Abstract
We consider a one-dimensional random walk in random environment in the Sinai's regime. Our main result is that logarithms of the transition probabilities, after a suitable rescaling, converge in distribution as time tends to infinity, to some functional of the Brownian motion. We compute the law of this functional when the initial and final points agree. Also, among other things, we estimate the probability of being at time t at distance at least z from the initial position, when z is larger than ln2 t, but still of logarithmic order in time.
References
Bovier, A., Eckhoff, M., Gayrard, V., Klein, M.: Metastability in stochastic dynamics of disordered mean-field models. Probab. Theory Relat. Fields 119, 99–161 (2001)
Comets, F., Gantert, N., Zeitouni, O.: Quenched, annealed and functional large deviations for one dimensional random walk in random environment. Probab. Theory Relat. Fields 118, 65–114 (2000)
Comets, F., Menshikov, M.V., Popov, S.Yu.: Lyapunov functions for random walks and strings in random environment. Ann. Probab. 26(4), 1433–1445 (1998)
Freidlin, M.I., Wentzell, A.D.: Random Perturbations of Dynamical Systems. Springer- Verlag, New York. Second edition 1998, 1984
Golosov, A.O.: Localization of random walks in one-dimensional random environments. Commun. Math. Phys. 92, 491–506 (1984)
Greven, A., den~Hollander, F.: Large deviations for a random walk in random environment. Ann. Probab. 22, 1381–1428 (1994)
Hu, Y.: Tightness of localization and return time in random environment. Stochastic Process. Appl. 86, 81–101 (2000)
Hu, Y., Shi, Z.: The limits of Sinai's simple random walk in random environment. Ann. Probab. 26(4), 1477–1521 (1998)
Hu, Y., Shi, Z.: The local time of simple random walk in random environment. J. Theoret. Probab. 11, 765–793 (1998)
Hughes, B.: Random Walks and Random Environments. Vol. 2. Random Environments. The Clarendon Press, Oxford University Press, New York, 1996
Kesten, H.: The limit distribution of Sinai's random walk in random environment. Physica A 138, 299–309 (1986)
Komlós, J., Major, P., Tusnády, G.: An approximation of partial sums of independent RV's and the sample DF. I. Z. Wahrsch. Verw. Gebiete 32, 111–131 (1975)
Laloux, L., Le Doussal, P.: Aging and diffusion in low-dimensional environments. Phys. Rev. E (3) 57, 6296–6326 (1998)
Mathieu, P.: Zero white noise limit through Dirichlet forms, with application to diffusions in a random media. Probab. Theory Relat. Fields 99, 549–580 (1994)
Miclo, L.: An example of application of discrete Hardy's inequalities. Markov Process. Relat. Fields 5, 319–330 (1999)
Revuz, D., Yor, M.: Continuous Martingales and Brownian Motion. Springer, Berlin, 1999
Rogers, L., Williams, D.: Diffusions, Markov Processes, and Martingales. Vol. 2. J. Wiley & Sons, New York, 1987
Saloff-Coste, L.: Lectures on finite markov chains. Lectures on probability theory and statistics (Saint-Flour, 1996), 301–413, Lecture Notes in Math., 1665, Springer, Berlin, 1997
Shi, Z.: Sinai's walk via stochastic calculus. In: Milieux Aléatoires (F. Comets, E. Pardoux, eds.) Panoramas et Synthèses, 12, Société Mathématique de France, Paris, 2001
Sinai, Ya.G.: The limiting behavior of one-dimensional random walk in random medium. Theory Probab. Appl. 27, 256–268 (1982)
Solomon, F.: Random walks in a random environment. Ann. Probab. 3, 1–31 (1975)
Zeitouni, O.: Lecture notes on random walks in random environment, 2002. Preliminary version is available at www-ee.technion.ac.il/˜zeitouni/ ps/notes1.ps
Author information
Authors and Affiliations
Corresponding author
Additional information
Partially supported by CNRS (UMR 7599 ``Probabilités et Modéles Aléatoires''), and by the ``Réseau Mathématique France-Brésil''
Partially supported by CNPq (300676/00–0 and 302981/02–0), COFECUB, and by the ``Rede Matemática Brasil-França''
Rights and permissions
About this article
Cite this article
Comets, F., Popov, S. Limit law for transition probabilities and moderate deviations for Sinai's random walk in random environment. Probab. Theory Relat. Fields 126, 571–609 (2003). https://doi.org/10.1007/s00440-003-0273-3
Received:
Revised:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00440-003-0273-3
Keywords
- Random environment
- Sinai's regime
- Elevation
- Moments of return
- t-stable points
- Spectral gap
- Metastability