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May 2010 Quenched invariance principle for the Knudsen stochastic billiard in a random tube
Francis Comets, Serguei Popov, Gunter M. Schütz, Marina Vachkovskaia
Ann. Probab. 38(3): 1019-1061 (May 2010). DOI: 10.1214/09-AOP504


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.


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Francis Comets. Serguei Popov. Gunter M. Schütz. Marina Vachkovskaia. "Quenched invariance principle for the Knudsen stochastic billiard in a random tube." Ann. Probab. 38 (3) 1019 - 1061, May 2010.


Published: May 2010
First available in Project Euclid: 2 June 2010

zbMATH: 1200.60091
MathSciNet: MR2674993
Digital Object Identifier: 10.1214/09-AOP504

Primary: 60K37
Secondary: 37D50 , 60J05 , 60J25

Keywords: Cosine law , invariance principle , Knudsen random walk , Random conductances , random medium , random walks in random environment , Stochastic homogenization

Rights: Copyright © 2010 Institute of Mathematical Statistics


Vol.38 • No. 3 • May 2010
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