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, p. 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 : https://doi.org/10.1214/09-AIHP205
Classification:  60K37,  60J50,  60J55,  60F10
Keywords: random walk in random environment, empirical distribution, local time, self-intersection local time
@article{AIHPB_2010__46_2_525_0,
     author = {Gantert, Nina and Peres, Yuval and Shi, Zhan},
     title = {The infinite valley for a recurrent random walk in random environment},
     journal = {Annales de l'I.H.P. Probabilit\'es et statistiques},
     publisher = {Gauthier-Villars},
     volume = {46},
     number = {2},
     year = {2010},
     pages = {525-536},
     doi = {10.1214/09-AIHP205},
     zbl = {1201.60096},
     mrnumber = {2667708},
     language = {en},
     url = {http://www.numdam.org/item/AIHPB_2010__46_2_525_0}
}
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/item/AIHPB_2010__46_2_525_0/

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