Quenched small deviation for the trajectory of a random walk with time-inhomogeneous random environment

TL;DR

This paper investigates the probability that a random walk in a time-inhomogeneous random environment stays close to a fixed trajectory, revealing that the decay rate differs from classical i.i.d. cases due to environmental randomness.

Contribution

It extends small deviation analysis to random walks in time-inhomogeneous environments, quantifying the impact of environmental randomness on deviation rates.

Findings

The decay rate of small deviation probabilities is smaller than in i.i.d. cases.

The rate is characterized by quenched and annealed variances.

Provides explicit formulas for the deviation probabilities in the inhomogeneous setting.

Abstract

We consider the small deviation probability for random walk with time-inhomogeneous random environment. Compared with the result in Mogul'ski\u{\i} (1974) for the i.i.d. random walk, the rate is smaller (due to the random environment), which is specified in terms of the quenched and annealed variance.