Strong transience of one-dimensional random walk in a random environment

TL;DR

This paper characterizes strong transience in one-dimensional random walks in random environments, showing its equivalence to transience under quenched measure and to ballisticity under averaged measure.

Contribution

It provides a complete characterization of strong transience, linking it to transience and ballisticity under different probability measures.

Findings

Under quenched measure, transience is equivalent to strong transience.

Under averaged measure, strong transience is equivalent to ballisticity.

The paper clarifies the relationship between transience, strong transience, and ballisticity.

Abstract

A transient stochastic process is considered strongly transient if conditioned on returning to the starting location, the expected time it takes to return the the starting location is finite. We characterize strong transience for a one-dimensional random walk in a random environment. We show that under the quenched measure transience is equivalent to strong transience, while under the averaged measure strong transience is equivalent to ballisticity (transience with non-zero limiting speed).