On slowdown and speedup of transient random walks in random environment

We consider one-dimensional random walks in random environment which are transient to the right. Our main interest is in the study of the sub-ballistic regime, where at time n the particle is typically at a distance of order O(n ^κ ) from the origin, ${\kappa \in (0, 1)}$ . We investigate the probabilities of moderate deviations from this behaviour. Specifically, we are interested in quenched and annealed probabilities of slowdown (at time n, the particle is at a distance of order ${O(n^{\nu_0})}$ from the origin, ${\nu_0 \in (0, \kappa)}$ ), and speedup (at time n, the particle is at a distance of order ${n^{\nu_1}}$ from the origin, ${\nu_1 \in (\kappa, 1)}$ ), for the current location of the particle and for the hitting times. Also, we study probabilities of backtracking: at time n, the particle is located around (?n ^ν ), thus making an unusual excursion to the left. For the slowdown, our results are valid in the ballistic case as well.