Discrete harmonic functions in Lipschitz domains

TL;DR

This paper establishes the existence and uniqueness of discrete harmonic functions in Lipschitz domains for certain random walks, using potential theory and comparison methods.

Contribution

It introduces a new approach to prove existence and uniqueness of discrete harmonic functions in Lipschitz domains for finite-range, centered, elliptic random walks.

Findings

Proves existence of discrete harmonic functions in Lipschitz domains.

Establishes uniqueness of these functions under specified conditions.

Develops a method based on comparison and potential theory.

Abstract

We prove the existence and uniqueness of a discrete nonnegative harmonic function for a random walk satisfying finite range, centering and ellipticity conditions, killed when leaving a globally Lipschitz domain in $\mathbb{Z}^d$. Our method is based on a systematic use of comparison arguments and discrete potential-theoretical techniques.

Equations

$$\sum _{e\in \Gamma }\pi (x,e)=1,\sum _{e\in \Gamma }\pi (x,e)e=0;e\in \Gamma ,x\in {\mathbb{Z}}^d.$$

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

$$\pi (x,e)\ge \alpha ,e\in \Gamma ,x\in {\mathbb{Z}}^d,$$

$$\mathcal{D}=\left\{(x_1,x^{\prime })\in \mathbb{R}\times {\mathbb{R}}^{d-1};x_1>\phi (x^{\prime })\right\}$$

$$|\phi (x^{\prime })-\phi (y^{\prime })|\le A|x^{\prime }-y^{\prime }|,x^{\prime },y^{\prime }\in {\mathbb{R}}^{d-1},$$

$$\tau =\inf \{n=0,1,\dots ;S_n\notin \mathcal{C}\}.$$

$$G_x^y=\sum _{n\in \mathbb{N}}{\mathbb{P}}_x(S_n=y,\tau >n).$$

$$\partial A=\{x\in A^c,x=z+e\text{mbox}forsomez\in A\text{mbox}ande\in \Gamma \}$$

$$h(x)={\mathbb{E}}^x\left(h(S_1),\tau >1\right),x\in \mathcal{C}.$$

$$h(x)={\int }_{{\partial }_m{E}_M}{k}_{\gamma }^x\mu (d\gamma ),$$

$$Lu(x)=\sum _{e\in \Gamma }\pi (x,e)u(x+e)-u(x).$$

$$\underset{B_R(y)}{\max }u\le C\underset{B_R(y)}{\min }u,$$

$$\max \left\{u(x),x\in \mathcal{C}\cap B_R(y)\right\}\le Cu(y+R{e}_1),R\ge C,$$

$$\max \left\{u(x),x\in \overline{\mathcal{C}\cap B_R(y)}\right\}\le \rho \max \left\{u(x),x\in \overline{\mathcal{C}\cap B_{2\sqrt{d}R}(y)}\right\},$$

$$\max \left\{u(x),x\in \overline{\mathcal{C}\cap Q_R(y)}\right\}\le \rho \max \left\{u(x),x\in \overline{\mathcal{C}\cap Q_{2R}(y)}\right\}.$$

$$v(x)\ge \lambda =\lambda (d,\alpha ,\Gamma ,A)>0,x\in \overline{Q_R}.$$

$$\underset{\overline{Q_R}}{\min }v\ge \gamma {\left(\frac{\left|\overline{Q_R}\cap \{v\ge 1\}\right|}{|\overline{Q_R}|}\right)}^{\frac{\log \gamma }{\log \delta }}$$

$$\left|\overline{Q_R}\cap \{v\ge 1\}\right|\ge \left|\overline{Q_R}\cap {\mathcal{C}}^{\prime }\right|\ge \mu |\overline{Q_R}|.$$

$$\underset{\overline{Q_R}}{\min }v\ge {\gamma }^{1+\frac{\log \mu }{\log \delta }},$$

$$u(x)\le C{\left(\frac{R}{\delta (x)}\right)}^{\gamma }u(y+R{e}_1),x\in \mathcal{C}\cap B_{2R}(y),$$

