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

Francis Comets; Serguei Popov

ESAIM: Probability and Statistics (2010)

  • Volume: 8, page 56-65
  • ISSN: 1292-8100

Abstract

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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))}.

How to cite

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Comets, Francis, and Popov, Serguei. "A note on quenched moderate deviations for Sinai's random walk in random environment." ESAIM: Probability and Statistics 8 (2010): 56-65. <http://eudml.org/doc/104322>.

@article{Comets2010,
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))\}. },
author = {Comets, Francis, Popov, Serguei},
journal = {ESAIM: Probability and Statistics},
keywords = {Random walk in random environment; Sinai's regime; t-stable point; moderate deviations.; -stable point; moderate deviations},
language = {eng},
month = {3},
pages = {56-65},
publisher = {EDP Sciences},
title = {A note on quenched moderate deviations for Sinai's random walk in random environment},
url = {http://eudml.org/doc/104322},
volume = {8},
year = {2010},
}

TY - JOUR
AU - Comets, Francis
AU - Popov, Serguei
TI - A note on quenched moderate deviations for Sinai's random walk in random environment
JO - ESAIM: Probability and Statistics
DA - 2010/3//
PB - EDP Sciences
VL - 8
SP - 56
EP - 65
AB - 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))}.
LA - eng
KW - Random walk in random environment; Sinai's regime; t-stable point; moderate deviations.; -stable point; moderate deviations
UR - http://eudml.org/doc/104322
ER -

References

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  1. F. Comets and S.Yu. Popov, Limit law for transition probabilities and moderate deviations for Sinai's random walk in random environment. Probab. Theory Relat. Fields 126 (2003) 571-609.  Zbl1027.60091
  2. A. Greven and F. den Hollander, Large deviations for a random walk in random environment. Ann. Probab.22 (1994) 1381-1428.  Zbl0820.60054
  3. Y. Hu and Z. Shi, The limits of Sinai's simple random walk in random environment. Ann. Probab.26 (1998) 1477-1521.  Zbl0936.60088
  4. 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  Zbl1066.60096
  5. B. Hughes, Random Walks and Random Environments. Vol. 2. Random Environments. The Clarendon Press, Oxford University Press, New York (1996).  Zbl0925.60076
  6. 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).  Zbl1031.60088
  7. A. Shiryaev, Probability. 2nd edn., Springer, New York (1989).  
  8. Ya.G. Sinai, The limiting behavior of one-dimensional random walk in random medium. Theory Probab. Appl.27 (1982) 256-268.  
  9. O. Zeitouni, Lecture Notes on Random Walks in Random Environment. (2003) Preliminary version at www-ee.technion.ac.il/~zeitouni/ps/notes1.ps  

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