Environment viewed from the particle and slowdown for ballistic RWRE in low dimensions

Abstract

We consider a random walk in a random environment on $\mathbb{Z}^d$ under ballisticity condition $(T)$. We show the existence of the invariant measure $Q$ with respect to the environment viewed from the particle for $d=2$ and $d=3$, which disproves a conjecture made in arXiv:1405.6819 regarding the two-dimensional case. We also prove tail estimates for the Radon-Nikodym derivative $dQ/dP$, where $P$ is the original distribution on the environment. Lastly, we provide nearly sharp tail bounds for regeneration times for $d=3$.