Random walks in time-inhomogeneous random environment conditioned to stay positive

TL;DR

This paper studies a random walk in a time-inhomogeneous random environment, constructs a conditioned process that stays positive using harmonic functions, and proves a quenched invariance principle for it.

Contribution

It introduces a method to condition the random walk to stay positive in a time-inhomogeneous environment and establishes a quenched invariance principle for this conditioned process.

Findings

Construction of a quenched harmonic function for the environment

Definition of the conditioned random walk via Doob's h-transform

Proof of a quenched invariance principle for the conditioned walk

Abstract

We consider a random walk $\{S_n\}_{n\in \mathbb{N}}$ in time-inhomogeneous random environment $\xi$. For almost each realization of $\xi$, we formulate a quenched harmonic function, based on which we can define the random walk in random environment conditioned to stay positive by the Doob's $h$-transform. Furthermore, we prove a quenched invariance principle for the conditioned random walk for almost each realization of $\xi$.