Dynamical Systems/Probability Theory
Behavior of random walks on in Gibbsian medium
Comptes Rendus. Mathématique, Volume 338 (2004) no. 11, pp. 895-898.

We consider random walks on in a random medium with {−L,…,−1,0,+1} as possible jumps, where L⩾1 is fixed. When the environment is defined by a Gibbs measure on a subshift of finite type, we show a dichotomy in the recurrent case between the pointwise functional CLT and the slow behavior described by Sinaï. In the transient cases and under natural integrability conditions, we prove the validity of the averaged CLT.

Nous étudions des marches aléatoires sur en milieu aléatoire avec {−L,…,−1,0,+1} comme sauts possibles, où L⩾1 est fixé. Pour un environnement défini par une mesure de Gibbs sur un sous-shift de type fini, nous montrons dans le cas récurrent une dichotomie entre le TCL fonctionnel ponctuel et le comportement lent décrit par Sinaï. Dans les cas transients et sous des conditions d'intégrabilité naturelles, nous montrons la validité du TCL en moyenne.

Received:
Accepted:
Published online:
DOI: 10.1016/j.crma.2004.03.030
Bremont, Julien 1

1 CMLA – E.N.S. de Cachan, 61, avenue du Président Wilson, 94235 Cachan, France
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Bremont, Julien. Behavior of random walks on $ \mathbb{Z}$ in Gibbsian medium. Comptes Rendus. Mathématique, Volume 338 (2004) no. 11, pp. 895-898. doi : 10.1016/j.crma.2004.03.030. http://www.numdam.org/articles/10.1016/j.crma.2004.03.030/

[1] Baladi, V. Positive Transfer Operators and Decay of Correlations, Adv. Ser. Nonlinear Dynam., vol. 16, World Scientific, 2000

[2] Physics in One Dimension (Bernasconi, J.; Schneider, T., eds.), Springer-Verlag, Berlin, 1981

[3] Bowen, R. Equilibrium States and the Ergodic Theory of Anosov Diffeomorphisms, Lecture Notes in Math., vol. 470, Springer-Verlag, Berlin, 1975

[4] Brémont, J. On some random walks on in random medium, Ann. Probab., Volume 30 (2002) no. 3, pp. 1266-1312

[5] J. Brémont, Thèse de doctorat, Marches aléatoires en milieu aléatoire sur ; dynamique d'applications localement contractantes sur le Cercle, Université de Rennes I, 2002

[6] J.-P. Conze, A. Raugi, Convergence of iterates of a transfer operator, application to dynamical systems and to Markov chains, ESAIM, Probab. Statist., 2003

[7] Golosov, A.O. Limit distributions for random walks in a random environment, Dokl. Akad. Nauk SSSR, Volume 271 (1983) no. 1, pp. 25-29

[8] Hall, P.; Heyde, C.C. Martingale Limit Theory and Its Applications, Academic Press, New York, 1980

[9] Kesten, H. The limit distribution of Sinaı̆'s random walk in random environment, Phys. A, Volume 138 (1986) no. 1–2, pp. 299-309

[10] Key, E. Recurrence and transience criteria for random walk in a random environment, Ann. Probab., Volume 12 (1984) no. 2, pp. 529-560

[11] Sinaı̆, Y.G. The limit behavior of a one-dimensional random walk in a random environment, Teor. Veroyatnost. i Primenen., Volume 27 (1982) no. 2, pp. 247-258

[12] Zeitouni, O. St. Flour lecture notes on random walks in random environment http://www-ee.technion.ac.il/~zeitouni (Technical report, available at)

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