Martin Boundary and Integral Representation for Harmonic Functions of Symmetric Stable Processes

Zhen-Qing Chen ^{∗} Departments of Mathematics and Statistics Cornell University Ithaca, NY 14853, USA Email: zchen@math.cornell.edu and Renming Song Department of Mathematics University of Illinois Urbana, IL 61801, USA Email: rsong@math.uiuc.edu

Abstract

Martin boundaries and integral representations of positive functions which are harmonic in a bounded domain D with respect to Brownian motion are well understood. Unlike the Brownian case, there are two different kinds of harmonicity with respect to a discontinuous symmetric stable process. One kind are functions harmonic in D with respect to the whole process X, and the other are functions harmonic in D with respect to the process X^{D} killed upon leaving D. In this paper we show that for bounded Lipschitz domains, the Martin boundary with respect to the killed stable process X^{D} can be identified with the Euclidean boundary. We further give integral representations for both kinds of positive harmonic functions. Also given is the conditional gauge theorem conditioned according to Martin kernels and the limiting behaviors of the h-conditional stable process, where h is a positive harmonic function of X^{D}. In the case when D is a bounded C^{1,1} domain, sharp estimate on the Martin kernel of D is obtained.

Keywords and phrases: Symmetric stable processes, harmonic functions, conditional stable processes, and Martin boundaries.

Running Title: Martin Boundary for Stable Processes

∗The research of this author is supported in part by an NSA grant

1 Introduction

Martin boundary and integral representation for harmonic functions of diffusions processes (or of elliptic differential operators) are well studied. However there is little detailed analysis of these for Markov processes with jumps (or for integro-differential operators). In this paper we take a closer look at an important class of discontinuous Markov processes—symmetric α-stable processes with 0 < α < 2, and study the notion and integral representation of harmonic functions for these processes, where some new phenomena arise. We hope that this paper can shed some new light on the potential theory of general Markov processes.

Symmetric stable processes constitute an important subfamily of Lévy processes. A symmetric α-stable process X on R^{n} is a Lévy process whose transition density p(t, x−y) relative to the Lebesgue measure is uniquely determined by its Fourier transform ∫_{R^{n}} e^{ix·ξ}p(t, x)dx = e^{−t|ξ|^{α}}. Here α must be in the interval (0, 2]. When α = 2, we get a Brownian motion running with a time clock twice as fast as the standard one. In this paper, symmetric stable processes are referred to the case when 0 < α < 2, unless otherwise specified.

Unlike the Brownian case, there are two different kinds of harmonicity with respect to symmetric stable processes, one kind are functions harmonic in D with respect to the whole process X, and the other are functions harmonic in D with respect to the process X^{D} killed upon leaving D. The theory of Martin kernel and Martin boundary for the killed process X^{D} is known from the general theory. This Martin boundary gives an integral representation for positive functions harmonic in a domain D with respect to the killed process X^{D}. We show that when D is a bounded Lipschitz domain, the Martin boundary with respect to killed symmetric stable process X^{D} in D coincide with the Euclidean boundary. It seems that integral representations of positive functions harmonic in a domain D with respect to the whole processes X have not been studied in the literature. In this paper, we present an integral representation for positive functions harmonic in a domain D with respect to the whole processes and this representation is shown to be unique. In particular, this implies that any harmonic function with respect to the whole process is uniquely determined by its values in D. In the case when D is a bounded C^{1,1} domain, sharp estimates on the Martin kernel are given. As a consequence of these estimates, we prove a conditional gauge theorem conditioned according to Martin kernel. We also study the limiting behavior of the h–conditioned symmetric stable process in D when h is a positive harmonic function of X^{D}, and the the limiting behavior of the h–conditioned symmetric stable process will provide a probabilistic interpretation to positive harmonic functions of X^{D}.

This paper is organized as follows. The definitions of harmonic and superharmonic functions with respect to symmetric stable processes are given in section 2. Some important facts about those harmonic functions are also given in section 2. Section 3 contains results on Martin boundary and conditional gauge theorem. Integral representations of positive functions harmonic in a domain D with respect to the whole processes are given in section 4.

In the sequel, we will use v^{+} and v^{−} to denote the positive and negative part of a realvalued Borel measurable function v, i.e., v^{+} = max{v, 0} and v^{−} = max{−v, 0}.

Acknowledgement. After the first draft of this paper was written, we met Krzysztof Bogdan and learned about his paper [6] at a Stochastic Analysis workshop held at the MSRI at Berkeley from March 22–27, 1998. Some of our results had been independently and simultaneously obtained by him in [6]. However the approach of this paper and that of [6] are different. We thank Krzysztof Bogdan for the very interesting and helpful discussions at the MSRI, Berkeley and at University of Washington, Seattle. We are also grateful to Chris Burdzy, Eugene Dynkin, Pat Fitzsimmons, Tom Kurtz and Sergei Kuznetsov for very helpful discussions about this paper. Parts of the research for this paper were conducted while the authors were visiting the CMS at University of Wisconsin at Madison and the MSRI in Berkeley. Financial support from two institutions are gratefully acknowledged.

2 Definitions and Preliminaries

In sections 2–4 of this paper, we always assume n ≥ 2. Let X = (Ω, F, X_{t}, F_{t}, P_{x}) be a symmetric α-stable process on R^{n} with 0 < α < 2, where {F_{t}, t ≥ 0} is the minimal admissible σ-fields generated by X. The process X is transient and we are going to use G to denote the potential of X. We know that the Green function of X is given by

G(x, y) = 2^{−α}π^{− \frac{n}{2}} Γ (\frac{n − α}{2}) Γ (\frac{α}{2})^{−1} |x − y|^{α−n}.

