Slowdown estimates for one-dimensional random walks in random environment with holding times

Amir Dembo,Ryoki Fukushima,Naoki Kubota

doi:10.1214/18-ecp191

Abstract

We consider a one dimensional random walk in random environment that is uniformly biased to one direction. In addition to the transition probability, the jump rate of the random walk is assumed to be spatially inhomogeneous and random. We study the probability that the random walk travels slower than its typical speed and determine its decay rate asymptotic.

Highlights

Setting and preliminariesLet (ω = {ω(x)}x∈Z, P) be indepnedent and identically distributed random variables taking values in (0, 1)
We consider a variant of this process whose jump rate is spatially inhomogeneous and random
We consider a continuous time random walk (X = {Xt }t≥0, {Pzω,μ }z∈Z ) on Z whose jump rates from x to x + 1 and x − 1 are given by ω(x)/μ(x) and (1 − ω(x))/μ(x), respectively

Summary

Introduction

When t is large, the inequality (1.15) holds for h = log t and fails for h = t, whereas per fixed t > 0 the left hand side of (1.15) is decreasing in h and its right hand side eventually increasing in h. In the case of positive and zero drift, that is, P-essinf ω(0) = 1/2, under an additional assumption that P(ω(0) = 1/2) > 0, the slowdown probability is shown to decay stretched exponentially with exponent 1/3.

Objectives

Results

Conclusion