Discrete harmonic functions on an orthant in Zd

Aymen Bouaziz, Sami Mustapha, Mohamed Sifi

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Aymen Bouaziz, Sami Mustapha, Mohamed Sifi. Discrete harmonic functions on an orthant in Zd. Electronic Communications in Probability, 2015, 20, pp.1-13. 10.1214/ECP.v20-4249. hal-01214538

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Distributed under a Creative Commons Attribution 4.0 International License
Discrete harmonic functions on an orthant in $\mathbb{Z}^d$

Aymen Bouaziz*

Sami Mustapha†

Mohamed Sifi‡

Abstract

We give a positive answer to a conjecture on the uniqueness of harmonic functions stated in [20]. 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 $\mathbb{Z}^d$.

**Keywords:** Discrete harmonic functions; Orthants; Martin Boundary.

**AMS MSC 2010:** Primary Primary 60G50; 31C35, Secondary 60G40; 30F10.

Submitted to ECP on April 21, 2015, final version accepted on June 23, 2015.

# 1 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]). In [4] Doney describes the harmonic functions and the Martin boundary of a random walk ($Z(n)$) on $\mathbb{Z}$ killed on the negative half-line $\{z : z < 0\}$. Alili and Doney [2] extend this result for the corresponding space-time random walk $S(n) = (Z(n), n)$. Kurkova and Malyshev [13] describe the Martin boundary for nearest neighbor random walks on $\mathbb{Z} \times \mathbb{N}$ and on $\mathbb{Z}_+^2$ with reflected conditions on the boundary. The recent results of Raschel [20] and Kurkova and Raschel [14] identify the Martin compactification for random walks in $\mathbb{Z}_+^2$ with jumps at distance at most 1 and absorbing boundary. All these results use methods that seem to be unlikely to apply in a general situation. The methods of Doney [4] and Alili and Doney [2] rely in an essential way on the one-dimensional structure of the process ($Z(n)$). Kurkova and Malyshev [13], Raschel [20] and Kurkova and Raschel [14] use an analytical method where the geometrical structure of the elliptic curve defined by the jump generating function of the random walk plays a crucial role. For small steps walks the method uses the fact that the corresponding elliptic curve is homeomorphic to the torus. In a very recent paper Fayolle and Raschel [6] show how to extend the method for random walks with arbitrary big jumps (in this case the algebraic curve is no more homeomorphic to the torus).

<footnote>*Université de Tunis El Manar. Faculté des Sciences de Tunis. LR11ES11 Laboratoire d'Analyse Mathématiques et Applications. 2092, Tunis, Tunisie. E-mail: bouazizaymen16@yahoo.com</footnote>

<footnote>†Centre de Mathématiques de Jussieu. Université Paris 6 Pierre et Marie Curie. Tour 46 5e étage Boîte 247. 4, Place Jussieu F-75252 PARIS CEDEX 05. France. E-mail: sam@math.jussieu.fr</footnote>

<footnote>‡Université de Tunis El Manar. Faculté des Sciences de Tunis. LR11ES11 Laboratoire d'Analyse Mathématiques et Applications. 2092, Tunis, Tunisie. E-mail: mohamed.sifi@fst.rnu.tn</footnote>
The aim of our paper is to offer a general approach allowing to study positive harmonic functions for random walks killed at the boundary of an orthant in $\mathbb{Z}^d$ without assuming invariance with respect the translations over the state space.

More precisely let $\Gamma = -\Gamma \subset \mathbb{Z}^d$, be a symmetric finite subset of $\mathbb{Z}^d$ containing all unit vectors in $\mathbb{Z}^d$, i.e. all the vectors $e_k = (0, \dots, 0, 1, 0, \dots, 0) \in \mathbb{Z}^d$, where the 1 is the k-th component; and $\pi : \mathbb{Z}^d \times \Gamma \to [0, 1]$ be such that

$$ \sum_{e \in \Gamma} \pi(x, e) = 1, \quad x \in \mathbb{Z}^d. $$

Then, we let $(S_j)_{j \in \mathbb{N}}$, be the Markov chain defined by

$$ \mathbb{P}[S_{j+1} = x + e/S_j = x] = \pi(x, e); \quad e \in \Gamma, x \in \mathbb{Z}^d, j = 0, 1, \dots $$

We shall impose the following three conditions on the step set $\Gamma$ and the probabilities $\pi(x, e)$:

(Finite range) $|\Gamma| < \infty$,

(Centering) $\sum_{e \in \Gamma} \pi(x, e)e = 0$,

(Ellipticity) there exists $\alpha > 0$ such that $\pi(x, \pm e) \ge \alpha$; $x \in \mathbb{Z}^d$, $|e| = 1$.

