Weak quenched limiting distributions for transient one-dimensional random walk in a random environment

Peterson, Jonathon; Samorodnitsky, Gennady

Annales de l'I.H.P. Probabilités et statistiques, Tome 49 (2013) no. 3, pp. 722-752.

Nous considérons une marche aléatoire unidimensionnelle dans un environnement i.i.d. Le comportement asymptotique d’une telle marche aléatoire dépend largement d’un paramètre crucial $\kappa$ qui détermine les fluctuations du processus. Si $0<\kappa <2$, alors les distributions moyennées des temps d’atteinte de la marche aléatoire convergent vers une loi $\kappa$-stable. Cependant, il a été récemment prouvé que dans ce cas là, il n’existe pas de distribution limite des temps d’atteinte à environnement fixé. C’est-à-dire, il n’est pas vrai que presque tout environnement fixé, les distributions des temps d’atteinte (centrés et normalisés de quelque manière que ce soit) convergent vers une distribution non dégénérée. Nous montrons néanmoins que les distributions à environnement fixé ont une limite au sens faible. Plus précisément, les distributions à environnement fixé des temps d’atteinte - vues comme des mesures de probabilité aléatoires sur $ℝ$ - convergent en distribution vers une mesure de probabilité aléatoire qui a d’intéressantes propriétés de stabilité. Nos résultats généralisent à la fois la limite des distributions moyennisées et la non existence de distributions limites à environnement fixé.

We consider a one-dimensional, transient random walk in a random i.i.d. environment. The asymptotic behaviour of such random walk depends to a large extent on a crucial parameter $\kappa >0$ that determines the fluctuations of the process. When $0<\kappa <2$, the averaged distributions of the hitting times of the random walk converge to a $\kappa$-stable distribution. However, it was shown recently that in this case there does not exist a quenched limiting distribution of the hitting times. That is, it is not true that for almost every fixed environment, the distributions of the hitting times (centered and scaled in any manner) converge to a non-degenerate distribution. We show, however, that the quenched distributions do have a limit in the weak sense. That is, the quenched distributions of the hitting times - viewed as a random probability measure on $ℝ$ - converge in distribution to a random probability measure, which has interesting stability properties. Our results generalize both the averaged limiting distribution and the non-existence of quenched limiting distributions.

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