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

ESAIM: Probability and Statistics (2010)

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

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topComets, 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

top- 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
- A. Greven and F. den Hollander, Large deviations for a random walk in random environment. Ann. Probab.22 (1994) 1381-1428. Zbl0820.60054
- Y. Hu and Z. Shi, The limits of Sinai's simple random walk in random environment. Ann. Probab.26 (1998) 1477-1521. Zbl0936.60088
- 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
- B. Hughes, Random Walks and Random Environments. Vol. 2. Random Environments. The Clarendon Press, Oxford University Press, New York (1996). Zbl0925.60076
- 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
- A. Shiryaev, Probability. 2nd edn., Springer, New York (1989).
- Ya.G. Sinai, The limiting behavior of one-dimensional random walk in random medium. Theory Probab. Appl.27 (1982) 256-268.
- 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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