We shall denote by:

$$ \mathcal{D} = \{(x_1, \dots, x_d) \in \mathbb{Z}^d; \ x_1 > 0, \dots, x_d > 0\} $$

and we shall be interested in characterizing the positive harmonic functions for the random walk killed at the boundary of the orthant $\mathcal{D}$, i.e. in functions $u : \bar{\mathcal{D}} \to \mathbb{R}$ such that:

(i) For any $x \in \mathcal{D}$, $u(x) = \sum_{e \in \Gamma} \pi(x, e)u(x + e)$,

(ii) if $x \in \mathcal{D}$, $u(x) > 0$,

(iii) if $x \in \partial \mathcal{D}$, $u(x) = 0$,

where we denote by

$$ \partial \mathcal{D} = \{x \in \mathcal{D}^c / x = z + e, \text{ for some } z \in \mathcal{D}, \text{ and } e \in \Gamma\}; \quad \text{and} \quad \bar{\mathcal{D}} = \mathcal{D} \cup \partial \mathcal{D}. $$

**Theorem 1.1.** Let $(S_j)_{j \in \mathbb{N}}$ be a spatially inhomogeneous random walk satisfying the above centering, finite range and ellipticity conditions. Then, up to a multiplicative constant, there exists a unique positive harmonic function for the random walk killed at the boundary.

Note that general results on harmonic functions for killed random walks in half spaces $\mathbb{Z}^+ \times \mathbb{Z}^{d-1}$ and orthants were obtained in the non-zero drift case in [8, 9, 14]. In this case, the Martin boundary is composed of infinitely many harmonic functions. Regarding random walks with zero drift, important results were obtained recently by Raschel in [20] in the quarter plane. In a sense our results can be seen as complementing those of [20] because they provide an answer to the conjecture stated in [20] in the case of random walks with finite support (cf. [20, Conjecture 1, §4.1]). Our methods, using ideas introduced in [16, 17], allow on the other hand to generalize from the quarter plane to orthants in higher dimensions and to treat the spatially inhomogeneous walks.

The paper is organized as follows. Section 2 gives in its first part the main technical ingredients in the proof of Theorem 1.1. A lower estimate of the harmonic measure
providing a Hölder continuity property at the boundary is established. Specifically, Proposition 1 gives a control of the growth of positive harmonic functions vanishing on the boundary. A Carleson-type estimate in then derived. In section 2.2 we establish the existence of a positive harmonic function on $\mathcal{D}$. Section 3 is devoted to prove the uniqueness of such positive harmonic function on $\mathcal{D}$.

# 2 Proof of Theorem 1.1

## 2.1 Key technical ingredients

The proof of Theorem 1.1 relies on a systematic use of two fundamental principles of potential theory, maximum principle and Harnack principle.

Let $A \subset \mathbb{Z}^d$ denote a bounded domain (i.e. a finite connected set of vertices of $\mathbb{Z}^d$). We let

$$ \partial A = \{x \in A^c / x = z + e, \text{ for some } z \in A, \text{ and } e \in \Gamma\}; \text{ and } \bar{A} = A \cup \partial A. $$

We say that a function $u : \bar{A} \to \mathbb{R}$ is harmonic on $A$ if

$$ u(x) = \sum_{e \in \Gamma} \pi(x, e)u(x + e), \quad x \in A. $$

The following maximum principle is immediate

**Theorem 2.1. (Maximum principle)** Let $A \subset \mathbb{Z}^d$ be a bounded domain in $\mathbb{Z}^d$ and $u : \bar{A} \to \mathbb{R}$ a harmonic function on $A$. Assume that $u \ge 0$ on $\partial A$. Then $u \ge 0$ in $A$.

The following theorem is a centered version of the Harnack inequality established by Lawler [15] for spatially inhomogeneous random walk with bounded symmetric increments.

**Theorem 2.2. (Harnack principle) (Cf. [11])** Let $u : B_{2R}(y) \to \mathbb{R}$ a nonnegative harmonic function on a discrete Euclidean ball centered at $y$ and with radius $2R$ ($R = 1, 2, \dots$ and $y \in \mathbb{Z}^d$). Then

$$ \max_{x \in B_R(y)} u(x) \le C \min_{x \in B_R(y)} u(x), $$

where $C = C(d, \alpha, \Gamma)$ is independent of $y \in \mathbb{Z}^d$, $R \ge 1$ and $u$.

In the proof of Theorem 1.1, together with Theorems 2.1 and 2.2, we need their parabolic versions.