For a domain D in R^{n}, let τ_{D} = inf{t > 0 : X_{t} ∉ D}. Adjoin a cemetery point ∂ to D and set

X_{t}^{D}(ω) = \{ \begin{matrix}X_{t}(ω) & if t < τ_{D}, \\ ∂ & if t ≥ τ_{D}. \end{matrix}

X^{D} is a strong Markov process with state space D_{∂} = D∪{∂}, which is called the subprocess of the symmetric α–stable process X killed upon leaving D, or simply the symmetric α–stable process in D. We are going to use G_{D} to denote the Green function of X^{D}.

For Brownian motion or other diffusion processes, there is only one kind of harmonicity on a domain D. However, for symmetric stable processes, there are two kinds of harmonic functions on D: functions which are harmonic in D with respect to the killed process X^{D} and functions which are harmonic in D with respect to the process X. The precise definitions of these two kinds of harmonic functions are as follows.

Definition 2.1 Let D be a domain in R^{n}. A locally integrable function f defined on D taking values in (−∞, ∞] and satisfying the condition ∫_{{|x|>1}∩D} |f(x)||x|^{−(n+α)}dx < ∞ is said to be

1) harmonic with respect to X^{D} if f is continuous in D and for each x ∈ D and each ball B(x, r) with \overline{B(x, r)} ⊂ D,

f(x) = E_{x}[f(X_{τ_{B(x,r)}}); τ_{B(x,r)} < τ_{D}];

2) superharmonic respect to X^{D} if f is lower semicontinuous in D and for each x ∈ D and each ball B(x, r) with \overline{B(x, r)} ⊂ D,

f(x) ≥ E_{x}[f(X_{τ_{B(x,r)}}); τ_{B(x,r)} < τ_{D}].

The next definition is taken from Landkof [17].

Definition 2.2 Let D be a domain in R^{n}. A locally integrable function f defined on R^{n} taking values in (−∞, ∞] and satisfying the condition ∫_{{|x|>1}} |f(x)||x|^{−(n+α)}dx < ∞ is said to be

1) harmonic in D with respect to X if f is continuous in D and for each x ∈ D and each ball B(x, r) with \overline{B(x, r)} ⊂ D,

f(x) = E_{x}[f(X_{τ_{B(x,r)}})];

2) superharmonic in D with respect to X if f is lower semicontinuous in D and for each x ∈ D and each ball B(x, r) with \overline{B(x, r)} ⊂ D,

f(x) ≥ E_{x}[f(X_{τ_{B(x,r)}})].

Remark 2.1 (1) If f is a lower semicontinuous function defined on D taking values in (−∞, ∞], then f is bounded from below on any subdomain whose closure is contained in D. Thus for such kind of function f which is locally integrable and satisfying ∫_{{|x|>1}} |f(x)||x|^{−(n+α)}dx < ∞, it follows from estimate (2.2) below that E_{x}[f^{−}(X_{τ_{B(x,r)}})] < ∞ for any ball B(x, r) with \overline{B(x, r)} ⊂ D. Therefore the expections in Definitions 2.1 and 2.2 are well defined.

(2) For a function f which is (super)harmonic with respect to X^{D}, if we extend it to be zero off the domain D, then the resulting function is (super)harmonic in D with respect to X.

(3) Conversely, if f is a non-negative superharmonic in D with respect to X, then clearly it is a superharmonic with respect to X^{D}.

We now record some facts, which will be used later, concerning bounded Lipschitz domains and the exit distributions of X from a domain U. Recall that a bounded domain D in R^{n} is said to be a bounded Lipschitz domain with Lipschitz characteristic constants (r_{0}, A_{0}) if for every z ∈ ∂D, there is a local coordinate system (ξ_{1}, ξ^{(1)}) ∈ R × R^{n−1} with origin sitting at z and there is a Lipschitz function f defined on R^{n−1} with Lipschitz constant A_{0} such that D ∩ B(z, r_{0}) = B(z, r_{0}) ∩ {ξ = (ξ_{1}, ξ^{(1)}) : ξ_{1} > f(ξ^{(1)})}. A domain U in R^{n} is said to satisfy the uniform exterior cone condition if there exist constants η > 0, r > 0 and a cone C = {x = (x_{1}, …, x_{n}) ∈ R^{n} : 0 < x_{n}, (x_{1}^{2} + ⋯ + x_{n−1}^{2})^{1/2} < ηx_{n}} such that for every z ∈ ∂U, there is a cone C_{z} with vertex z, isometric to C and satisfying C_{z} ∩ B(z, r) ⊂ U^{c}. It is well known that bounded Lipschitz domains satisfy the uniform exterior cone condition. Bogdan showed in [5] that for a bounded domain U satisfying the uniform exterior cone condition,

P_{x}(X_{τ_{U}} ∈ ∂U) = 0 for all x ∈ U.  (2.1)

In [8], Chen and Song showed that if U is a bounded C^{1,1} domain in R^{n}, there is a Poisson kernel K_{U}(x, z) defined on U ×(R^{n}\\overline{U}) such that for any bounded Borel measurable function ϕ, E_{x}[ϕ(X_{τ_{U}})] = ∫_{U^{c}} ϕ(z)K_{U}(x, z)dz, where dz is the Lebesgue measure on R^{n}. Furthermore there exists a C = C(U, α) > 1 such that for x ∈ U and z ∈ \overline{U}^{c},

\frac{δ(x)^{α/2}}{C δ(z)^{α/2}(1 + δ(z))^{α/2}} \frac{1}{|x − z|^{n}} ≤ K_{U}(x, z) ≤ \frac{C δ(x)^{α/2}}{δ(z)^{α/2}(1 + δ(z))^{α/2}} \frac{1}{|x − z|^{n}} ,  (2.2)

where δ(y) = dist(y, ∂U) is the Euclidean distance from point y to the set ∂U. Here a domain U is C^{1,1} means that for every z ∈ ∂D, there exists a r > 0 such that B(z, r) ∩ ∂D is the graph of a function whose first derivatives are Lipschitz.

