The infinite valley for a recurrent random walk in random environment
Annales de l'I.H.P. Probabilités et statistiques, Volume 46 (2010) no. 2, pp. 525-536.

We consider a one-dimensional recurrent random walk in random environment (RWRE). We show that the - suitably centered - empirical distributions of the RWRE converge weakly to a certain limit law which describes the stationary distribution of a random walk in an infinite valley. The construction of the infinite valley goes back to Golosov, see Comm. Math. Phys. 92 (1984) 491-506. As a consequence, we show weak convergence for both the maximal local time and the self-intersection local time of the RWRE and also determine the exact constant in the almost sure upper limit of the maximal local time.

Nous prouvons que les mesures empiriques d'une marche aléatoire unidimensionnelle en environnement aléatoire convergent étroitement vers la loi stationnaire d'une marche aléatoire dans une vallée infinie. La construction de cette vallée infinie revient à Golosov, voir Comm. Math. Phys. 92 (1984) 491-506. En applications, nous obtenons la convergence étroite du maximum des temps locaux et du temps local d'intersections de la marche aléatoire en environnement aléatoire; de plus, nous identifions la constante représentant la “limsup” presque sûre du maximum des temps locaux.

DOI: 10.1214/09-AIHP205
Classification: 60K37,  60J50,  60J55,  60F10
Keywords: random walk in random environment, empirical distribution, local time, self-intersection local time
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Gantert, Nina; Peres, Yuval; Shi, Zhan. The infinite valley for a recurrent random walk in random environment. Annales de l'I.H.P. Probabilités et statistiques, Volume 46 (2010) no. 2, pp. 525-536. doi : 10.1214/09-AIHP205. http://www.numdam.org/articles/10.1214/09-AIHP205/

[1] J. Bertoin. Splitting at the infimum and excursions in half-lines for random walks and Lévy processes. Stochastic Process. Appl. 47 (1993) 17-35. | MR | Zbl

[2] R. Billing. Häufig besuchte Punkte der Irrfahrt in zufälliger Umgebung. Diploma thesis, Johannes Gutenberg-Universität Mainz, 2009.

[3] A. Dembo, N. Gantert, Y. Peres and Z. Shi. Valleys and the maximal local time for random walk in random environment. Probab. Theory Related Fields 137 (2007) 443-473. | MR | Zbl

[4] P. G. Doyle and E. J. Snell. Probability: Random Walks and Electrical Networks. Carus Math. Monographs 22. Math. Assoc. Amer., Washington, DC, 1984. | MR | Zbl

[5] N. Gantert and Z. Shi. Many visits to a single site by a transient random walk in random environment. Stochastic Process. Appl. 99 (2002) 159-176. | MR | Zbl

[6] A. O. Golosov. Localization of random walks in one-dimensional random environments. Comm. Math. Phys. 92 (1984) 491-506. | MR | Zbl

[7] P. Révész. Random Walk in Random and Non-Random Environments, 2nd edition. World Scientific, Hackensack, NJ, 2005. | MR | Zbl

[8] Z. Shi. A local time curiosity in random environment. Stochastic Process. Appl. 76 (1998) 231-250. | MR | Zbl

[9] Z. Shi. Sinai's walk via stochastic calculus. In Milieu aléatoires 53-74. F. Comets and E. Pardoux (Eds). Panoramas et Synthèses 12. Soc. Math. France, Paris, 2001. | MR | Zbl

[10] Y. G. Sinai. The limiting behavior of a one-dimensional random walk in a random medium. Theory Probab. Appl. 27 (1982) 256-268. | MR | Zbl

[11] F. Solomon. Random walks in random environment. Ann. Probab. 3 (1975) 1-31. | MR | Zbl

[12] O. Zeitouni. Random walks in random environment. In XXXI Summer School in Probability, St Flour (2001) 193-312. Lecture Notes in Math. 1837. Springer, Berlin, 2004. | MR | Zbl

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