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March 2011 Harmonic functions, h-transform and large deviations for random walks in random environments in dimensions four and higher
Atilla Yilmaz
Ann. Probab. 39(2): 471-506 (March 2011). DOI: 10.1214/10-AOP556

Abstract

We consider large deviations for nearest-neighbor random walk in a uniformly elliptic i.i.d. environment on ℤd. There exist variational formulae for the quenched and averaged rate functions Iq and Ia, obtained by Rosenbluth and Varadhan, respectively. Iq and Ia 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.

Citation

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Atilla Yilmaz. "Harmonic functions, h-transform and large deviations for random walks in random environments in dimensions four and higher." Ann. Probab. 39 (2) 471 - 506, March 2011. https://doi.org/10.1214/10-AOP556

Information

Published: March 2011
First available in Project Euclid: 25 February 2011

zbMATH: 1225.60160
MathSciNet: MR2789504
Digital Object Identifier: 10.1214/10-AOP556

Subjects:
Primary: 60F10 , 60K37 , 82C41

Keywords: Doob h-transform , Harmonic functions , large deviations , random environment , Random walk

Rights: Copyright © 2011 Institute of Mathematical Statistics

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Vol.39 • No. 2 • March 2011
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