# Quenched invariance principle for the Knudsen stochastic billiard in a random tube - Mathematics > Probability

Quenched invariance principle for the Knudsen stochastic billiard in a random tube - Mathematics > Probability - Download this document for free, or read online. Document in PDF available to download.

Abstract: We consider a stochastic billiard in a random tube which stretches toinfinity in the direction of the first coordinate. This random tube isstationary and ergodic, and also it is supposed to be in some sense wellbehaved. The stochastic billiard can be described as follows: when strictlyinside the tube, the particle moves straight with constant speed. Upon hittingthe boundary, it is reflected randomly, according to the cosine law: thedensity of the outgoing direction is proportional to the cosine of the anglebetween this direction and the normal vector. We also consider thediscrete-time random walk formed by the particle-s positions at the moments ofhitting the boundary. Under the condition of existence of the second moment ofthe projected jump length with respect to the stationary measure for theenvironment seen from the particle, we prove the quenched invariance principlesfor the projected trajectories of the random walk and the stochastic billiard.

Author: ** Francis Comets, Serguei Popov, Gunter M. Schütz, Marina Vachkovskaia**

Source: https://arxiv.org/