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Local Limit Theorems for random walks in a 1D random environment

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Abstract

We consider random walks in a one-dimensional i.i.d. random environment with jumps to the nearest neighbours. For almost all environments, we prove a quenched Local Limit Theorem (LLT) for the position of the walk if the diffusivity condition is satisfied. As a corollary, we obtain the annealed version of the LLT and a new proof of the theorem of Lalley which states that the distribution of the environment viewed from the particle has a limit for a. e. environment.

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Authors and Affiliations

  1. Department of Mathematics, Institute of Physical Science and Technology, University of Maryland, College Park, MD, 20742, USA

    D. Dolgopyat

  2. School of Mathematical Sciences, Queen Mary University of London, Mile End Road, London, E1 4NS, UK

    I. Goldsheid

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  1. D. Dolgopyat
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  2. I. Goldsheid
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Correspondence to I. Goldsheid.

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Dolgopyat, D., Goldsheid, I. Local Limit Theorems for random walks in a 1D random environment. Arch. Math. 101, 191–200 (2013). https://doi.org/10.1007/s00013-013-0547-7

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  • DOI: https://doi.org/10.1007/s00013-013-0547-7

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