Université Paris 6
Pierre et Marie Curie
Université Paris 7
Denis Diderot

CNRS U.M.R. 7599
``Probabilités et Modèles Aléatoires''

Sinai's walk via stochastic calculus


Code(s) de Classification MSC:

Résumé: Sinai's walk is a recurrent nearest-neighbour random walk on $Z$ in random environment, and is reputed for its exotic slow movement. The present paper summarizes the approach via stochastic calculus in the study of Sinai's walk. The main tool is the Ray-Knight theorem which describes the local time process of Brownian motion stopped at some special random times. The method is very powerful. For example, it allows to (i) establish all the possible Levy classes for Sinai's walk; (ii) determine the escape rate of favourite sites. It is interesting to mention that the latter problem remains open for the usual random walk. A number of unanswered questions, which concern various asymptotic properties of Sinai's walk, are listed at the end of the paper.

Mots Clés: Random walk in random environment ; diffusion in a random potential ; Ray-Knight theorem

Date: 2001-05-18

Prépublication numéro: PMA-660