Archive for Rational Mechanics and Analysis

, Volume 191, Issue 3, pp 497–537

Billiards in a General Domain with Random Reflections

  • Francis Comets
  • Serguei Popov
  • Gunter M. Schütz
  • Marina Vachkovskaia
Article
  • 100 Downloads

Abstract

We study stochastic billiards on general tables: a particle moves according to its constant velocity inside some domain \(\fancyscript{D}\subset {\mathbb{R}}^d\) until it hits the boundary and bounces randomly inside, according to some reflection law. We assume that the boundary of the domain is locally Lipschitz and almost everywhere continuously differentiable. The angle of the outgoing velocity with the inner normal vector has a specified, absolutely continuous density. We construct the discrete time and the continuous time processes recording the sequence of hitting points on the boundary and the pair location/velocity. We mainly focus on the case of bounded domains. Then, we prove exponential ergodicity of these two Markov processes, we study their invariant distribution and their normal (Gaussian) fluctuations. Of particular interest is the case of the cosine reflection law: the stationary distributions for the two processes are uniform in this case, the discrete time chain is reversible though the continuous time process is quasi-reversible. Also in this case, we give a natural construction of a chord “picked at random” in \(\fancyscript{D}\) , and we study the angle of intersection of the process with a (d − 1)-dimensional manifold contained in \(\fancyscript{D}\) .

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

Copyright information

© Springer-Verlag 2008

Authors and Affiliations

  • Francis Comets
    • 1
  • Serguei Popov
    • 2
  • Gunter M. Schütz
    • 3
  • Marina Vachkovskaia
    • 4
  1. 1.UFR de MathématiquesUniversité Paris 7Paris Cedex 05France
  2. 2.Instituto de Matemática e EstatísticaUniversidade de São PauloSão PauloBrasil
  3. 3.Institut für FestkörperforschungForschungszentrum Jülich GmbHJülichDeutschland
  4. 4.Instituto de MatemáticaEstatística e Computação CientíficaCampinasBrasil

Personalised recommendations