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

Comets, Francis, and Popov, Serguei. "Ballistic regime for random walks in random environment with unbounded jumps and Knudsen billiards." Annales de l'I.H.P. Probabilités et statistiques 48.3 (2012): 721-744. <http://eudml.org/doc/272065>.

@article{Comets2012,
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 $\mathbb \{R\}^\{d\}$, $d\ge 3$, 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.},
author = {Comets, Francis, Popov, Serguei},
journal = {Annales de l'I.H.P. Probabilités et statistiques},
keywords = {cosine law; stochastic billiard; Knudsen random walk; random medium; random walk in random environment; unbounded jumps; stationary ergodic environment; regenerative structure; point of view of the particle},
language = {eng},
number = {3},
pages = {721-744},
publisher = {Gauthier-Villars},
title = {Ballistic regime for random walks in random environment with unbounded jumps and Knudsen billiards},
url = {http://eudml.org/doc/272065},
volume = {48},
year = {2012},
}

TY - JOUR
AU - Comets, Francis
AU - Popov, Serguei
TI - Ballistic regime for random walks in random environment with unbounded jumps and Knudsen billiards
JO - Annales de l'I.H.P. Probabilités et statistiques
PY - 2012
PB - Gauthier-Villars
VL - 48
IS - 3
SP - 721
EP - 744
AB - 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 $\mathbb {R}^{d}$, $d\ge 3$, 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.
LA - eng
KW - cosine law; stochastic billiard; Knudsen random walk; random medium; random walk in random environment; unbounded jumps; stationary ergodic environment; regenerative structure; point of view of the particle
UR - http://eudml.org/doc/272065
ER -