$$\delta (x)<\left(1-{\left(\frac{1+\rho }{2}\right)}^{\frac{1}{\gamma }}\right)\frac{2R-|x-y|}{8\sqrt{d}}$$

$$u(x)\le \max \left\{u(x^{\prime }),x^{\prime }\in \mathcal{C}\cap \overline{B_{\frac{3}{2}\delta (x)}(x_0)}\right\}\le \rho \max \left\{u(x^{\prime }),x^{\prime }\in \mathcal{C}\cap \overline{B_{3\sqrt{d}\delta (x)}(x_0)}\right\}.$$

$$u(z)=\max \left\{u(x^{\prime }),x^{\prime }\in \mathcal{C}\cap \overline{B_{3\sqrt{d}\delta (x)}(x_0)}\right\}.$$

$$(2R-|x-y|)\le (2R-|z-y|)+8\sqrt{d}\delta (x).$$

$$(2R-|x-y|)\le {\left(\frac{1+\rho }{2}\right)}^{-\frac{1}{\gamma }}(2R-|z-y|).$$

$$(2R-|x-y|{)}^{\gamma }u(x)\le {\left(\frac{1+\rho }{2}\right)}^{-1}(2R-|z-y|{)}^{\gamma }u(x),$$

$$(2R-|x-y|{)}^{\gamma }u(x)$$

$${\theta }_0=\frac{2\rho }{1+\rho }<1.$$

$$\delta (x)\ge \left(1-{\left(\frac{1+\rho }{2}\right)}^{\frac{1}{\gamma }}\right)\frac{2R-|x-y|}{8\sqrt{d}}.$$

$$(2R-|x-y|{)}^{\gamma }u(x)\le {\epsilon }_0^{-\gamma }\delta (x{)}^{\gamma }u(x)\le {\epsilon }_0^{-\gamma }\underset{x^{\prime }\in \mathcal{C}\cap B_{2R}(y)}{\max }\delta (x^{\prime }{)}^{\gamma }u(x^{\prime })$$

Full text

1. Introduction and main results

Random walks conditioned to live in domains $\mathcal{C}\subset\mathbb{Z}^{d}$ are of growing interest because of the range of their applications in enumerative combinatorics, in probability theory and in harmonic analysis (cf. [7], [9], [11], [17], [18], [30]). Doob $h$-transforms, where $h$ is harmonic for the random walk, positive within $\mathcal{C}$ and vanishing on its boundary $\partial\mathcal{C}$, are used to perform such conditioning. It is therefore crucial to identify the set of all positive harmonic functions associated with a killed random walk.

General results for homogeneous random walks with non-zero drift killed at the boundary of a half-space or an orthant were obtained in [20], [22], [25]. For random walks with zero drift, only few results are available [6], [11], [19], [30], [31]. The first systematical result was obtained by K. Raschel, who introduced in [31] a new approach based on the investigation of a functional equation satisfied by the generating function of the values taken by the harmonic function. This approach allows him to establish the existence of positive harmonic functions for random walks with small steps and zero drift killed at the boundary of the quadrant $\mathbb{N}^{2}$. It should be also mentioned that [31] provides explicit expressions for these harmonic functions.

In a recent work Ignatiouk-Robert [21] investigated the properties of harmonic functions for random walks in via ladder heights. Applying her general results to random walk in a convex cone she deduced the uniqueness (up to a multiplicative constant) of the harmonic function constructed by Denisov and Wachtel in [11] under some moment condition on the jumps. Alternative constructions of this harmonic function are proposed by Denisov and Wachtel in [12]. These new constructions allow them to remove quite restrictive extendability assumption imposed in [11]. In [32] Raschel and Tarrago studied the behavior of the Green function for random walks in convex cone which gives the uniqueness of the harmonic function (see also [14].

Regarding spatially inhomogeneous random walks the problem is more difficult.

Uniqueness of positive harmonic functions for random walks with symmetric spatially inhomogeneous increments, killed at the boundary of a half space, was established in [28] and more recently in the case of an orthant [8].

