Diffusion in random environment and the renewal theorem

Abstract

According to a theorem of Schumacher and Brox, for a diffusion X in a Brownian environment, it holds that (X_t−b_logt)/log²t→0 in probability, as t→∞, where b is a stochastic process having an explicit description and depending only on the environment. We compute the distribution of the number of 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 nonrigorous paper by Le Doussal, Monthus and Fisher [Phys. Rev. E 59 (1999) 4795–4840] 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. Finally, we comment on the information these results give about the behavior of the diffusion.