| For a diffusion X in a one-dimensional random environment, with the environment belonging to a specific class, it was proved by S. Schumacher that (Xt - blog t)/a(t) → 0 in probability, as t → infinity, where b is a stochastic process having an explicit description and depending only on the environment, and a(t) is a specific deterministic function. In the case that the environment is a Brownian motion path, we compute the distribution of the sign changes for b on an interval [1, x] and study some of the consequences of the computation; in particular, we get the probability of b keeping the same sign on that interval. These results have been announced in 1999 in a non-rigorous paper by P. Le Doussal, C. Monthus, and D. Fisher, and were treated with a Renormalization Group analysis. We prove that this analysis can be made rigorous using a path decomposition for the Brownian environment and renewal theory. Then we consider the case that the environment is the path of a spectrally one sided stable process, and we derive the one-dimensional distributions of the process b.; Another issue that we study concerns recurrent random walks on R . For any such walk (Sn)n ≥ 0, there are sequences (gn) n ≥ 1 of positive real numbers converging to infinity for which limn→ +infinity gn |Sn| = 0. For two classes of random walks, we give a criterion characterizing the increasing to infinity sequences that satisfy the above relation. This extends a result proved by K. L. Chung and P. Erdos in 1947, and our proof is simpler. |