It is well known that for any domain D, there exists an increasing sequence of bounded C^{∞}-smooth domains \{D_{k}\}_{k≥1} such that \overline{D_{k}} ⊂ D_{k+1} for k ≥ 1 and that ⋃_{k=1}^{∞} D_{k} = D (see, for example, Lemma 2.4 of [7]).

Theorem 2.1 Suppose that D is a bounded domain in R^{n}. If h is superharmonic in D with respect to X, then for any domain D_{1} ⊂ \overline{D_{1}} ⊂ D, E_{x}[h^{−}(X_{τ_{D_{1}}})] < ∞ and

h(x) ≥ E_{x}[h(X_{τ_{D_{1}}})] for every x ∈ D_{1}.

Proof. For a fixed ϵ > 0 and each x ∈ D_{1} we put

r(x) = \frac{1}{2}δ(x, ∂D_{1}) ∧ ϵ, B(x) = B(x, r(x)),

where δ(x, ∂D_{1}) denotes the Euclidean distance between x and ∂D_{1}. Define a sequence of stopping times {T_{m}, m ≥ 1} as follows:

T_{1} = inf{t > 0 : X(t) ∉ B(X_{0})},

and for m ≥ 2,

T_{m} = \{ \begin{matrix}T_{m−1} + T_{B(X_{T_{m−1}})} ∘ θ_{T_{m−1}} & if X_{T_{m−1}}) ∈ D_{1}, \\ τ_{D_{1}} & otherwise. \end{matrix}

The superharmonicity of h and the strong Markov property imply that

h(X_{T_{m−1}}) ≥ E_{x}[h(X_{T_{m}})|F_{T_{m−1}}].

Thus {h(X_{T_{m}}), m ≥ 1} is a supermartingale under P_{x}.

We claim that for each x ∈ D_{1},

P_{x}( lim_{m→∞} T_{m} = τ_{D_{1}}) = 1.

It is clear that P_{x}–a.s., T_{m} ↑ and T_{m} ≤ τ_{D_{1}} < ∞ because D is bounded. Let T_{∞} = lim_{m→∞} T_{m}. Then X_{T_{∞}} = lim_{m→∞} X_{T_{m}} by quasi left continuity. On the set {T_{∞} < τ_{D_{1}}}, we have X_{T_{∞}} ∈ D_{1}. Then for all sufficiently large values of m, we have δ(X_{T_{m−1}}, ∂D_{1}) > \frac{1}{2}δ(X_{T_{∞}}, ∂D_{1}) > 0 and |X_{T_{m−1}} − X_{T_{m}}| < \frac{1}{4}δ(X_{T_{∞}}, ∂D_{1}) ∧ ϵ. But by the definition of T_{m}, |X_{T_{m−1}} − X_{T_{m}}| > \frac{1}{2}δ(X_{T_{m−1}}, ∂D_{1}) ∧ ϵ. These inequalities are incompatible. Hence P_{x}(T_{∞} < τ_{D_{1}}) = 0.

Put A = \{τ_{D_{1}} = T_{m} for some t ≥ 1\}. Since h is bounded from below on \overline{D_{1}}, we have by Fatou’s lemma

lim inf_{m→∞} E_{x}[h(X_{T_{m}}); T_{m} < τ_{D_{1}}] ≥ E_{x}[lim inf_{m→∞} h(X_{T_{m}}); A^{c}] ≥ E_{x}[h(X_{τ_{D_{1}}}); A^{c}].

Take two smooth domains D_{2} and D_{3} such that \overline{D_{1}} ⊂ D_{2} ⊂ \overline{D_{2}} ⊂ D_{3} ⊂ \overline{D_{3}} ⊂ D, then h is bounded from below on \overline{D_{3}}. Since

h(X_{τ_{D_{1}}}) = h(X_{τ_{D}})1_{{X_{τ_{D_{1}}} ∈D_{3}}} + h(X_{τ_{D_{1}}})1_{{X_{τ_{D_{1}}} ∉D_{3}}}
= h(X_{τ_{D_{1}}})1_{{X_{τ_{D_{1}}} ∈D_{3}}} + h(X_{τ_{D_{2}}})1_{{X_{τ_{D_{1}}} ∉D_{3}}},

we have by estimate (2.2) with D_{2} in place of U and the integrability assumption about h in Definition 2.2 that

E_{x}[h^{−}(X_{τ_{D_{1}}})] < ∞.

Thus by Fatou’s Lemma

h(x) ≥ lim inf_{m→∞} E_{x}[h(X_{T_{m}})]
≥ lim inf_{m→∞} E_{x}[h(X_{τ_{D_{1}}}), T_{m} = τ_{D_{1}}] + lim inf_{m→∞} E_{x}[h(X_{T_{m}}), T_{m} < τ_{D_{1}}]
≥ E_{x}[h(X_{τ_{D_{1}}}); A] + E_{x}[h(X_{τ_{D_{1}}}); A^{c}]
= E_{x}[h(X_{τ_{D_{1}}})].

This completes the proof.

