We consider the continuous time, one-dimensional random walk in random environment in Sinai’s regime. We show that the probability for the particle to be, at time $t$ and in a typical environment, at a distance larger than $t^a$ ($0\<a\<1$) from its initial position, is $exp\{-\mathrm{Const}·t^a/[(1-a)lnt](1+o\left(1\right))\}$.

We consider the continuous time, one-dimensional random walk in random environment in Sinai's regime. We show that the probability for the particle to be, at time t and in a typical environment, at a distance larger than ta (0<a<1) from its initial position, is exp{-Const ⋅ ta/[(1 - a)lnt](1 + o(1))}.

We consider random walks where each path is equipped with a random weight which is stationary and independent in space and time. We show that under some assumptions the arising probability distributions are in a sense uniformly absolutely continuous with respect to the usual probability distribution for symmetric random walks.