$t$-Martin boundary of killed random walks in the quadrant

TL;DR

This paper characterizes the $t$-Martin boundary for two-dimensional small steps random walks killed in the quarter plane, providing explicit formulas for $t$-harmonic functions and revealing three distinct regimes for the boundary.

Contribution

It introduces a uniform approach to compute the $t$-Martin boundary and explicit $t$-harmonic functions for these random walks, identifying three regimes.

Findings

Three regimes for the $t$-Martin boundary identified

Explicit expressions for discrete $t$-harmonic functions derived

Approach is uniform in $t$, applicable across regimes

Abstract

We compute the $t$-Martin boundary of two-dimensional small steps random walks killed at the boundary of the quarter plane. We further provide explicit expressions for the (generating functions of the) discrete $t$-harmonic functions. Our approach is uniform in $t$, and shows that there are three regimes for the Martin boundary.