The Annals of Probability

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

Atilla Yilmaz

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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.

Article information

Source
Ann. Probab., Volume 39, Number 2 (2011), 471-506.

Dates
First available in Project Euclid: 25 February 2011

Permanent link to this document
https://projecteuclid.org/euclid.aop/1298669171

Digital Object Identifier
doi:10.1214/10-AOP556

Mathematical Reviews number (MathSciNet)
MR2789504

Zentralblatt MATH identifier
1225.60160

Subjects
Primary: 60K37: Processes in random environments 60F10: Large deviations 82C41: Dynamics of random walks, random surfaces, lattice animals, etc. [See also 60G50]

Keywords
Random walk random environment large deviations harmonic functions Doob h-transform

Citation

Yilmaz, Atilla. Harmonic functions, h-transform and large deviations for random walks in random environments in dimensions four and higher. Ann. Probab. 39 (2011), no. 2, 471--506. doi:10.1214/10-AOP556. https://projecteuclid.org/euclid.aop/1298669171


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