Abstract
We consider random walks in a balanced random environment in \({\mathbb{Z}^d}\), d ≥ 2. We first prove an invariance principle (for d ≥ 2) and the transience of the random walks when d ≥ 3 (recurrence when d = 2) in an ergodic environment which is not uniformly elliptic but satisfies certain moment condition. Then, using percolation arguments, we show that under mere ellipticity, the above results hold for random walks in i.i.d. balanced environments.
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X. Guo’s work was partially supported by NSF grant DMS-0804133.
O. Zeitouni’s work was partially supported by NSF grant DMS-0804133, the Israel Science Foundation and the Herman P. Taubman chair of Mathematics at the Weizmann Institute.
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Guo, X., Zeitouni, O. Quenched invariance principle for random walks in balanced random environment. Probab. Theory Relat. Fields 152, 207–230 (2012). https://doi.org/10.1007/s00440-010-0320-9
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DOI: https://doi.org/10.1007/s00440-010-0320-9
Keywords
- Random walks in random environments
- Homogenization
- Maximum principle
- Mean value inequality
- Percolation
Mathematics Subject Classification (2000)
- 60K37