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Many visits to a single site by a transient random walk in random environment

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  • Gantert, Nina
  • Shi, Zhan

Abstract

We consider a transient random walk on in random environment, and study the almost sure asymptotics of the supremum of its local time. Our main result states that if the random walk has zero speed, there is a (random) sequence of sites and a (random) sequence of times such that the walk spends a positive fraction of the times at these sites. This was known for a recurrent random walk in random environment (Random Walk in Random and Non-Random Environments, World Scientific, Singapore, 1990; Stochastic Process. Appl. 76 (1998) 231). Our method of proof is different and relies on the connection of random walk in random environment with branching processes in random environment used in Kesten et al. (Compositio Math. 30 (1975) 145).

Suggested Citation

  • Gantert, Nina & Shi, Zhan, 2002. "Many visits to a single site by a transient random walk in random environment," Stochastic Processes and their Applications, Elsevier, vol. 99(2), pages 159-176, June.
    • Handle: RePEc:eee:spapps:v:99:y:2002:i:2:p:159-176
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    References listed on IDEAS

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    1. Afanasyev, V. I., 2001. "On the maximum of a subcritical branching process in a random environment," Stochastic Processes and their Applications, Elsevier, vol. 93(1), pages 87-107, May.
    2. Shi, Zhan, 1998. "A local time curiosity in random environment," Stochastic Processes and their Applications, Elsevier, vol. 76(2), pages 231-250, August.
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    Cited by:

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    2. Andreoletti, Pierre, 2006. "On the concentration of Sinai's walk," Stochastic Processes and their Applications, Elsevier, vol. 116(10), pages 1377-1408, October.
    3. Andreoletti, Pierre, 2007. "Almost sure estimates for the concentration neighborhood of Sinai's walk," Stochastic Processes and their Applications, Elsevier, vol. 117(10), pages 1473-1490, October.
    4. Zindy, Olivier, 2008. "Upper limits of Sinai's walk in random scenery," Stochastic Processes and their Applications, Elsevier, vol. 118(6), pages 981-1003, June.

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