Ballistic regime for random walks in random environment with unbounded jumps and Knudsen billiards

TL;DR

This paper establishes a law of large numbers with positive speed for random walks in ergodic environments with unbounded jumps and applies these results to analyze Knudsen billiards with drift in random tubes, demonstrating asymptotic behavior.

Contribution

It proves the law of large numbers for random walks with unbounded jumps under strong transience and extends these results to Knudsen billiards with drift in random environments.

Findings

Law of large numbers with positive speed for the random walk

Ergodicity of the environment viewed from the particle

Law of large numbers for the stochastic billiard with drift

Abstract

We consider a random walk in a stationary ergodic environment in $\mathbb Z$, with unbounded jumps. In addition to uniform ellipticity and a bound on the tails of the possible jumps, we assume a condition of strong transience to the right which implies that there are no "traps". We prove the law of large numbers with positive speed, as well as the ergodicity of the environment seen from the particle. Then, we consider Knudsen stochastic billiard with a drift in a random tube in ${\mathbb R}^d$, $d\geq 3$, which serves as environment. The tube is infinite in the first direction, and is a stationary and ergodic process indexed by the first coordinate. A particle is moving in straight line inside the tube, and has random bounces upon hitting the boundary, according to the following modification of the cosine reflection law: the jumps in the positive direction are always accepted while the jumps in the negative direction may be rejected. Using the results for the random walk in random environment together with an appropriate coupling, we deduce the law of large numbers for the stochastic billiard with a drift.