Annales de l'Institut Henri Poincaré, Probabilités et Statistiques

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

Francis Comets and Serguei Popov

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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 $\mathbb{R}^{d}$, $d\geq3$, 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.

Résumé

Nous considérons une marche aléatoire dans un milieu stationnaire ergodique sur $\mathbb{Z}$, avec des sauts non bornés. En plus de l’uniforme ellipticité et d’une borne uniforme sur la queue de la loi des sauts, nous supposons une condition de transience forte qui garantit l’absence de “pièges.” Nous montrons la loi des grands nombres avec vitesse strictement positive, ainsi que l’ergodicité de l’environnement vu de la particule. Par ailleurs, nous étudions aussi le billard stochastique de Knudsen avec dérive dans un tube aléatoire dans $\mathbb{R}^{d}$, $d\geq3$, qui constitue l’environnement. Le tube est infini dans la première direction, et, vu comme un processus indéxé par la première coordonnée, il est supposé stationnaire ergodique. Une particule se déplace en ligne droite à l’intérieur du tube, avec des rebonds aléatoires sur le bord, selon la modification suivante de la loi de reflexion en cosinus: les sauts dans la direction positive sont toujours acceptés, tandis que ceux dans l’autre direction peuvent être rejetés. En utilisant les résultats pour la marche aléatoire en milieu aléatoire et un couplage approprié, nous obtenons la loi des grands nombres pour le billard stochastique avec dérive.

Article information

Source
Ann. Inst. H. Poincaré Probab. Statist., Volume 48, Number 3 (2012), 721-744.

Dates
First available in Project Euclid: 26 June 2012

Permanent link to this document
https://projecteuclid.org/euclid.aihp/1340714870

Digital Object Identifier
doi:10.1214/11-AIHP439

Mathematical Reviews number (MathSciNet)
MR2976561

Zentralblatt MATH identifier
1247.60139

Subjects
Primary: Primary 60K37 secondary 37D50 60J25: Continuous-time Markov processes on general state spaces

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

Citation

Comets, Francis; Popov, Serguei. Ballistic regime for random walks in random environment with unbounded jumps and Knudsen billiards. Ann. Inst. H. Poincaré Probab. Statist. 48 (2012), no. 3, 721--744. doi:10.1214/11-AIHP439. https://projecteuclid.org/euclid.aihp/1340714870


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