The main purpose of the present paper is to extend the results of [8] for the whole class of spatially inhomogeneous centered random walks satisfying finite span and ellipticity conditions and killed when leaving a globally Lipschitz unbounded domain in $\mathbb{Z}^{d}$.

Consider $\Gamma\subset\mathbb{Z}^{d}$ a finite subset of $\mathbb{Z}^{d}$ and let $\pi:\mathbb{Z}^{d}\times\Gamma\rightarrow[0,1]$ such that

[TABLE]

Then, we let $\{S(n),\,n\in\mathbb{N}\}=(S_{n})_{n\in\mathbb{N}}$ be the Markov chain on $\mathbb{Z}^{d}$ defined by

[TABLE]

$(S_{n})_{n\in\mathbb{N}}$ is a centered random walk with bounded increments which becomes spatially homogeneous if we assume the probabilities $\pi(x,e)$ are independent of $x$. We shall assume that the set $\Gamma$ contains all unit vectors in $\mathbb{Z}^{d}$, i.e. all the vectors $e_{k}=(0,\ldots,0,1,0,\ldots 0)\in\mathbb{Z}^{d}$, where the $1$ is the $k$-th component. We shall impose to the random walk $(S_{n})_{n\in\mathbb{N}}$ to satisfy the following uniform ellipticity condition:

[TABLE]

for some $\alpha>0$.

We shall denote by:

$\bullet\quad\mathcal{C}$ a globally Lipschitz domain of $\mathbb{Z}^{d}$ that is, a domain $\mathcal{C}=\mathcal{D}\cap\mathbb{Z}^{d}$ where

[TABLE]

for some Lipschitz function on $\mathbb{R}^{d-1}$ satisfying

[TABLE]

for some $A>0$, where $|.|$ denote the Euclidean norm. We shall assume that $\varphi(0)=0$.

$\bullet\quad\tau$ the first exit time from $\mathcal{C}$, i.e.,

[TABLE]

$\bullet\quad G_{x}^{y}$, $x,y\in\mathcal{C}$, the Green function defined by

[TABLE]

We are interested in positive functions $h$ which are discrete harmonic for the random walk $(S_{j})_{j\in\mathbb{N}}$ killed at the boundary of $\mathcal{C}$, i.e. in functions $h:\overline{\mathcal{C}}\rightarrow\mathbb{R}_{+}$ such that:

• i)

For any $x\in\mathcal{C}$, $\,h(x)=\displaystyle\sum_{e\in\Gamma}\pi(x,e)h(x+e)$;

• ii)

If $x\in\partial\mathcal{C}$, then $h(x)=0$;

• iii)

If $x\in\mathcal{C}$, then $h(x)>0$;

where $\,\overline{\mathcal{C}}=\partial\mathcal{C}\cup\mathcal{C}$. The boundary of a set $A\subset\mathbb{Z}^{d}$ is defined by

[TABLE]

and $\overline{A}=A\cup\partial A$. In terms of the first exit time of the random walk from $\mathcal{C}$, we have that

[TABLE]

Theorem 1.1**.**

Let $(S_{n})_{n\in\mathbb{N}}$ be a centered random walk satisfying the above finite support and ellipticity conditions. Assume that $\mathcal{C}$ is a globally Lipschitz domain of $\mathbb{Z}^{d}$. Then, up to a multiplicative constant, there exists a unique positive function, harmonic for the random walk killed at the boundary.

The previous result has an important consequence on the Martin boundary theory attached to the random walk $(S_{n})_{n\in\mathbb{N}}$ killed on the boundary of $\mathcal{C}$. Recall that for a transient Markov chain on a countable state space $E$, the Martin compactification of $E$ is the unique smallest compactification $E_{M}$ of the discrete set $E$ for which the Martin kernels $y\rightarrow\displaystyle k_{y}^{x}=G_{y}^{x}/G_{y}^{x_{0}}$ (where $x_{0}$ is a given reference state in $E$) extend continuously for all $x\in E$. The minimal Martin boundary $\partial_{m}E_{M}$ is the set of all those $\gamma\in\partial E_{M}$ for which the function $x\rightarrow k_{\gamma}^{x}$ is minimal harmonic. Recall that a harmonic function $h$ is minimal if $0\leq g\leq h$ with $g$ harmonic implies $g=ch$ with some $c>0$. By the Poisson-Martin boundary representation theorem, every nonnegative harmonic function $h$ can be written as

[TABLE]

for a some positive Borel measure $\mu$ on $\partial_{m}E_{M}$ (cf. [13], [27], [29]).

