, H., Stochastic search in a convex region, Probab. Theory Relat. Fields, 77, 99-116, (1988) · Zbl 0617.60086 |

[20] | Lalley, S.; Robbins, H., Asymptotically minimax stochastic search strategies in the plane, Proc. Nat. Acad. Sci. USA, 84, 2111-2112, (1987) · Zbl 0613.90108 |

[21] | Malek, K.; Coppens, M.-O., Effects of surface roughness on self- and transport diffusion in porous media in the Knudsen regime, Phys. Rev. Lett., 87, 125505, (2001) |

[22] | Meyn S.P., Tweedie R.L. (1993) Markov Chains and Stochastic Stability. Springer, Berlin · Zbl 0925.60001 |

[23] | Morgan F. (1988) Geometric Measure. Theory A beginners guide. Academic Press, San Diego · Zbl 0671.49043 |

[24] | Roberts A.W., Varberg D.E. (1973) Convex Functions. Academic Press, New York · Zbl 0271.26009 |

[25] | Romeijn, H. E., A general framework for approximate sampling with an application to generating points on the boundary of bounded convex regions, Statistica Neerlandica, 52, 42-59, (1998) · Zbl 0961.60503 |

[26] | Russ S., Zschiegner S., Bunde A., Kärger J. Lambert diffusion in porous media in the Knudsen regime: equivalence of self- and transport diffusion. Phys. Rev. E72 030101(R) (2005) |

[27] | Stoyan D., Kendall W.S., Mecke J. (1987) Stochastic Geometry and its Applications. Wiley, Chichester · Zbl 0622.60019 |

[28] | Thorisson H. (2000) Coupling, Stationarity, and Regeneration. Springer, Berlin · Zbl 0949.60007 |

This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.