Localization of favorite points for diffusion in random environment

For a diffusion X_t in a one-dimensional Wiener medium W, it is known that there is a certain process b_x(W) that depends only on the environment W, so that X_t-b_{logt}(W) converges in distribution as t goes to infinity. We prove that, modulo a relatively small time change, the process {b_x(W):x>0}is followed closely by the process {F_X(e^x): x>0}, with F_X(t) denoting the point with the most local time for the diffusion at time t.Comment: 23 pages, 3 figure

One dimensional diffusion in an asymmetric random environment

According to a theorem of S. Schumacher, for a diffusion X in an environment determined by a stable process that belongs to an appropriate class and has index a, it holds that X_t/(log t)^a converges in distribution, as t goes to infinity, to a random variable having an explicit description in terms of the environment. We compute the density of this random variable in the case the stable process is spectrally one-sided. This computation extends a result of H. Kesten and quantifies the bias that the asymmetry of the environment causes to the behavior of the diffusion.Comment: 14 pages. To appear in Annales de l'Institut Henri Poincare, Probability and Statistic

Estimates for Bellman functions related to dyadic-like maximal operators on weighted spaces

We provide some new estimates for Bellman type functions for the dyadic maximal opeator on $R^n$ and of maximal operators on martingales related to weighted spaces. Using a type of symmetrization principle, introduced for the dyadic maximal operator in earlier works of the authors we introduce certain conditions on the weight that imply estimate for the maximal operator on the corresponding weighted space. Also using a well known estimate for the maximal operator by a double maximal operators on different m easures related to the weight we give new estimates for the above Bellman type functions.Comment: 10 pages. arXiv admin note: text overlap with arXiv:1511.0611

A note on recurrent random walks

For any recurrent random walk (Sn)n[greater-or-equal, slanted]1 on , there are increasing sequences (gn)n[greater-or-equal, slanted]1 converging to infinity for which (gnSn)n[greater-or-equal, slanted]1 has at least one finite accumulation point. For one class of random walks, we give a criterion on (gn)n[greater-or-equal, slanted]1 and the distribution of S1 determining the set of accumulation points for (gnSn)n[greater-or-equal, slanted]1. This extends, with a simpler proof, a result of Chung and Erdös. Finally, for recurrent, symmetric random walks, we give a criterion characterizing the increasing sequences (gn)n[greater-or-equal, slanted]1 of positive numbers for which .Random walk Recurrence Stable distributions Symmetric distributions