An immediate consequence of Theorem 1.1 is the following.

Theorem 1.2**.**

For all transient random walks satisfying centering, finite support and ellipticity conditions and all global Lipschitz domains of $\mathbb{Z}^{d}$, the minimal Martin boundary is reduced to one point.

We conclude this introduction with some comments which may be helpful in placing the results of this paper in their proper perspective.

(i) The proof of Theorem 1.1 given in [8] uses in a crucial way the parabolic Harnack principle. We noted in [8] that a more satisfactory approach should dispense with parabolic information and restrict to elliptic tools. A way to get round the difficulties encountered in [8] is to use a lower estimate for superharmonic extensions of discrete positive harmonic functions derived by Kuo and Trudinger in [23]. This lower estimate encompasses three powerful ingredients: the Aleksandrov-Bakel’man-Pucci’s maximum principle, a barrier technique and a Calderón-Zygmund covering argument. Going trough the superharmonic extension gives an alternative to the use of [8, Lemma 2.5] and provides a purely elliptic derivation of [8, Proposition 2.6] . An advantage of this approach is that it allows us to relax the assumptions $0\in\Gamma$ and $\Gamma=-\Gamma$ made in [8].

(ii) In case of homogeneous symmetric random walks on unbounded Lipschitz domains, the main results of this paper follows from [19]. Although the work of Gyrya and Saloff-Coste concerns diffusion on Dirichlet spaces, to derive the desired results for symmetric random walks, it suffices to consider the corresponding cable process (see [3, §2]). Since the harmonic functions for cable process and the random walk on the corresponding graph are essentially the same one has all the desired results (namely Theorem 1.1, Theorem 1.2 and Theorem 2.3).

(iii) Spatially inhomogeneous random walks can be considered as the discrete analogues of diffusions generated by second-order differential operators in nondivergence form. As in [8], the main tools in this paper are discrete versions of Carleson estimate and boundary Harnack inequality (cf. [4], [5], [15], [16]).

(iv) We restrict ourselves in this paper to random walks in Lipschitz domains. However, the proofs given below should work for a larger class of domains, for instance uniform or inner uniform domains (cf. [1]).

2. Proof of Theorem 1.1

2.1. Harnack principle

We say that a function $u:\overline{A}=A\cup\partial A\rightarrow\mathbb{R}$ is harmonic in $A\subset\mathbb{Z}^{d}$ if $Lu=0$ in $A$, where $L$ is the difference operator defined by

[TABLE]

In addition to an obvious maximum principle, harmonic functions satisfy, when they are positive, a Harnack principle. For convenience this principle is formulated in balls. The discrete Euclidean ball of center $y\in\mathbb{Z}^{d}$ and radius $R\geq 1$ is denoted $B_{R}(y)$ and simply $B_{R}$ when $y$ is clearly understood. We shall also have to use cubes. The cube of center $y\in\mathbb{Z}^{d}$ and sides $2R$, parallel to the coordinate axes is denoted $Q_{R}(y)$ and simply $Q_{R}$ when $y$ is clear. The following theorem (see [23] and [24]) is a centered version of Harnack principle established by Lawler [26] for random walks with symmetric bounded increments (as well homogeneous and inhomogeneous).

Theorem 2.1**.**

(Harnack principle)* Assume that $u$ is a nonnegative harmonic function associated to a random walk satisfying centering, finite support and uniform ellipticity conditions in a ball $B_{2R}(y)$. Then*

[TABLE]

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

2.2. Carleson estimate

The classical Carleson estimate [10] asserts that a positive harmonic function vanishing on a portion of the boundary is bounded, up to a smaller portion, by the value at a fixed point in the domain with a multiplicative constant independent of the function.

