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

Author(s)

Comets, Francis ^[1] ; Popov, Serguei ^[2]

Affiliation

^[1] Univ Paris 07, UFR Math, F-75205 Paris 13 - France

^[2] Univ Campinas UNICAMP, Inst Math Stat & Sci Computat, Dept Stat, BR-13083859 Campinas, SP - Brazil

• Document type: Journal article
• Source: ANNALES DE L INSTITUT HENRI POINCARE-PROBABILITES ET STATISTIQUES; v. 48, n. 3, p. 721-744, AUG 2012.
• Web of Science Citations: 5

Abstract

We consider a random walk in a stationary ergodic environment in 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 R-d, d >= 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. (AU)

• FAPESP's process: 09/08665-6 - Random walks on trees and branching random walks
• Grantee: Serguei Popov
• Support Opportunities: Regular Research Grants