Limit of the environment viewed from Sinaï’s walk

Francis Comets, Oleg Loukianov and Dasha Loukianova

Stochastic Processes and their Applications, 2024, vol. 168, issue C

Abstract: For Sinaï’s walk (Xk) we show that the empirical measure of the environment seen from the particle (ω̄k) converges in law to some random measure S∞. This limit measure is explicitly given in terms of the infinite valley, which construction goes back to Golosov (1984). As a consequence an “in law” ergodic theorem holds: 1n∑k=1nF(ω̄k)⟶ℒ∫ΩFdS∞.When the last limit is deterministic, it holds in probability. This allows some extensions to the recurrent case of the ballistic “environment’s method” dating back to Kozlov and Molchanov (1984). In particular, we show an LLN and a mixed CLT for the sums ∑k=1nf(ΔXk) where f is bounded and depending on the steps ΔXk≔Xk+1−Xk.

Keywords: Random walk in random environment; Recurrent regime; Localisation; Environment viewed from the particle (search for similar items in EconPapers)
Date: 2024
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DOI: 10.1016/j.spa.2023.104266

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