Theorem 2.2**.**

Assume that $u$ is a nonnegative harmonic function in $\mathcal{C}\cap B_{3R}(y)$. Assume that $u=0$ on $\partial\mathcal{C}\cap B_{2R}(y)$. Then

[TABLE]

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

The proof of Theorem 1.1 relies on the following Proposition.

Proposition 2.1**.**

Let $y\in\partial\mathcal{C}$ and $R$ large enough. Let $u$ be a nonnegative harmonic function in $\mathcal{C}\cap B_{3\sqrt{d}R}(y)$ which vanishes on $\partial\mathcal{C}\cap B_{2\sqrt{d}R}(y)$. Then

[TABLE]

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

Proof.

To prove (2.2) we first observe that it suffices to show that

[TABLE]

Without loss of generality, we assume $y=0$ and $\max\left\{u(x),\;x\in\overline{\mathcal{C}\cap Q_{2R}}\right\}=1$. Then considering the function $v:\overline{Q_{2R}}\rightarrow\mathbb{R}$ defined by $v=1-u$ in $\overline{\mathcal{C}\cap Q_{2R}}$ and $v=1$ on $\overline{Q_{2R}}\,\backslash\,\overline{\mathcal{C}\cap Q_{2R}}$, we see that (2.3) reduces to the following lower estimate

[TABLE]

Since $v$ is superharmonic in $Q_{2R}$ (i.e $Lv\leq 0$ in $Q_{2R}$) we can use the estimate [24, 3.24] and deduce that

[TABLE]

where $0<\gamma,\delta<1$ are two positive constants depending on $d,\alpha$ and $\Gamma$ and where the notation $|S|$ is used to denote the cardinality of a subset $S\subset\mathbb{Z}^{d}$.

On the other hand, the fact that $\mathcal{C}$ is Lipschitz allows us to find a circular cone $\mathcal{C}^{\prime}$ with vertex at the origin such that $\mathcal{C}^{\prime}\subset\mathcal{C}^{c}$. It follows then that there exists a positive constant $\mu$ (depending on $A$) such that for $R$ large enough

[TABLE]

We conclude from (2.5) and (2.6) that

[TABLE]

which implies (2.4) and completes the proof of (2.3).∎

*Proof of Theorem 2.2. * To prove the Carleson estimate (2.1) we first observe that the uniform ellipticity assumption implies that $u(\xi)\leq Ce^{C\,|\xi-\zeta|}u(\zeta)$, $\xi,\zeta\in\mathcal{C}\cap B_{3R}(y)$; where $C=C(d,\alpha,\Gamma,A)>0$. This local Harnack principle allows us to assume that the distance of $x$ from $\partial\mathcal{C}$ is sufficiently large. We shall denote by $\delta(x)$ ($x\>\in\>\mathcal{C}\cap B_{2R}(y)$) this distance and suppose that $\delta(x)\geq C$. The fact that $\mathcal{C}$ is Lipschitz combined with Harnack principle (Theorem 2.1) imply that

[TABLE]

where $\gamma$ and $C$ are positive constants depending on $d,\alpha,\Gamma$ and $A$.

Let $x\in\mathcal{C}\cap{B_{2R}(y)}$, and let us assume that

[TABLE]

where $\rho$ is the constant obtained in (2.2) and $\gamma$ the exponent that appears in (2.7). Let $x_{0}\in\partial\mathcal{C}$ such that $|x-x_{0}|=\delta(x)$. It follows easily from (2.8) and the fact that $\delta(x)$ is sufficiently large that $\overline{B_{3\sqrt{d}\delta(x)}(x_{0})}\subset B_{2R}(y)$. By Proposition 2.1 applied to the harmonic function $u$ in the domain $B_{{3}\sqrt{d}\delta(x)}(x_{0})\cap\mathcal{C}$, we have

[TABLE]

Let $z\in\mathcal{C}\cap\overline{B_{3\sqrt{d}\delta(x)}(x_{0})}$ satisfying

[TABLE]

