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

TL;DR

This paper proves a quenched invariance principle for a stochastic billiard in a stationary ergodic random tube, demonstrating that the particle's scaled trajectory converges to Brownian motion under certain conditions.

Contribution

It establishes the quenched invariance principle for both the stochastic billiard and the associated boundary hitting random walk in a random tube, extending previous results to this setting.

Findings

Quenched invariance principle proven for the stochastic billiard

Invariance principle also holds for the boundary hitting random walk

Results depend on the existence of a second moment of the projected jump length

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.