Abstract
We consider one-dimensional random walks in random environment which are transient to the right. Our main interest is in the study of the sub-ballistic regime, where at time n the particle is typically at a distance of order O(n κ) from the origin, \({\kappa \in (0, 1)}\) . We investigate the probabilities of moderate deviations from this behaviour. Specifically, we are interested in quenched and annealed probabilities of slowdown (at time n, the particle is at a distance of order \({O(n^{\nu_0})}\) from the origin, \({\nu_0 \in (0, \kappa)}\)), and speedup (at time n, the particle is at a distance of order \({n^{\nu_1}}\) from the origin, \({\nu_1 \in (\kappa, 1)}\)), for the current location of the particle and for the hitting times. Also, we study probabilities of backtracking: at time n, the particle is located around (−n ν), thus making an unusual excursion to the left. For the slowdown, our results are valid in the ballistic case as well.
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Fribergh, A., Gantert, N. & Popov, S. On slowdown and speedup of transient random walks in random environment. Probab. Theory Relat. Fields 147, 43–88 (2010). https://doi.org/10.1007/s00440-009-0201-2
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DOI: https://doi.org/10.1007/s00440-009-0201-2
Keywords
- Slowdown
- Speedup
- Moderate deviations
- Transience
Mathematics Subject Classification (2000)
- Primary 60K37