Let $B = A \times \{a \le k \le b\} \subset \mathbb{Z}^d \times \mathbb{Z}$ where $A \subset \mathbb{Z}^d$ is a bounded domain of $\mathbb{Z}^d$ and $a < b \in \mathbb{Z}$. We let:

$$ \partial_{\ell} B = \cup_{a<k<b} \partial A \times \{k\} $$

$$ \partial_p B = \partial_{\ell} B \cup (\bar{A} \times \{a\}) $$

$$ \bar{B} = B \cup \partial_p B $$

$\partial_p B$ is the parabolic boundary of $B$ and $\partial_{\ell} B$ its lateral boundary. We say that $u : \bar{B} \to \mathbb{R}$ is caloric in $B$ if

$$ u(x, k+1) = \sum_{e \in \Gamma} \pi(x, e)u(x+e, k), \quad (x, k) \in A \times \{a \le k < b\}. $$

Caloric functions share with harmonic functions the fundamental property that there exist quantitative relations between their values at different points. Maximum principle extends so that we have a control of the values of a caloric function inside the cylinder $B$ by its values on the parabolic boundary $\partial_p B$. Harnack principle generalizes for
nonnegative caloric functions on two conditions: being at distance R – in the parabolic sense – from the parabolic boundary and observing a waiting period of time ≈ R². This requires to be at a distance R² from the bottom and R from the lateral boundary in addition to the waiting time. We then obtain a control of the values of the function in the lower sub-cylinder $B_R(y) \times \{s - 3R^2 \le k \le s - 2R^2\}$ by those in the upper sub-cylinder $B_R(y) \times \{s - R^2 \le k \le s\}$.

**Theorem 2.3. (Parabolic maximum principle)** Let $B = A \times \{a \le k \le b\} \subset \mathbb{Z}^d \times \mathbb{Z}$, where $a < b \in \mathbb{Z}$ and $A \subset \mathbb{Z}^d$ is a bounded domain in $\mathbb{Z}^d$, and $u: \bar{B} \to \mathbb{R}$ a caloric function in B. If $u \ge 0$ on $\partial_p B$ then $u \ge 0$ on B.

**Theorem 2.4. (Cf. [12])** Let $u$ be a nonnegative caloric function on $B_{2R}(y) \times \{s - 4R^2 \le k \le s\}$, where $(y, s) \in \mathbb{Z}^d \times \mathbb{Z}$ and $R \ge 1$. Then

$$ \max \{u(x, k); x \in B_R(y), s - 3R^2 < k < s - 2R^2\} \\ \le C \min \{u(x, k); x \in B_R(y), s - R^2 < k < s\}, $$

where $C = C(d, \alpha, \Gamma)$ is independent of $(y, s) \in \mathbb{Z}^d \times \mathbb{Z}$, $R \ge 1$ and $u$.

## 2.2 A lower estimate of the harmonic measure

For a finite set $A \subset \mathbb{Z}^d$ we denote by $\tau_A$ the exit time from A. We define the harmonic measure $h_A^x(E)$ ($E \subset \partial A$) of $(S_j)_{j \in \mathbb{N}}$ in A at the point $x \in A$, by

$$ h_A^x(E) = P_x[S_{\tau_A} \in E]. $$

**Lemma 2.5.** Let $y \in \partial D$ and $R \ge 1$ denote a large integer. Then

$$ h_{B_{2R}(y) \cap D}^x[\partial D \cap B_{2R}(y)] \ge \theta, \quad x \in B_R(y) \cap D, \qquad (2.1) $$

where $\theta = \theta(d, \alpha, \Gamma) > 0$ is independent of $y$ and $R$.

The proof of (2.1) we give below is the only place where caloric functions are used. It may seem strange that an issue regarding harmonic functions requires going through caloric functions. Indeed, it would have been more natural to estimate the left hand side of (2.1) by $h_{B_{2R}(y)}^x(\partial B_{2R}(y) \cap D^c)$ (using the maximum principle) and then establish the desired lower estimate for $h_{B_{2R}(y)}^x(\partial B_{2R}(y) \cap D^c)$. Such an approach may lead to the conclusion provided we have good estimates of the Green kernel as in [7, p. 511]; or by using a barrier argument as in [3, p. 157], but this is difficult to implement for inhomogeneous walks. A way to get round the difficulty is to apply Harnack principle to a harmonic extension of $h_{B_{2R}(y)}^x(\partial B_{2R}(y) \cap D^c)$, but harmonic functions are difficult to extend; hence the idea of going through a caloric function defined on a cylinder outside of $D \times \mathbb{Z}$ 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].

