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

Published online by Cambridge University Press:  15 September 2004

Francis Comets  and
Serguei Popov
Francis Comets
Affiliation:
Université Paris 7, UFR de Mathématiques, case 7012, 2, place Jussieu, 75251 Paris Cedex 05, France; e-mail: Partially supported by CNRS (UMR 7599 “Probabilités et Modèles Aléatoires”) and by the “Réseau Mathématiques France-Brésil”.
Serguei Popov
Affiliation:
Instituto de Matemática e Estatística, Universidade de São Paulo, Rua do Matão 1010, CEP 05508–090, São Paulo SP, Brasil; e-mail: Partially supported by CNPq (302981/02–0), and by the “Rede Matemática Brasil-França".
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Abstract

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 ta (0<a<1) from its initial position, is exp{-Const ⋅ ta/[(1 - a)lnt](1 + o(1))}.

Keywords

Random walk in random environmentSinai's regimet-stable pointmoderate deviations.
Type
Research Article
Information
ESAIM: Probability and Statistics , Volume 8 , August 2004 , pp. 56 - 65
Copyright
© EDP Sciences, SMAI, 2004

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References

Comets, F. and Popov, S.Yu., Limit law for transition probabilities and moderate deviations for Sinai's random walk in random environment. Probab. Theory Relat. Fields 126 (2003) 571-609. CrossRef
Greven, A. and den Hollander, F., Large deviations for a random walk in random environment. Ann. Probab. 22 (1994) 1381-1428. CrossRef
Hu, Y. and Shi, Z., The limits of Sinai's simple random walk in random environment. Ann. Probab. 26 (1998) 1477-1521.
Y. Hu and Z. Shi, Moderate deviations for diffusions with Brownian potentials. (2003) Preprint PMA–792 available at www.proba.jussieu.fr/mathdoc/preprints/index.html#2003
B. Hughes, Random Walks and Random Environments. Vol. 2. Random Environments. The Clarendon Press, Oxford University Press, New York (1996).
Z. Shi, Sinai's Walk via Stochastic Calculus, in Milieux Aléatoires, F. Comets and E. Pardoux Eds., Société Mathématique de France, Paris, Panoramas et Synthèses 12 (2001).
A. Shiryaev, Probability. 2nd edn., Springer, New York (1989).
Sinai, Ya.G., The limiting behavior of one-dimensional random walk in random medium. Theory Probab. Appl. 27 (1982) 256-268. CrossRef
O. Zeitouni, Lecture Notes on Random Walks in Random Environment. (2003) Preliminary version at www-ee.technion.ac.il/~zeitouni/ps/notes1.ps