Theorem 2.2 Suppose that D is a bounded domain in R^{n}. If h is a harmonic in D with respect to X, then for any domain D_{1} ⊂ \overline{D_{1}} ⊂ D, h(X_{τ_{D_{1}}}) is P_{x}-integrable and

h(x) = E_{x}[h(X_{τ_{D_{1}}})] for every x ∈ D_{1}.  (2.3)

Proof. We can always take a smooth domain D_{2} such that D_{1} ⊂ \overline{D_{1}} ⊂ D_{2} ⊂ \overline{D_{2}} ⊂ D. If we could prove that for any x ∈ D_{2}, h(X_{τ_{D_{2}}}) is P_{x}-integrable and

h(x) = E_{x}[h(X_{τ_{D_{2}}})].

then by strong Markov property we immediately get h(X_{τ_{D_{1}}}) is P_{x}-integrable and

h(x) = E_{x}[h(X_{τ_{D_{1}}})], x ∈ D_{1}.

Therefore we can assume, without loss of generality, that D_{1} is a smooth domain.

Define T_{m} as in the proof of the previous theorem, then in this case {h(X_{T_{m}}), m ≥ 1} is a martingale under P_{x} for any x ∈ D_{1}. By (2.1) with D_{1} in place of U, we have P_{x}(τ_{D_{1}} = T_{m} for some m ≥ 1) = 1. Since h is bounded on D_{1}, we have

|E_{x}[h(X_{T_{m}}), T_{m} < τ_{D_{1}}]| ≤ CP_{x}(T_{m} < τ_{D_{1}}) → 0.

Take a domain D_{2} such that \overline{D_{1}} ⊂ D_{2} ⊂ \overline{D_{2}} ⊂ D, then h is continuous and therefore bounded on \overline{D_{2}}. By the estimate (2.2) with D_{1} in place of U and the integrability assumption about h in Definition 2.2 we have E_{x}[|h|(X_{τ_{D_{1}}})] < ∞. Thus by the dominated convergence theorem

lim_{m→∞} E_{x}[h(X_{τ_{D_{1}}}), T_{m} = τ_{D_{1}}] = E_{x}[h(X_{τ_{D_{1}}})].

Therefore

h(x) = lim_{m→∞} E_{x}[h(X_{T_{m}})]
= lim_{m→∞} E_{x}[h(X_{τ_{D_{1}}}), T_{m} = τ_{D_{1}}] + lim_{m→∞} E_{x}[h(X_{T_{m}}), T_{m} < τ_{D_{1}}]
= E_{x}[h(X_{τ_{D_{1}}})].

Similarly, we have the following result for functions harmonic with respect to X^{D}.

Theorem 2.3 Suppose that D is a bounded domain in R^{n}. If h is superharmonic in D with respect to X^{D}, then for any domain D_{1} ⊂ \overline{D_{1}} ⊂ D, E_{x}[h^{−}(X_{τ_{D_{1}}}^{D})] < ∞ and

h(x) ≥ E_{x}[h(X_{τ_{D_{1}}}^{D})] for every x ∈ D_{1}.

If h is harmonic in D with respect to X^{D}, then for any domain D_{1} ⊂ \overline{D_{1}} ⊂ D, h(X_{τ_{D_{1}}}^{D}) is P_{x}-integrable and

h(x) = E_{x}[h(X_{τ_{D_{1}}}^{D})] for every x ∈ D_{1}.

Theorem 2.4 Suppose that D is a bounded domain and h is harmonic in D with respect to X and continuous on \overline{D}, then h(X_{τ_{D}}) is P_{x}-integrable and

h(x) = E_{x}[h(X_{τ_{D}})], for each x ∈ D.

Proof. Take an increasing sequence of smooth domains \{D_{m}\}_{m≥1} such that \overline{D_{m}} ⊂ D_{m+1} and ⋃_{m=1}^{∞} D_{m} = D. Set τ_{m} = τ_{D_{m}}. Then τ_{m} ↑ τ_{D} and lim_{m→∞} X_{τ_{m}} = X_{τ_{D}} by quasi-left continuity of X. Set

A = {τ_{m} = τ_{D} for some m ≥ 1}.  (2.4)

From Theorem 2.2 we know that for any m ≥ 1,

h(x) = E_{x}[h(X_{τ_{m}})], x ∈ D_{m}.

Since h is continuous on \overline{D}, we have by dominated convergence theorem that

lim_{m→∞} E_{x}[h(X_{τ_{m}}), τ_{m} < τ_{D}] = E_{x}[h(X_{τ_{D}}), A^{c}].  (2.5)

Since h is continuous on \overline{D}, we can find two smooth domains U_{1} and U_{2} such that \overline{D} ⊂ U_{1} ⊂ \overline{U_{1}} ⊂ U_{2} and that h is bounded on \overline{U_{2}}. Since

|h|(X_{τ_{D}}) = |h|(X_{τ_{D}})1_{{X_{τ_{D}} ∈U_{2}}} + |h|(X_{τ_{D}})1_{{X_{τ_{D}} ∉U_{2}}}
= |h|(X_{τ_{D}})1_{{X_{τ_{D}} ∈U_{2}}} + |h|(X_{τ_{U_{1}}})1_{{X_{τ_{U_{1}}} ∉U_{2}}},

we have by estimate (2.2) with U_{1} in place of U that

E_{x}[|h|(X_{τ_{D}})] < ∞.

Thus by the dominated convergence theorem

h(x) = lim_{m→∞} E_{x}[h(X_{τ_{m}})]
= lim_{m→∞} E_{x}[h(X_{τ_{D}}), τ_{m} = τ_{D}] + lim_{m→∞} E_{x}[h(X_{τ_{m}}), τ_{m} < τ_{D}]
= E_{x}[h(X_{τ_{D}})].