We have

[TABLE]

Hence, thanks to (2.8)

[TABLE]

It follows that

[TABLE]

and therefore, by (2.9)

[TABLE]

where

[TABLE]

It remains to consider the case where

[TABLE]

In follows from (2.11) that

[TABLE]

where $8\sqrt{d}\varepsilon_{0}=\displaystyle 1-\left(\frac{1+\rho}{2}\right)^{\frac{1}{\gamma}}$ and, thanks to (2.7),

[TABLE]

Putting together (2.2) and (2.12) and taking the supremum over $\mathcal{C}\cap B_{2R}(y)$, we deduce that

[TABLE]

Using the fact that $(2R-|x-y|)\approx R$ for $x\,\in\mathcal{C}\cap B_{R}(y)$ we deduce the estimate (2.1).$\hfill\Box$

2.3. Boundary Harnack principle

Carleson estimate can be extended to the ratio $u/v$ of positive harmonic functions.

Theorem 2.3**.**

(Boundary Harnack principle)* Let $y\in\partial\mathcal{C}$ and $K>0$ large enough. Assume that $u$ and $v$ are two nonnegative harmonic functions in $\mathcal{C}\cap B_{3KR}(y)$. Assume that $u,\,v=0$ on $\partial\mathcal{C}\cap B_{2KR}(y)$. Then*

[TABLE]

where $C=C(d,\alpha,\Gamma,A,K)>0$.

The above formulation of the boundary Harnack principle follows the classical formulation but the proof of (2.13) which will be given below shows that the assumption $v=0$ on $\partial\mathcal{C}\cap B_{2R}(y)$ is not needed so that (2.1) constitutes a special case of (2.13).

The estimate (2.13) is an immediate consequence of the lower estimate contained in the following lemma.

For $y\in\partial\mathcal{C}$ and $R\geq r\geq 1$, we shall denote by

[TABLE]

For $R\geq r\geq 1$, the boundary of $\mathcal{C}_{R,r}$ is the union of three sets: the “bottom” $\partial\mathcal{C}_{R,r}\cap\mathcal{C}^{c}$, the “lateral side” $\partial\mathcal{C}_{R,r}\cap\{x\in\mathcal{C},\>0\leq\delta(x)\leq r\}$ and the “top” $\partial\mathcal{C}_{R,r}\cap\mathcal{D}_{R,r}$.

Lemma 2.1**.**

There exists a constant $K_{0}>0$ such that for all $K\geq K_{0}$ and for all $y\in\partial\mathcal{C},$ $r\geq 1$,

[TABLE]

where $\tau_{\mathcal{C}_{Kr,r}(y)}$ denotes the exit time from $\mathcal{C}_{Kr,r}(y)$.

Proof of Theorem 2.3. In order to derive estimate (2.13) from (2.14), we first observe that it is always possible to assume that $u(y+Re_{1})=v(y+Re_{1})=1$. For a large $R$, Carleson estimate (2.1) implies that the function $u$ is dominated by a positive constant $c_{0}$ in the region $B_{KR}(y)\cap\mathcal{C}$. This constant $c_{0}$ can be chosen so that by Harnack principle the lower estimate $v\geq\frac{1}{c_{0}}$ holds on $\mathcal{D}_{KR,R}(y)$. Let $v_{0}=c_{0}v$ and $u_{0}=\displaystyle\frac{u}{c_{0}}-v_{0}$. Let $x\in B_{R}(y)\cap\mathcal{C}$. We have:

[TABLE]

where the second inequality follows from (2.14). We deduce then that

[TABLE]

So that

[TABLE]

which completes the proof of (2.13). $\hfill\Box$

Proof of Lemma 2.1. To prove estimate (2.14) it suffices to show that if $u,\,v:\overline{\mathcal{C}_{Kr,r}(y)}\rightarrow\mathbb{R}$ (where $y\in\partial\mathcal{C}$, $r\geq 1$ are fixed) satisfy

[TABLE]

[TABLE]

then we have

[TABLE]

provided that $K\geq K_{0}$ is large enough.

