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

CNRS U.M.R. 7599
| ||

``Probabilités et Modèles Aléatoires''
| ||

**Auteur(s): **

**Code(s) de Classification MSC:**

- 60G50 Sums of independent random variables
- 60J60 Diffusion processes, See also {58G32}

**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*