Law of large numbers for ballistic random walks in dynamic random environments under lateral decoupling

TL;DR

This paper proves a strong law of large numbers for ballistic random walks in certain dynamic environments, showing that the walk's position converges linearly over time under specific decoupling conditions.

Contribution

It introduces a novel approach to establish the law of large numbers for random walks in dynamic environments with space-time correlations, under decoupling assumptions.

Findings

Law of large numbers holds for walks with speed exceeding environment dependence bounds

Applicable to environments with strong space-time correlations like zero-range and exclusion processes

Provides a framework for analyzing ballistic behavior in complex dynamic environments

Abstract

We establish a strong law of large numbers for one-dimensional continuous-time random walks in dynamic random environments under two main assumptions: the environment is required to satisfy a decoupling inequality that can be interpreted as a bound on the speed of dependence propagation, while the random walk is assumed to move ballistically with a speed larger than this bound. Applications include environments with strong space-time correlations such as the zero-range process and the asymmetric exclusion process.