Remark 2.2 If D is a bounded domain satisfying the uniform exterior cone condition, then the conclusion of Theorem 2.4 holds for any harmonic function h in D with respect to X that is bounded in a neighborhood of \overline{D}. This is because in this case by (2.1) P_{x}(A) = 1 for x ∈ D, where A is the set defined in (2.4) and the term in (2.5) vanishes. The rest of the argument goes through without the continuous assumption on h up to the boundary ∂D.

Obviously there are plenty of bounded functions which are harmonic in D with respect to the whole processes X. The following results says that, when D is a bounded domain satisfying the uniform exterior cone condition, the only bounded function which is harmonic in D with respect to X^{D} is constant zero.

Theorem 2.5 Suppose that D is a bounded domain in R^{n} satisfying the uniform exterior cone condition. If h is a bounded function harmonic in D with respect to X^{D}, then h must be identically zero.

Proof. Take an increasing sequence of smooth domains D_{m} such that D_{m} ⊂ \overline{D_{m}} ⊂ D_{m+1} ⊂ \overline{D_{m+1}} ⊂ D and set τ_{m} = τ_{D_{m}}. Then τ_{m} ↑ τ_{D}. By (2.1), we know that P_{x}(τ_{D} = τ_{m} for some m ≥ 1) = 1 for x ∈ D. From Theorem 2.2 we get that

|h(x)| = |E_{x}[h(X_{τ_{m}}^{D})]|
≤ C P_{x}(τ_{m} < τ_{D}) → 0.

The proof is now complete.

3 Martin Boundary

Superharmonic and harmonic functions with respect to X^{D} have been studied in the context of general theory of Markov processes and their potential theory (see, for instance, Kunita-Watanabe [15]). From the general theory, we know that positive harmonic functions with respect to X^{D} admit Martin representations. However, no particular attention was paid to the special case of harmonic functions with respect to the killed stable process. For instance, the relationship between the Martin boundary of X^{D} and the Euclidean boundary ∂D of D has not been studied.

In this section we assume that D is a bounded Lipschitz domain. In the first part of this section we are going to show that the Martin boundary of X^{D} and the Euclidean boundary ∂D coincide. Our proof of the identification between the Martin boundary and the Euclidean boundary is similar to the argument of Bass–Burdzy [2] in the Brownian motion case.

Fix x_{0} ∈ D and set

M_{D}(x, y) = \frac{G_{D}(x, y)}{G_{D}(x_{0}, y)}, x, y ∈ D.

The Martin boundary is the set ∂_{M}D = D^{∗} \ D, where D^{∗} is the smallest compact set for which M_{D}(x, y) is continuous in y in the extended sense.

Lemma 3.1 Suppose that D is a bounded Lipschitz domain. Then for any γ > 0,

lim_{x→∂D} G_{D}(x, y) = 0

uniformly on D_{γ} = {y ∈ D : δ(y, ∂D) ≥ γ}.

Proof. Suppose that C = {x = (x_{1}, …, x_{n}) ∈ R^{n} : 0 < x_{n}, (x_{1}^{2} + ⋯ + x_{n−1}^{2})^{1/2} < ηx_{n}} is a cone with vertex at the origin O. For any r > 0, set

C_{r} = C ∩ B(O, r)

and

T_{C_{r}} = inf\{t > 0, X_{t} ∈ \overline{C_{r}}\}.

One can easily show (similar to the proof of Proposition 1.19 of [11]) that for any t > 0, the function

x ↦ P_{x}(t < T_{C_{r}})

is upper semi–continuous in R^{n}. Thus for any s > 0, we have

lim sup_{x→O} P_{x}(T_{C_{r}} > s) ≤ P^{O}(T_{C_{r}} > s) = 0

since P^{O}(T_{C_{r}} = 0) = 1. Now use the fact that D satisfies the uniform exterior cone condition we can easily see that

lim_{x→z∈∂D} P_{x}(τ_{D} > s) = 0

uniformly in z ∈ ∂D, i.e., for any ϵ > 0 there exists δ′ > 0 such that

P_{x}(τ_{D} > s) < ϵ,  if δ(x, ∂D) < δ′.  (3.1)

We know that

G_{D}(x, y) = G(x, y) − E_{x}[G(X_{τ_{D}}, y)]

and that G(x, y) is bounded D^{c} × D_{γ}. Now use the fact (3.1) and argue along the line of the proof of Theorem 1.23 of [11] we easily arrive at our conclusion.

Take a positive number ϵ < δ(x_{0}, ∂D)/4.

Lemma 3.2 Suppose x ∈ D with |x − x_{0}| > 4ϵ. There exists a constant c_{1} = c_{1}(ϵ, D, x, x_{0}) such that

M_{D}(x, y) ≤ c_{1} for y ∈ D \ (\overline{B(x_{0}, ϵ)} ∪ \overline{B(x, ϵ)}) .

Proof. Pick y_{0} ∈ ∂B(x_{0}, 2ϵ). By the explicit formula for the Green function of balls (see [3] for instance) we know that

G_{D}(x_{0}, y_{0}) ≥ G_{B(x_{0},3ϵ)}(x_{0}, y_{0}) ≥ δ(ϵ) > 0.