**Proof.** Let $z = y - \frac{R}{2}e_i$ ($i = 1, \dots, d$) (the direction $e_i$ to be chosen in an appropriate way depending on the piece of the boundary to which $y$ belongs) and $u_1, u_2$ and $u_3$ the three caloric functions defined by:

$$ \bullet u_1 : (B_{2R}(y) \cap D) \times \{-4R^2 \le k \le 0\} \to \mathbb{R}, $$

$$ u_1(x, k) = h_{B_{2R}(y) \cap D}^x(\partial D \cap B_{2R}(y)). $$
• $u_2 : \Omega = B_{2R}(y) \times \{-4R^2 \le k \le 0\} \to \mathbb{R}$, the solution of the parabolic boundary problem

$$
\left\{
\begin{array}{ll}
u_2(x, k+1) = \sum_{e \in \Gamma} \pi(x, e) u_2(x+e, k), & \text{if } (x, k) \in B_{2R}(y) \times \{-4R^2 \le k < 0\} \\
\\
u_2(x, -4R^2) = \mathbf{1}_{B_{R/8}(z)}(x) \\
\\
u_2|_{\partial_p \Omega \setminus B_{\frac{R}{8}}(z)} = 0
\end{array}
\right.
$$

• $u_3 : \Omega' = B_{R/8}(z) \times \{-4R^2 \le k \le 0\} \to \mathbb{R}$, the solution of the parabolic boundary problem

$$
\left\{
\begin{array}{ll}
u_3(x, k+1) = \sum_{e \in \Gamma} \pi(x, e) u_3(x+e, k), & \text{if } (x, k) \in B_{R/8}(z) \times \{-4R^2 \le k < 0\} \\
\\
u_3(x, -4R^2) = \mathbf{1}_{B_{R/8}(z)}(x) \\
\\
u_3|_{\partial_t \Omega'} = 0
\end{array}
\right.
$$

The parabolic maximum principle implies:

$$
u_1(x, k) \geq u_2(x, k), \quad (x, k) \in (\mathcal{D} \cap B_{2R}(y)) \times \{-4R^2 \leq k \leq 0\}
$$

$$
u_2(x, k) \geq u_3(x, k), \quad (x, k) \in B_{R/8}(z) \times \{-4R^2 \leq k \leq 0\}
$$

Combining with parabolic Harnack principle gives that for $x \in B_R(y) \cap \mathcal{D}$:

$$
h_{B_{2R}(y) \cap \mathcal{D}}^{x} [\partial \mathcal{D} \cap B_{2R}(y)] \geq c u_{3}(z, -R^{2}) \geq \theta > 0,
$$

simply because the caloric function $u_3$ can be extended to a larger domain ( e.g. $B_{\frac{R}{8}}(z) \times \{-10R^2 \le k \le 0\}$) so that $u_3 = 1$ on $B_{\frac{R}{8}}(z) \times \{-10R^2 \le k \le -4R^2\}$. $\square$

2.3 The Hölder continuity at the boundary

An important consequence of the lower bound of Lemma 2.5 is that:

**Proposition 2.6.** Let $y \in \partial \mathcal{D}$ and $R \ge 1$ denote a large integer. Let $u$ be a nonnegative harmonic function in $\mathcal{D} \cap B_{2R}(y)$ which vanishes on $\partial \mathcal{D} \cap B_{2R}(y)$. Then

$$
\max_{x \in \mathcal{D} \cap B_R(y)} u(x) \le \rho \max_{x \in \mathcal{D} \cap B_{2R}(y)} u(x) \quad (2.2)
$$

with $0 < \rho = \rho(d, \alpha, \Gamma) < 1$.

Proof. Let $x \in \mathcal{D} \cap B_R(y)$. Let $\tau$ denote the exit time from $\mathcal{D} \cap B_{2R}(y)$.
We have

$$
\begin{align*}
u(x) &= \sum_{z \in \partial(\mathcal{D} \cap B_{2R}(y))} u(z) \mathbb{P}_x [S_\tau = z] \\
&= \sum_{z \in \partial(\mathcal{D} \cap B_{2R}(y)) \setminus (\partial \mathcal{D} \cap B_{2R}(y))} u(z) \mathbb{P}_x [S_\tau = z].
\end{align*}
$$

Hence

$$
\begin{align*}
u(x) &\le \mathbb{P}_x [S_\tau \in \partial (\mathcal{D} \cap B_{2R}(y)) \setminus (\partial \mathcal{D} \cap B_{2R}(y))] \max_{x \in \mathcal{D} \cap B_{2R}(y)} u(x) \\
&= (1 - \mathbb{P}_x [S_\tau \in \partial \mathcal{D} \cap B_{2R}(y)]) \max_{x \in \mathcal{D} \cap B_{2R}(y)} u(x).
\end{align*}
$$
Using Lemma 2.5 and taking supremum on $x \in \mathcal{D} \cap B_R(y)$ we deduce that:

$$ \max_{x \in \mathcal{D} \cap B_R(y)} u(x) \le (1 - \theta) \max_{x \in \mathcal{D} \cap B_{2R}(y)} u(x), $$

which implies (2.2) with $\rho = 1 - \theta$. $\square$

## 2.4 The Carleson principle

This principle provides control of the bound of nonnegative harmonic functions vanishing on a portion of the boundary.

