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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.

MSC:

37D50 Hyperbolic systems with singularities (billiards, etc.) (MSC2010)
60D05 Geometric probability and stochastic geometry
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