Martin boundary of a killed random walk on a quadrant

TL;DR

This paper fully characterizes the Martin boundary of killed random walks in the quadrant, showing the Martin compactification is homeomorphic to a specific closure in Euclidean space, using ratio limit and large deviation methods.

Contribution

It provides a complete description of the Martin boundary for killed random walks on the quadrant, a novel result in potential theory for such stochastic processes.

Findings

Martin boundary characterized explicitly

Martin compactification homeomorphic to a specific closure

Method combines ratio limit theorem and large deviation techniques

Abstract

A complete representation of the Martin boundary of killed random walks on the quadrant ${\mathbb{N}}^*\times{\mathbb{N}}^*$ is obtained. It is proved that the corresponding full Martin compactification of the quadrant ${\mathbb{N}}^*\times{\mathbb{N}}^*$ is homeomorphic to the closure of the set $\{w={z}/{(1+|z|)}:z\in{\mathbb{N}}^*\times{\mathbb{N}}^*\}$ in ${\mathbb{R}}^2$. The method is based on a ratio limit theorem for local processes and large deviation techniques.