Aging and quenched localization for one-dimensional random walks in random environment in the sub-ballistic regime

Abstract

We consider transient one-dimensional random walks in a random environment with zero asymptotic speed. An aging phenomenon involving the generalized Arcsine law is proved using the localization of the walk at the foot of “valleys“ of height logt. In the quenched setting, we also sharply estimate the distribution of the walk at time t.