Random walks in the quarter plane, discrete harmonic functions and conformal mappings

TL;DR

This paper introduces a novel method for determining discrete harmonic functions in the quarter plane using functional equations, with applications to exit time distributions and proving uniqueness in zero drift cases.

Contribution

It presents a new approach based on solving functional equations for generating functions, providing explicit formulas and uniqueness results for harmonic functions.

Findings

Derived a simple expression for the harmonic function governing exit time tail distribution.

Proved uniqueness of discrete harmonic functions in the zero drift case.

Connected harmonic functions to conformal mappings and functional equations.

Abstract

We propose a new approach for finding discrete harmonic functions in the quarter plane with Dirichlet conditions. It is based on solving functional equations that are satisfied by the generating functions of the values taken by the harmonic functions. As a first application of our results, we obtain a simple expression for the harmonic function that governs the asymptotic tail distribution of the first exit time for random walks from the quarter plane. As another corollary, we prove, in the zero drift case, the uniqueness of the discrete harmonic function.