**Theorem 2.7.** Let $R \ge 1$ denote a large integer. Assume that $u$ is a nonnegative harmonic function in $\mathcal{D}$. Assume that $u = 0$ on $\partial \mathcal{D} \cap B_{2R}$ (where $B_R$ is the discrete Euclidean ball centered on 0 of radius $R$). Then

$$ \sup \{u(x), x \in \mathcal{D} \cap B_R\} \le C u(Re), \qquad (2.3) $$

where $e = (1, ..., 1)$ and $C = C(d, \alpha, \Gamma) > 0$.

*Proof.* To prove (2.3) we first observe that the ellipticity assumption implies that $u(x) \le Ce^{C|x-y|}u(y)$, $x, y \in \mathcal{D}$; where $C = C(d, \alpha, \Gamma) > 0$. This "local" Harnack principle allows us to assume that the distance of $x = (x_1, ..., x_d) \in \mathcal{D} \cap B_{2R}$ to the boundary $\partial \mathcal{D}$ is sufficiently large. Without loss of generality we may assume that $\min_{1 \le i \le d} x_i = x_1 \ge C$. On the other hand, the Harnack principle implies (by the classical chain argument) that

$$ u(x) \le C \left(\frac{R}{x_1}\right)^{\gamma} u(Re); \quad x \in B_{2R} \cap \mathcal{D}, \qquad (2.4) $$

where $\gamma$ and $C$ are positive constants depending only on $d, \alpha$ and $\Gamma$. Let $x_0 = (0, x_2, ..., x_d) \in \partial \mathcal{D}$. By Proposition 2.6 applied to the positive harmonic function in $B_{2x_1}(x_0) \cap \mathcal{D}$, we have

$$ u(x) \le \rho u(z), $$

where $z \in \overline{\mathcal{D} \cap B_{2x_1}(x_0)}$ satisfies

$$ u(z) = \max\{u(y), y \in \overline{\mathcal{D} \cap B_{2x_1}(x_0)}\}. $$

We distinguish two cases: (i) where $x_1 \le \frac{1}{2^N}(2R - |x|)$ and (ii) where $x_1 > \frac{1}{2^N}(2R - |x|)$. The constant $N$ that appears in the definition of these conditions is appropriately large and will be chosen later.

**Case (i).** In this case we observe that $z \in \mathcal{D} \cap B_{2R}$. This is because:

$$ \begin{aligned} |z| &\le |z - x| + |x| \le 8x_1 + |x| \\ &\le \frac{1}{2^{N-3}}(2R - |x|) + |x| \\ &\le \frac{R}{2^{N-4}} + (1 - \frac{1}{2^{N-3}})|x| \\ &= \frac{R}{2^{N-4}} + (1 - \frac{1}{2^{N-3}})2R \le 2R. \end{aligned} $$

We have

$$ 2R - |z| \ge 2R - |x| - 8x_1 \ge (1 - \frac{1}{2^{N-3}})(2R - |x|). $$
From which it follows that

$$ (2R - |x|)^{\gamma} u(x) \le (1 - \frac{1}{2^{N-3}})^{-\gamma} (2R - |z|)^{\gamma} \rho u(z) \\ \le (1 - \frac{1}{2^{N-3}})^{-\gamma} \rho \max_{x \in D \cap B_{2R}} (2R - |x|)^{\gamma} u(x) $$

and we choose $N$ large enough such that $(1 - \frac{1}{2^{N-3}})^{-\gamma} \rho = \rho' < 1$.

Case (ii). In this case we observe that:

$$ (2R - |x|)^{\gamma} u(x) \le 2^{\gamma N} x_1^{\gamma} u(x) \le C 2^{\gamma N} R^{\gamma} u(Re) $$

where the last inequality follows from (2.4).

Combining case (i) and case (ii) we deduce that for all $x \in D \cap B_{2R}$

$$ (2R - |x|)^{\gamma} u(x) \le \rho' \max_{x \in D \cap B_{2R}} (2R - |x|)^{\gamma} u(x) + C 2^{\gamma N} R^{\gamma} u(Re). $$

Using the fact that $\rho' < 1$ and $(2R - |x|) \approx R$ for $x \in D \cap B_R$ we deduce the estimate (2.3). □

