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 ${\mathbb{Z}}^d$. 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 $d\ge 4$ and the walk satisfies the so-called (T) condition of Sznitman, they have been previously shown to be equal on an open set ${\mathcal{A}}_{eq}$. For every $\xi \in {\mathcal{A}}_{eq}$, 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.

Published in at http://dx.doi.org/10.1214/10-AOP556 the Annals of Probability (http://www.imstat.org/aop/) by the Institute of Mathematical Statistics (http://www.imstat.org)

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Cited by in corpus (1)

• Homogenization of a class of one-dimensional nonconvex viscous Hamilton-Jacobi equations with random potential