Harmonic functions, h-transform and large deviations for random walks in random environments in dimensions four and higher

Abstract

We consider large deviations for nearest-neighbor random walk in a uniformly elliptic i.i.d. environment on ℤ^d. There exist variational formulae for the quenched and averaged rate functions I_q and I_a, obtained by Rosenbluth and Varadhan, respectively. I_q and I_a are not identically equal. However, when d ≥ 4 and the walk satisfies the so-called (T) condition of Sznitman, they have been previously shown to be equal on an open set $\mathcal{A}_{\mathit {eq}}$.

For every $\xi\in\mathcal{A}_{\mathit {eq}}$, we prove the existence of a positive solution to a Laplace-like equation involving ξ and the original transition kernel of the walk. We then use this solution to define a new transition kernel via the h-transform technique of Doob. This new kernel corresponds to the unique minimizer of Varadhan’s variational formula at ξ. It also corresponds to the unique minimizer of Rosenbluth’s variational formula, provided that the latter is slightly modified.