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Quenched invariance principle for the Knudsen stochastic billiard in a random tube. (English) Zbl 1200.60091

The authors 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. They 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, the authors prove the quenched invariance principles for the projected trajectories of the random walk and the stochastic billiard.

MSC:

60K37 Processes in random environments
37D50 Hyperbolic systems with singularities (billiards, etc.) (MSC2010)
60J05 Discrete-time Markov processes on general state spaces
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