On the other hand, we know that G_{D}(x, y_{0}) ≤ 2^{−α}π^{−n/2}Γ((n−α)/2)Γ(α/2)^{−1}|x−y_{0}|^{α−n}. Therefore G_{D}(x, y_{0}) is bounded above by a constant depending on ϵ in x ∈ D with |x−x_{0}| > 4ϵ. Thus M_{D}(x, y_{0}) is bounded above in x ∈ D with |x − x_{0}| > 4ϵ. But from the boundary Harnack principle (see [5]) we get that M_{D}(x, y) is comparable to M_{D}(x, y_{0}) for all points y in D \ (B(x, ϵ) ∪ B(x_{0}, ϵ)). The lemma follows.

Lemma 3.3 Let x, x_{0}, ϵ be as above. Then M_{D}(x, y) is a Hölder continuous function of y in D \ (\overline{B(x_{0}, ϵ)} ∪ \overline{B(x, ϵ)}) with Hölder exponent and coefficient depending only on x, x_{0}, ϵ and D.

Proof. For any set A, we define that

Osc_{A} f = sup_{y∈A} f(y) − inf_{y∈A} f(y).

Let f(y) = M_{D}(x, y). Let y_{0} ∈ D_{ϵ} = D \ (\overline{B(x_{0}, ϵ)} ∪ \overline{B(x, ϵ)}). Since by Lemma 3.2 f is bounded by c_{1} on D_{ϵ}, Osc_{D_{ϵ}} f ≤ c_{1}. So it suffices to show that there exists ρ = ρ(ϵ, D, x, x_{0}) < 1 such that

Osc_{D∩B(y_{0},r)} f ≤ ρ Osc_{D∩B(y_{0},2r)} f for r < ϵ/4.  (3.2)

Suppose r < ϵ/4, and let g be the ratio of any two positive harmonic functions on D_{ϵ/4} vanishing continuously on D^{c}. By considering ag + b for suitable a and b, we may assume

sup_{D∩B(y_{0},2r)} g = 1, inf_{D∩B(y_{0},2r)} g = 0.

If sup_{D∩B(y_{0},r)} g ≤ 1/2, then since inf_{D∩B(y_{0},2r)} g ≥ 0, we have

Osc_{D∩B(y_{0},r)} g ≤ \frac{1}{2} = \frac{1}{2}Osc_{D∩B(y_{0},2r)} g.

If sup_{D∩B(y_{0},r)} g ≥ 1/2, there exists a point y_{1} in D ∩ B(y_{0}, r) with g(y_{1}) ≥ 1/2. But then by the boundary Harnack principle with V = \{x : δ(x, D) < ϵ\} \ (\overline{B}(x_{0}, r) ∪ \overline{B}(x, r)) and K = \overline{D} \ (\overline{B}(x_{0}, r) ∪ \overline{B}(x, r)) , there exists a constant c_{2} = c_{2}(ϵ, D, x, x_{0}) ∈ (0, 1) such that

inf_{D∩B(y_{0},r)} g ≥ c_{2}g(y_{1}).

Since sup_{D∩B(y_{0},r)} g ≤ 1, in this case we have

Osc_{D∩B(y_{0},r)} g ≤ 1 − \frac{c_{2}}{2} = (1 − \frac{c_{2}}{2} ) Osc_{D∩B(y_{0},2r)} g.

So we have (3.1) with ρ = max \{\frac{1}{2}, 1 − \frac{c_{2}}{2} \}. Therefore M_{D}(x, y) is a (globlly) Hölder continuous in y ∈ D \ (\overline{B(x_{0}, ϵ)} ∪ \overline{B(x, ϵ)}).

A direct consequence of Lemma 3.3 is that M_{D}(x, y) = G_{D}(x, y)/G_{D}(x_{0}, y) converges when y → z ∈ ∂D. Let the limit be denoted as M_{D}(x, z). This implies that the Martin boundary of D can identified with a subset of ∂D.

It is also well known that for a bounded Lipschitz domain D with Lipschitz characteristic constants (r_{0}, A_{0}), there exists κ = κ(A_{0}) ∈ (0, 1) such that for every ϵ ∈ (0, r_{0}) and z ∈ ∂D, there is a point A_{ϵ}(z) ∈ D ∩ B(z, r) such that B(A_{ϵ}(z), κr) ⊂ D ∩ B(z, r). It is not difficult to show the following (cf. Lemma 6 of Bogdan [6]).

Lemma 3.4 For any z ∈ ∂D, M_{D}(·, z) is harmonic with respect to X^{D}.

Proof. Clearly any fixed x ∈ D and and r < δ(x, ∂D),

M_{D}(x, y) = E_{x} [M_{D}(X_{τ_{B(x,r)}}, y); τ_{B(x,r)} < τ_{D}] for y ∈ D \ \overline{B(x, r)}.

In particular,

M_{D}(x, A_{ϵ}(z)) = E_{x} [M_{D}(X_{τ_{B(x,r)}}, A_{ϵ}(z)); τ_{B(x,r)} < τ_{D}]  (3.3)

for any 0 < ϵ < min{r, r_{0}}. By Fatou’s lemma,

M_{D}(x, z) ≥ E_{x} [M_{D}(X_{τ_{B(x,r)}}, z); τ_{B(x,r)} < τ_{D}] .

Therefore M_{D}(X_{τ_{B(x,r)}}, z) is P_{x}-integrable. Put

ϵ_{0} = min \{\frac{δ(x_{0}, ∂D)}{4}, \frac{r_{0}}{2} , \frac{r}{4}\} ,

Then by Lemma 13 of [5] we get that there exists C_{1} = C_{1}(D) > 0 such that for any y ∈ D ∩ B(z, ϵ_{0}) and ϵ ∈ (0, ϵ_{0}),

M_{D}(w, A_{ϵ}(z)) ≤ C_{1}M_{D}(w, y) for w ∈ D \ B(z, 2ϵ).

