Martin boundary of a killed random walk on $\Z_+^d$

TL;DR

This paper characterizes the Martin boundary for a killed random walk in multi-dimensional lattice space, linking boundary points to harmonic functions of associated Markov chains and providing explicit descriptions for product spaces.

Contribution

It introduces a novel representation of the Martin boundary for killed random walks using harmonic functions of induced Markov chains and extends results to Cartesian product spaces.

Findings

Martin boundary points correspond to harmonic functions of induced chains.

Full Martin compactification is described for product spaces.

Logarithmic asymptotics of Green functions are established.

Abstract

The Martin compactification is investigated for a d-dimensional random walk which is killed when at least one of it's coordinates becomes zero or negative. The limits of the Martin kernel are represented in terms of the harmonic functions of the associated induced Markov chains. It is shown that any sequence of points x_n with lim_n |x_n| = \infty and lim_n x_n/|x_n|= q is fundamental in the Martin compactification if up to the multiplication by constants, the induced Markov chain corresponding to the direction q has a unique positive harmonic function. The full Martin compactification is obtained for Cartesian products of one-dimensional random walks. The methods involve a ratio limit theorem and a large deviation principle for sample paths of scaled processes leading to the logarithmic asymptotics of the Green function.