Logarithmic speeds for one-dimensional perturbed random walk in random environment

M. V. Menshikov, Andrew R. Wade

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We study the random walk in random environment on Z+ = f0; 1; 2; : : :g, where the environment is subject to a vanishing (random) perturbation. The two particular cases that we consider are: (i) random walk in random environment perturbed from Sinai's regime; (ii) simple random walk with random perturbation. We give almost sure results on how far the random walker is from the origin, for almost every environment. We give both upper and lower almost sure bounds. These bounds are of order (log t), for 2 (1;1), depending on the perturbation. In addition, in the ergodic cases, we give results on the rate of decay of the stationary distribution.
Original languageEnglish
Pages (from-to)389-416
Number of pages28
JournalStochastic Processes and their Applications
Issue number3
Publication statusPublished - Mar 2008


  • random walk
  • random environment
  • logarithmic speeds
  • almost sure behaviour
  • slow transience

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