Letting y → z we get that for any ϵ ∈ (0, ϵ_{0}),

M_{D}(w, A_{ϵ}(z)) ≤ C_{1}M_{D}(w, z) for w ∈ D \ B(z, 2ϵ).  (3.4)

For w ∈ D ∩B(z, 2ϵ), |w−x| > 3r/2 and thus by the explicit formula for K_{B(x,r)} we know that there is a constant C_{2} = C_{2}(r) > 0 such that K_{B(x,r)}(x, w) ≤ C_{2} for w ∈ D ∩ B(z, 2ϵ). Hence for any w ∈ D ∩ B(z, 2ϵ),

E_{x} [M_{D}(X_{B(x,r)}, A_{ϵ}(z)); X_{B(x,r)} ∈ D ∩ B(z, 2ϵ)]
≤ \frac{C_{2}}{G_{D}(x_{0}, A_{ϵ}(z))} ∫_{D∩B(z,2ϵ)} G(w, A_{ϵ}(z))dw
≤ \frac{C_{3}}{G_{D}(x_{0}, A_{ϵ}(z))} ∫_{D∩B(z,2ϵ)} |w − A_{ϵ}(z)|^{α−n}dw
≤ \frac{C_{4}}{G_{D}(x_{0}, A_{ϵ}(z))}ϵ^{α}.

From Lemma 5 of [5] we know that there exists a constant C_{5} = C_{5}(D, x_{0}) > 0 and positive number γ = γ(D) < α such that G_{D}(x_{0}, A_{ϵ}(z)) ≥ C_{5}ϵ^{γ}. Therefore

E_{x} [M_{D}(X_{B(x,r)}, A_{ϵ}(z)); X_{B(x,r)} ∈ D ∩ B(z, 2ϵ)] ≤ C_{4}C_{5}^{−1}ϵ^{α−γ}.  (3.5)

Now combine (3.4) and (3.5) we see that the family of functions {M_{D}(X_{B(x,r)}), A_{ϵ}(z)) : 0 < ϵ < ϵ_{0}} is P_{x}-uniformly integrable. Letting ϵ → 0 in (3.3) yields

M_{D}(x, z) = E_{x} [M_{D}(X_{τ_{B(x,r)}}, z); τ_{B(x,r)} < τ_{D}] for any r < δ(x, ∂D).

Thus M_{D}(·, z) is harmonic with respect to X^{D}.

The next result tells that each Euclidean boundary point corresponds to a different nonnegative harmonic function. Hence the Martin boundary can not be identified with a proper subset of the Euclidean boundary.

Lemma 3.5 If M_{D}(·, z_{1}) ≡ M_{D}(·, z_{2}) for z_{1}, z_{2} ∈ ∂D, then z_{1} = z_{2}.

Proof. Let ϵ > 0 be such that

ϵ < min \{\frac{δ(x_{0}, ∂D)}{4}, \frac{r_{0}}{2} \} ,

where r_{0} comes from the Lipschitz characteristic constants (r_{0}, A_{0}) of D. First we are going to show that M_{D}(x, w) → 0 uniformly in w ∈ ∂D as δ(x, ∂D \ B(w, 3ϵ)) → 0. In fact, for any given η > 0, by Lemma 3.1 there is a β = β(η, A_{0}, ϵ) > 0 such that G_{D}(x, A_{ϵ}(w)) < η for w ∈ ∂D and x ∈ D with δ(x, ∂D) < β. Let D_{0} be a smooth domain such that

\{x ∈ D; δ(x, ∂D) ≥ \frac{κϵ}{2}\} ⊂ D_{0} ⊂ \overline{D_{0}} ⊂ D.

Then for all w ∈ ∂D, δ(A_{ϵ}(w), ∂D_{0}) > κϵ/2 and so by Theorem 1.1 of [8]

G_{D}(x_{0}, A_{ϵ}(w)) ≥ G_{D_{0}}(x_{0}, A_{ϵ}(w))
≥ C(D_{0}) min \{\frac{1}{|x_{0} − A_{ϵ}(w)|^{n−α}} , \frac{δ(x_{0}, ∂D_{0})^{α/2}δ(A_{ϵ}(w), ∂D_{0})^{α/2}}{|x_{0} − A_{ϵ}(w)|^{n}} \}
≥ C(D_{0}) min \{\frac{1}{d_{D}^{n−α}} , \frac{(3ϵ)^{α/2}(κϵ/2)^{α/2}}{d_{D}^{n}} \}
:= C_{1} > 0,

where d_{D} is the diameter of D. Therefore

M_{D}(x, A_{ϵ}(w)) < η/C_{1}, ∀w ∈ ∂D,

whenever δ(x, ∂D) < β. Fix w ∈ ∂D. Clearly |x − w| > 2ϵ for any x ∈ D with δ(x, ∂D \ B(w, 3ϵ)) < ϵ. Now by Lemma 13 of [5] with r = ϵ, we get that there is a C_{2} = C_{2}(A_{0}) > 0 such that

M_{D}(x, y) ≤ C_{2}M_{D}(x, A_{ϵ}(w)) for y ∈ D ∩ B(w, ϵ)

whenever x ∈ D satisfies δ(x, ∂D \ B(w, 3ϵ)) < ϵ. Therefore

M_{D}(x, y) ≤ \frac{C_{2}}{C_{1}} η for y ∈ D ∩ B(w, ϵ)

whenever x ∈ D satisfies δ(x, ∂D \ B(w, 3ϵ)) < ϵ ∧ β. Letting y → w we get that

M_{D}(x, w) ≤ \frac{C_{2}}{C_{1}} η

for x ∈ D with δ(x, ∂D \ B(w, 3ϵ)) < ϵ ∧ β. Therefore

M_{D}(x, w) → 0 uniformly in w ∈ ∂D as δ(x, ∂D \ B(w, 3ϵ)) → 0.  (3.6)

