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
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.
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