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

TL;DR

This paper investigates large deviations for random walks in high-dimensional random environments, establishing a connection between harmonic functions, h-transforms, and variational formulas to identify unique minimizers under certain conditions.

Contribution

It proves the existence of solutions to a Laplace-like equation for specific directions and uses h-transforms to characterize the unique minimizers of large deviation rate functions in dimensions four and higher.

Findings

Existence of positive solutions to a Laplace-like equation for directions in the open set where quenched and averaged rate functions coincide.

Construction of a new transition kernel via h-transform that minimizes the variational formulas.

Identification of the unique minimizer of large deviation rate functions using harmonic functions and h-transforms.

Abstract

We consider large deviations for nearest-neighbor random walk in a uniformly elliptic i.i.d. environment on $\mathbb{Z}^d$. 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 $d\geq4$ 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 $\xi$ 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 $\xi$. It also corresponds to the unique minimizer of Rosenbluth's variational formula, provided that the latter is slightly modified.