Regularity of biased 1D random walks in random environment

2018/02/22 by Faggionato, Alessandra, Salvi, Michele
#60Fxx #60K37 #82D30 #FOS: Mathematics #Probability (math.PR)

paper · doi:10.48550/arxiv.1802.07874

Abstract

We study the asymptotic properties of nearest-neighbor random walks in 1d random environment under the influence of an external field of intensity λ∈ℝ. For ergodic shift-invariant environments, we show that the limiting velocity v(λ) is always increasing and that it is everywhere analytic except at most in two points λ_- and λ₊. When λ_- and λ₊ are distinct, v(λ) might fail to be continuous. We refine the assumptions in \citeZ for having a recentered CLT with diffusivity σ²(λ) and give explicit conditions for σ²(λ) to be analytic. For the random conductance model we show that, in contrast with the deterministic case, σ²(λ) is not monotone on the positive (resp.~negative) half-line and that it is not differentiable at λ=0. For this model we also prove the Einstein Relation, both in discrete and continuous time, extending the result of \citeLD16.

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