Harmonic functions for singular quadrant walks

TL;DR

This paper develops a method to compute positive harmonic functions for singular quarter-plane random walks, deriving explicit formulas for escape probabilities and providing a probabilistic interpretation using Martin boundary theory.

Contribution

It applies the compensation approach to singular walks, deriving explicit formulas for escape probabilities and characterizing all positive harmonic functions.

Findings

Explicit formulas for escape probabilities involving Fibonacci numbers.

Probabilistic interpretation of harmonic functions for singular walks.

Asymptotic analysis of Green functions in the quarter plane.

Abstract

We consider discrete (time and space) random walks confined to the quarter plane, with jumps only in directions $(i,j)$ with $i+j \geq 0$ and small negative jumps, i.e., $i,j \geq -1$. These walks are called singular, and were recently intensively studied from a combinatorial point of view. In this paper, we show how the compensation approach introduced in the 90ies by Adan, Wessels and Zijm may be applied to compute positive harmonic functions with Dirichlet boundary conditions. In particular, in case the random walks have a drift with positive coordinates, we derive an explicit formula for the escape probability, which is the probability to tend to infinity without reaching the boundary axes. These formulas typically involve famous recurrent sequences, such as the Fibonacci numbers. As a second step, we propose a probabilistic interpretation of the previously constructed harmonic functions and prove that they allow to compute all positive harmonic functions of these singular walks. To that purpose, we derive the asymptotics of the Green functions in all directions of the quarter plane and use Martin boundary theory.