## 2.5 Existence of a positive harmonic function on D

In order to establish the existence of a positive harmonic function on $D$ vanishing on $\partial D$, let:

$$ u_l(x) = \alpha_l(G_{l+1}(x, e) - G_l(x, e)), \quad x \in B_{2l} \cap D, $$

where $G_l(x, e)$; $l = 1, 2, \dots$ is the Green function of $(S_n)_{n \in \mathbb{N}}$ in the discrete ball $B_{2l}$, with pole at $e = (1, \dots, 1)$ and where the $\alpha_l$ are chosen so that $u_l(e) = 1$; $l = 1, 2, \dots$. Harnack principle combined with Carleson estimate (2.3) implies that the $u_l$ satisfy

$$ u_l(x) \le C, \quad x \in B_{2k} \cap D; \quad l > k, $$

with a constant $C = C(k)$ depending only on $k$. The diagonal process then allows us to deduce the existence of a positive harmonic function defined globally in $D$ and vanishing on $\partial D$. More precisely, for each $k = 1, 2, \dots$ there exists an extraction $\varphi_k$ so that $(u_{\varphi_k(n)})_{n \ge 1}$ converges on $B_{2k} \cap D$. An easy induction on $k$ gives a sequence of harmonic functions $(u_{\varphi_1 \circ \varphi_2 \circ \dots \circ \varphi_k(n)})_{n \ge 1}$ converging on $B_{2k} \cap D$. Taking $\psi(n) = \varphi_1 \circ \dots \circ \varphi_n(n)$ and setting $u(y) = \lim_{n \to \infty} u_{\psi(n)}(y)$ gives the required harmonic function. A similar construction of the harmonic function $u$ can be found in [10, p. 104].

# 3 Uniqueness of the positive harmonic function

## 3.1 Boundary Harnack principle

The previous section provides us a positive harmonic function in $D$ vanishing on $\partial D$. It remains to show that up to a multiplicative constant, this function is unique. One way that this can be seen is by using boundary Harnack principle.

**Theorem 3.1.** Assume that $u$ and $v$ are two positive harmonic functions in $D$. Assume that $u = 0$ on $\partial D$. Then

$$ \sup_{x \in D \cap B_{R/2}} \frac{u(x)}{v(x)} \le C \frac{u(Re)}{v(Re)}, \quad R \ge C, \qquad (3.1) $$

where $C = C(d, \alpha, \Gamma) > 0$.
Proof. Using (2.3) we see that

$$u(x) \leq Cu(Re), \quad x \in \partial B_R \cap \mathcal{D}.$$

Hence the maximum principle implies that

$$u(x) \leq Cu(Re)h_{B_R \cap \mathcal{D}}^{x} [\partial B_R \cap \mathcal{D}] \quad (3.2)$$

for all $x \in B_R \cap \mathcal{D}$. For $x \in (\partial B_R \cap \mathcal{D}) \cap \{x_i \geq \epsilon R, i = 1, \dots, d\}$ (for an appropriate small $\epsilon > 0$), Harnack principle gives

$$v(x) \geq cv(Re),$$

and so the maximum principle implies that

$$v(x) \geq cv(Re)h_{B_R \cap \mathcal{D}}^{x} [(\partial B_R \cap \mathcal{D}) \cap \{x_i \geq \epsilon R, i = 1, \dots, d\}] \quad (3.3)$$

for all $x \in B_R \cap \mathcal{D}$. The proof will follow from the following claim.

Claim. Let $x = (x_1, \dots, x_d) \in B_{R/2} \cap \mathcal{D}$ with $0 < x_i < \epsilon R$ for some $i = 1, \dots, d$. Then

$$h_{B_R \cap \mathcal{D}}^{x} [\partial B_R \cap \mathcal{D}] \leq C h_{B_R \cap \mathcal{D}}^{x} [(\partial B_R \cap \mathcal{D}) \cap \{x_i \geq \epsilon R, i = 1, \dots, d\}] \quad (3.4)$$

provided that $\epsilon$ is small enough.

Let us take the claim for granted and show how to finish the proof. We can assume, in the proof of (3.1), that $\text{dist}(x, \partial \mathcal{D}) \leq \epsilon R$ (otherwise (3.1) is an immediate consequence of Harnack principle). Combining (3.2) and (3.4) we deduce that

$$\frac{u(x)}{u(Re)} \leq C h_{B_R \cap \mathcal{D}}^{x} [(\partial B_R \cap \mathcal{D}) \cap \{x_i \geq \epsilon R, i = 1, \dots, d\}].$$

Since

$$h_{B_R \cap \mathcal{D}}^{x} [(\partial B_R \cap \mathcal{D}) \cap \{x_i \geq \epsilon R, i = 1, \dots, d\}] \leq C \frac{v(x)}{v(Re)},$$

by (3.3), our inequality is verified.

