HARMONIC FUNCTIONS, h-TRANSFORM AND LARGE DEVIATIONS FOR RANDOM WALKS IN RANDOM ENVIRONMENTS IN DIMENSIONS FOUR AND HIGHER

• 成果类型: Article
• 署名作者: Yilmaz, Atilla
• 署名单位: University of California System; University of California Berkeley
• 刊物名称: ANNALS OF PROBABILITY
• ISSN/ISSBN: 0091-1798
• DOI: 10.1214/10-AOP556
• 发表日期: 2011
• 页码: 471-506
• 关键词: quenched large deviations; mixing random environment; large numbers; LAW

摘要

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 A(eq). For every xi is an element of 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.