First we prove that under (2.15) the function $u$ satisfies

[TABLE]

for appropriate constants $\alpha,\,\beta>0.$

The proof of (2.18) relies on the following construction.

We assume $K$ large enough and we define $\tilde{u}:{\overline{B_{Mr}(y)\cap\mathcal{C}}}\longrightarrow\mathbb{R}$ by

[TABLE]

where $0<M\leq K$ is chosen so that $\tilde{y}=y+(M-1)re_{1}$ satisfies

[TABLE]

Let $\mathcal{U}=\left(B_{Mr}(y)\cap\mathcal{C}\right)\cap B_{2r}(\tilde{y})$ and $w:\overline{B_{2r}(\tilde{y})}\longrightarrow\mathbb{R}$ be defined by

[TABLE]

It is easy to see that $w$ is superharmonic. Let $\tilde{z}=y+(M-2)re_{1}$. By the same argument used in the proof of the lower estimate (2.4) combined with Harnack principle, we see that $w(\tilde{z})$ satisfies a lower estimate $w(\tilde{z})\geq c$. It follows then that $\tilde{u}(\tilde{z})\geq c$.

Since $u\geq 1$ on $\partial\mathcal{C}_{Kr,r}(y)\cap\mathcal{D}_{Kr,r}(y)$, we deduce by the maximum principle that $u\geq\tilde{u}$ on $B_{r}(y)\cap\mathcal{C}$. Combining with Harnack inequality we de deduce (2.18).

It follows from (2.18) that if $x\in\overline{\mathcal{C}_{r,r}(y)}\,\backslash\,\mathcal{C}_{r,r/K}(y)$ then we have

[TABLE]

Let us now prove that there exists $N>0$ such that

[TABLE]

Let $j=1,\ldots,\lfloor\frac{K-1}{2}\rfloor$ and let $x_{j}\in\partial\mathcal{C}_{(2j-1)r,r}(y)$ be such that

[TABLE]

Let $\mathcal{U}_{j}=B_{2r}(x_{j})\cap{\mathcal{C}_{(2j+1)r,r}(y)}$ and $\tau_{\mathcal{U}_{j}}$ be the exit time from $\mathcal{U}_{j}$. By the same argument used in the proof of (2.4) we see that

[TABLE]

Using (2.16) (in particular, the fact that $v\leq 0$ on $\partial\mathcal{C}_{Kr,r}(y)\cap\mathcal{D}_{Kr,r}(y)$) we deduce then that

[TABLE]

where $0<\theta<1$. Hence

[TABLE]

Iterating this estimate we obtain

[TABLE]

which proves (2.20). It follows from (2.20) that

[TABLE]

provided that $K$ is large enough.

From the previous considerations it follows that

[TABLE]

with

[TABLE]

thanks to (2.19) and, thanks to (2.21),

[TABLE]

with

[TABLE]

In particular, we have

[TABLE]

It follows that $u_{1}$, $v_{1}$ satisfy the same assumptions as $u$, $v$ with $r$ replaced by $r/K$. We can then iterate and define $u_{i}$, $v_{i}$ such that

[TABLE]

$i=1,2\ldots$. We deduce then that

[TABLE]

Let $x\in B_{r}(y)$ and $\tilde{x}\in\partial\mathcal{C}$ satisfying $\delta(x)=|x-\tilde{x}|$. Then

[TABLE]

Replacing $K$ by $K+2$ in the previous considerations we deduce that $u\geq v$ on $S(\tilde{x})$ that contains $x$. This shows that $u(x)\geq v(x)$ and completes the proof of (2.17). $\hfill\Box$

Proof of Theorem 1.1. The proof is the same as [8]. We observe that instead of Carleson estimate, we can simply use the estimate

[TABLE]

which follows from uniform ellipticity. The advantage of this estimate is that it works for all connected infinite domains, and not just for domains satisfying Carleson estimate. It is already enough to run the diagonal process argument used in [8].

As in [8] the uniqueness can be deduced by essentially the same method as in [1, Proof of Theorem 3] and [2, Lemma 6.2]. $\hfill\Box$