□

Proof of the claim. Let $x = (x_1, \dots, x_d) \in B_{R/2} \cap \mathcal{D}$. Without loss of generality we can assume that $x_1 = \text{dist}(x, \partial \mathcal{D}) < \epsilon R$. Using, as in §2.4, a sequence of balls, connecting $x$ to $\frac{R}{2}e$ and observing that

$$h_{B_R \cap \mathcal{D}}^{\frac{R}{2}e} [\partial B_R \cap \mathcal{D}] \geq c$$

(with a similar proof as in §2.2), we deduce by Harnack principle that

$$1 \leq C_1 \left(\frac{R}{x_1}\right)^{\gamma} h_{B_R \cap \mathcal{D}}^{x} [\partial B_R \cap \mathcal{D}].$$

Hence

$$h_{B_R \cap \mathcal{D}}^{x} [\partial B_R \cap \mathcal{D}] \geq \frac{1}{C_1} \left(\frac{x_1}{R}\right)^{\gamma}.$$

On the other hand, the same considerations, as in §2.3, show that

$$h_{B_R \cap \mathcal{D}}^{x} [(\partial B_R \cap \mathcal{D}) \cap \{x_i \leq \epsilon R, i = 1, \dots, d\}] \leq C_2 \rho^{\frac{R}{x_1}},$$

where $0 < \rho < 1$. Setting $\rho = e^{-r}$, for an appropriate $r > 0$ and choosing $\epsilon$ sufficiently small and satisfying

$$C_2 \epsilon^{-\gamma} e^{-\frac{r}{\epsilon}} < \frac{1}{C_1}$$

gives the inequality (3.4).
## 3.2 The denouement

Let us consider $u_1$ and $u_2$, two positive harmonic functions on $\mathcal{D}$ vanishing on $\partial\mathcal{D}$. By the boundary Harnack principle, we have

$$ \frac{1}{C} \frac{u_1(x)}{u_1(Re)} \le \frac{u_2(x)}{u_2(Re)} \le C \frac{u_1(x)}{u_1(Re)}, \quad x \in B_{R/2} \cap \mathcal{D}, \quad R \ge C, $$

where $C > 1$ is an appropriate large positive constant. Assume that $u_1$ and $u_2$ are normalized so that $u_1(e) = u_2(e) = 1$. It follows then that

$$ \frac{1}{C} u_1(Re) \le u_2(Re) \le C u_1(Re), \quad R \ge C. $$

Then by letting $R \to +\infty$ we deduce that:

$$ \frac{1}{C^2} \le \frac{u_2(x)}{u_1(x)} \le C^2, \quad x \in \mathcal{D}. $$

Define

$$ u_3(x) = u_2(x) + \frac{1}{C^2 - 1} (u_2(x) - u_1(x)), \quad x \in \mathcal{D}. $$

We have

$$ u_3(x) = \frac{C^2}{C^2 - 1} \left( u_2(x) - \frac{1}{C^2} u_1(x) \right) \ge 0, \quad x \in \mathcal{D}, $$

and clearly $u_3$ is harmonic function, vanishing on the boundary $\partial\mathcal{D}$ and satisfies the normalization condition $u_3(e) = 1$. We iterate and define (for $p \ge 4$)

$$ u_p(x) = u_{p-1}(x) + \frac{1}{C^2-1} (u_{p-1}(x) - u_1(x)). $$

An easy computation shows that $u_p$ is a non-negative harmonic function on $\mathcal{D}$, vanishing on the boundary $\partial\mathcal{D}$, satisfying the normalization condition $u_p(e) = 1$ and such that

$$ \frac{1}{C^2} \le \frac{u_p(x)}{u_1(x)} \le C^2, \quad x \in \mathcal{D}. $$

On the other hand, combining

$$ u_p(x) = \frac{C^2}{C^2 - 1} u_{p-1}(x) - \frac{1}{C^2 - 1} u_1(x) $$

$$ u_1(x) = \frac{C^2}{C^2 - 1} u_1(x) - \frac{1}{C^2 - 1} u_1(x). $$

We deduce that

$$ u_p(x) - u_1(x) = \frac{C^2}{C^2-1} (u_{p-1}(x) - u_1(x)), \quad x \in \mathcal{D}, p \ge 3. $$

From which it follows that

$$ u_p(x) = \left( \frac{C^2}{C^2-1} \right)^{p-2} (u_2(x) - u_1(x)) + u_1(x). $$

Dividing by $u_1(x)$ we deduce that:

$$ \frac{1}{C^2} \le \left( \frac{C^2}{C^2-1} \right)^{p-2} \left( -1 + \frac{u_2(x)}{u_1(x)} \right) + 1 \le C^2, \quad x \in \mathcal{D}, p \ge 4. $$

Letting $p \to +\infty$ we deduce that we necessarily have

$$ u_1(x) = u_2(x), \quad x \in \mathcal{D}. $$

The previous construction of the sequence $(u_p)$ is inspired by the proof of Lemma 6.2 of [1].
## 3.3 Examples

Here we illustrate our main result by two examples of a spatially inhomogeneous random walk in the quarter plane $Z_+^2$.

