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A note on quenched moderate deviations for Sinai’s random walk in random environment

Francis CometsSerguei Popov — 2004

ESAIM: Probability and Statistics

We consider the continuous time, one-dimensional random walk in random environment in Sinai’s regime. We show that the probability for the particle to be, at time t and in a typical environment, at a distance larger than t a ( 0 < a < 1 ) from its initial position, is exp { - Const · t a / [ ( 1 - a ) ln t ] ( 1 + o ( 1 ) ) } .

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

Francis CometsSerguei Popov — 2012

Annales de l'I.H.P. Probabilités et statistiques

We consider a random walk in a stationary ergodic environment in , 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 d , d 3 , which serves as environment....

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