# Quenched invariance principle for the Knudsen stochastic billiard in a random tube

Abstract : We consider a stochastic billiard in a random tube which stretches to infinity in the direction of the first coordinate. This random tube is stationary and ergodic, and also it is supposed to be in some sense well-behaved. The stochastic billiard can be described as follows: when strictly inside the tube, the particle moves straight with constant speed. Upon hitting the boundary, it is reflected randomly, according to the cosine law: the density of the outgoing direction is proportional to the cosine of the angle between this direction and the normal vector. We also consider the discrete-time random walk formed by the particle's positions at the moments of hitting the boundary. Under the condition of existence of the second moment of the projected jump length with respect to the stationary measure for the environment seen from the particle, we prove the quenched invariance principles for the projected trajectories of the random walk and the stochastic billiard.
Type de document :
Pré-publication, Document de travail
52 pages, 5 figures. 2008
Domaine :

https://hal.archives-ouvertes.fr/hal-00337910
Contributeur : Francis Comets <>
Soumis le : lundi 10 novembre 2008 - 10:42:36
Dernière modification le : mardi 30 mai 2017 - 01:07:52

### Identifiants

• HAL Id : hal-00337910, version 1
• ARXIV : 0811.0366

### Citation

Francis Comets, Serguei Popov, Gunter M. Schütz, Marina Vachkovskaia. Quenched invariance principle for the Knudsen stochastic billiard in a random tube. 52 pages, 5 figures. 2008. 〈hal-00337910〉

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