Discrete harmonic functions on an orthant in $\mathbb{Z}^d$

Mustapha Sami,Aymen Bouaziz,Mohamed Sifi

doi:10.1214/ecp.v20-4249

Abstract

We give a positive answer to a conjecture on the uniqueness of harmonic functions in the quarter plane stated by K. Raschel. More precisely we prove the existence and uniqueness of a positive discrete harmonic function for a random walk satisfying finite range, centering and ellipticity conditions, killed at the boundary of an orthant in Zd. Our methodsallow on the other hand to generalize from the quarter plane to orthants in higher dimensions and to treat the spatially inhomogeneous walks.

Highlights

An explicit description of the Martin compactification for random walks is usually a non-trivial problem and the most of the existing results are obtained for so-called homogeneous random walks, when the transition probabilities of the process are invariant with respect to the translations over the state space E
[4] Doney describes the harmonic functions and the Martin boundary of a random walk (Z(n)) on Z killed on the negative half-line {z : z < 0}
Kurkova and Malyshev [13] describe the Martin boundary for nearest neighbor random walks on Z × N and on Z2+ with reflected conditions on the boundary

Summary

Introduction

An explicit description of the Martin compactification for random walks is usually a non-trivial problem and the most of the existing results are obtained for so-called homogeneous random walks, when the transition probabilities of the process are invariant with respect to the translations over the state space E (see [5], [18], [22]). (Parabolic maximum principle) Let B = A × {a ≤ k ≤ b} ⊂ Zd × Z, where a < b ∈ Z and A ⊂ Zd is a bounded domain in Zd, and u : B → R a caloric function in B. A way to get round the difficulty is to apply Harnack principle to a harmonic extension of hxB2R(y)(∂B2R(y) ∩ Dc), but harmonic functions are difficult to extend; the idea of going through a caloric function defined on a cylinder outside of D × Z and much easier to extend so that it is possible to apply parabolic Harnack principle The idea of such a construction is inspired by the proof of lemma 4.1 in [21].

The Carleson principle

Existence of a positive harmonic function on D

Boundary Harnack principle

The denouement