The Annals of Probability
- Ann. Probab.
- Volume 38, Number 3 (2010), 1019-1061.
Quenched invariance principle for the Knudsen stochastic billiard in a random tube
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.
Ann. Probab. Volume 38, Number 3 (2010), 1019-1061.
First available in Project Euclid: 2 June 2010
Permanent link to this document
Digital Object Identifier
Mathematical Reviews number (MathSciNet)
Zentralblatt MATH identifier
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
Comets, Francis; Popov, Serguei; Schütz, Gunter M.; Vachkovskaia, Marina. Quenched invariance principle for the Knudsen stochastic billiard in a random tube. Ann. Probab. 38 (2010), no. 3, 1019--1061. doi:10.1214/09-AOP504. http://projecteuclid.org/euclid.aop/1275486187.