Sinai's Walk : a Statistical Aspect

In this paper we are interested in Sinai’s walk i.e., a one dimensional random walk in random environment with three conditions on the random environment: two necessaries hypothesis to get a recurrent process (see [Solomon(1975)]) which is not a simple random walk and an hypothesis of regularity which allows us to have a good control on the fluctuations of the random environment. The asymptotic behavior of such walk has been understood by [Sinai(1982)] : this walk is subdiffusive and at an instant n it is localized in the neighborhood of a well defined point of the lattice. It is well known, see (Zeitouni [2001] for a survey) that this behavior is strongly dependent of the random environment or, equivalently, by the associated random potential defined Section 1.2. The question we solve here is the following: given a single trajectory of a random walk (Xl, 1 ≤ l ≤ n) where the time n is fixed, can we estimate the trajectory of the random potential where the walk lives ? Let us remark that the law of this potential is unknown as-well. In their paper, [Adelman and Enriquez(2004)] are interested in the question of the distribution of the random environment that could be deduced from a single trajectory of the walk, on the other hand, our purpose is to get an approximation of the trajectory of the random potential. In the paper [V. Baldazzi and Monasson(2006)] the authors are interested in a method to predict the sequence of DNA molecules. They model the unzipping of the molecule as a one-dimensional biased random walk for the fork position (number of open base pair) k in this landscape. The elementary opening (k → k + 1) and closing (k → k − 1) transitions happen with a probability that depends on the unknown sequence. This probability of transition follows an Arrhénius law wich is closed to the one we discuss here. The question they answer is: given an unzipping signal can we predict