The Annals of Probability

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

Francis Comets, Serguei Popov, Gunter M. Schütz, and Marina Vachkovskaia

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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.

Article information

Source
Ann. Probab. Volume 38, Number 3 (2010), 1019-1061.

Dates
First available: 2 June 2010

Permanent link to this document
http://projecteuclid.org/euclid.aop/1275486187

Digital Object Identifier
doi:10.1214/09-AOP504

Zentralblatt MATH identifier
05757650

Mathematical Reviews number (MathSciNet)
MR2674993

Subjects
Primary: 60K37: Processes in random environments
Secondary: 37D50: Hyperbolic systems with singularities (billiards, etc.) 60J05: Discrete-time Markov processes on general state spaces 60J25: Continuous-time Markov processes on general state spaces

Keywords
Cosine law Knudsen random walk stochastic homogenization invariance principle random medium random conductances random walks in random environment

Citation

Comets, Francis; Popov, Serguei; Schütz, Gunter M.; Vachkovskaia, Marina. Quenched invariance principle for the Knudsen stochastic billiard in a random tube. The Annals of Probability 38 (2010), no. 3, 1019--1061. doi:10.1214/09-AOP504. http://projecteuclid.org/euclid.aop/1275486187.


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