*Example 1.* We consider a random walk with alternating transition probabilities defined as follows. On the lines $\mathcal{L}_k = \{(x, x + 2k); x \in \mathbb{Z}\} (k \in \mathbb{Z})$ the transition probabilities are those of the simple random walk, see 1.

<figure><img src="image_2.png" /><figcaption>Figure 1: The simple random walk in the quarter plane.</figcaption></figure>

On the lines $\mathcal{L}_k' = \{(x, x + 2k + 1); x \in \mathbb{Z}\}$ the walk is allowed to visit its eight nearest neighbors with probabilities $p_{i,j} = \frac{1}{8}, -1 \le i, j \le 1$, see 2.

The resulting random walk with killing at $\partial Z_+^2$ is represented on 3.

Easy computation shows that the function

$$u(x, y) = xy \tag{3.5}$$

satisfy, as in the case of the simple random walk, conditions i), ii) and iii) of §1 with respect to our random walk and is then the unique harmonic function found in §2.

The previous example can be generalized as follows. Let $(S_n)_{n \in \mathbb{N}}$ denote a random walk as in 2, but where the transition probabilities $p_{ij} = p_{ij}(x, y)$ depend on the position $(x, y)$. Assume that

$$p_{1,0}; \ p_{-1,0}; \ p_{0,-1}; \ p_{-1,-1} \ge \alpha > 0$$

(i.e the ellipticity condition is satisfied) and that $(S_n)_{n \in \mathbb{N}}$ is centered:

$$\sum_{-1 \le i,j \le 1} ip_{ij} = \sum_{-1 \le i,j \le 1} j p_{ij} = 0. \tag{3.6}$$
<figure><img src="image_3.png" /><figcaption>Figure 2: Walks with small steps in the quarter plane.</figcaption></figure>

Then $u$ as defined by (3.5) is harmonic for the random walk $(S_n)_{n \in \mathbb{N}}$, if it satisfies the finite difference equation

$$ \sum_{-1 \le i,j \le 1} p_{ij}(x,y)u(x+i, y+j) = u(x,y). \quad (3.7) $$

We observe that the right hand side of (3.7) is given by

$$ \sum_{-1 \le i,j \le 1} p_{ij}(x,y)(x+i)(y+j) = \left( \sum_{-1 \le i,j \le 1} p_{ij}(x,y) \right) xy + \left( \sum_{-1 \le i,j \le 1} j p_{ij}(x,y) \right) x + \left( \sum_{-1 \le i,j \le 1} i p_{ij}(x,y) \right) y + \sum_{-1 \le i,j \le 1} ijp_{ij}(x,y). $$

Using (3.6) and the fact that $\sum_{-1 \le i,j \le 1} p_{ij} = 1$, we see that equation (3.7) is satisfied if and only if

$$ \sum_{-1 \le i,j \le 1} ijp_{ij}(x,y) = 0. $$

This condition is obviously satisfied by the random walk represented on 3.
<figure><img src="image_4.png" /><figcaption>Figure 3: The walk has alternating transition probabilities inside the quarter plane and is killed on the boundary of $Z_+^2$ which consists of $\{(x, 0); x \in \mathbb{N}^*\} \cup \{(0, y); y \in \mathbb{N}^*\}$.</figcaption></figure>

A second example. We now give a second example, more sophisticated than the previous one and whose harmonic function is not of the product form $xy$. We start from the random walk in the Weyl chamber of the the Lie algebra $sl_3(C)$ (cf. [20, §3.5]) and we reflect this walk with respect to the diagonal $x = y$. The obtained walk is assigned to the even lines $\mathcal{L}_k$ where it is allowed to visit its six neighbors with equal probabilities

$$p_{1,0} = p_{-1,0} = p_{0,1} = p_{0,-1} = p_{1,-1} = p_{-1,1} = 1/6.$$

On the odd lines $\mathcal{L}_k'$ we modify these probabilities as follows:

$$p_{1,0} = p_{-1,1} = p_{0,-1} = \frac{2-\alpha}{6}, \quad p_{0,1} = p_{-1,0} = p_{1,-1} = \frac{\alpha}{6},$$

where $0 < \alpha < 1$ is fixed. Easy computations show that

$$u(x, y) = \frac{xy(x + y)}{2},$$

(the same function as in [20, §3.5]) is harmonic for our inhomogeneous walk. Finally, we observe that we can allow $\alpha$ to take different values on the odd lines as it varies within a range ensuring ellipticity.
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**Acknowledgments.** We are grateful to the referee for carefully reading the manuscript and making useful comments and suggestions, which led us to improve substantially the paper.
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