Suppose that M_{D}(·, w) = M_{D}(·, z) for some w, z ∈ ∂D, w ≠ z, and let ϵ < |w − z|/8. By the above argument, M_{D}(x, w) → 0 uniformly when δ(x, ∂D \ B(w, 2ϵ)) → 0 or when δ(x, ∂D\B(z, 2ϵ)) → 0. Therefore M_{D}(x, w) → 0 uniformly as δ(x, ∂D) → 0. Since M_{D}(·, w) is a non-negative harmonic function with respect to X^{D} which continuously vanishes on ∂D, it must be identically zero by Theorem 2.5. This contradicts the fact that M_{D}(x_{0}, w) = 1. The proof is now complete.

Combining the lemmas above we get the following result.

Theorem 3.6 The Martin boundary of D can be identified with its Euclidean boundary ∂D.

Theorem 3.7 For each z ∈ ∂D, M_{D}(x, z) is minimal harmonic with respect to X^{D}.

Proof. Fix z ∈ ∂D and suppose h ≤ M_{D}(·, z), where h is a positive harmonic function with respect to X^{D}. By Theorem 3.6 we know that there is a measure μ on ∂D such that

h(·) = ∫_{∂D} M_{D}(·, w)μ(dw).

If μ is not a multiple of the point mass at z, then there is a finite measure ν ≤ μ such that δ(z, supp(ν)) > 0. Let

u(·) = ∫_{∂D} M_{D}(·, w)ν(dw).

Then u is a positive harmonic function with respect to X^{D} bounded by M_{D}(·, z).

Recall from (3.6 in the proof of Lemma 3.5 that M_{D}(x, z) → 0 uniformly as δ(x, ∂D \ B(z, ϵ) → 0. So the same is true of u. But for each w ∈ supp(ν), we can see that M_{D}(x, w) → 0 uniformly as δ(x, ∂D ∩ B(z, 2ϵ) → 0 provided 2ϵ < δ(z, supp(ν)). So it follows by the dominated convergence theorem that u(x) → 0 as δ(x, ∂D ∩ B(z, 2ϵ) → 0. But then u is a positive harmonic function of X^{D} which continuously vanishes on ∂D. This implies that ν is 0, or that μ = cδ_{z} for some c.

From Theorem 3.6 and the general theory of Martin representation (cf. [15]), we have Theorem 3.8 If D is a bounded Lipschitz domain in R^{n}, then the restriction to D of any positive superharmonic function f with respect to X^{D} can be written uniquely as

f(x) = ∫_{D} G_{D}(x, y)ν(dy) + ∫_{∂D} M_{D}(x, z)μ(dz),  (3.7)

where ν and μ are finite measures on D and ∂D respectively.

When D is a bounded C^{1,1} domain, we can say more about the Martin kernel of D.

Theorem 3.9 Suppose that D is a bounded C^{1,1} domain, then M_{D}(·, ·) is continuous function on D × ∂D. Furthermore, there exists a constant C = C(D, α) > 0 such that for any x ∈ D and z ∈ ∂D,

\frac{δ(x, ∂D)^{α/2}}{C|x − z|^{n}} ≤ M_{D}(x, z) ≤ \frac{Cδ(x, ∂D)^{α/2}}{|x − z|^{n}} .

Proof. The joint continuity of M_{D} follows from the definition of Martin kernel and Lemma 3.3. The estimates on M_{D} follows easily from Theorems 1.1 and 1.2 of Chen and Song [8].

From Theorems 1.1–1.2 of Chen and Song [8] and Theorem 3.9 above we have

Theorem 3.10 (3G Theorem). Suppose that D is a bounded C^{1,1} domain, then there exists C = C(D, α) > 0 such that

\frac{G_{D}(x, y)M_{D}(y, z)}{M_{D}(x, z)} ≤ C \frac{|x − z|^{n−α}}{|x − y|^{n−α}|y − z|^{n−α}} , x, y ∈ D, z ∈ ∂D.

Proof. The proof is the same as the proof of Theorem 1.6 of Chen and Song [8] and we omit it here.

Using Theorems 3.9 and 3.10, we can prove a conditional gauge theorem, which complements the two conditional gauge theorems established in Chen and Song [9]. Before we state and prove the conditional gauge theorem, we need to do some preparations first.

Definition 3.1 A Borel measurable function q on R^{n} is said to be in the Kato class K_{n,α} if

lim_{r↓0} sup_{x∈R^{n}} ∫_{|x−y|≤r} \frac{|q(y)|}{|x − y|^{n−α}} dy = 0.  (3.8)

For q ∈ K_{n,α}, set

e_{q}(t) = exp (∫_{0}^{t} q(X_{s})ds) .

From Chen and Song [9], we know that the following semigroup

T_{t}f(x) = E_{x}[e_{q}(t)f(X_{t}); t < τ_{D}], x ∈ D,

admits an integral kernel k_{q}(t, x, y). The function

g(x) := E_{x}[e_{q}(τ_{D})]

is called the gauge function of (D, q). It is shown in [12] that either g is identically identically infinite or g is bounded on D. In the latter case, (D, q) is said to be gaugeable. When (D, q) is gaugeable, it can be showb (see [9]) that

V_{q}(x, y) = ∫_{0}^{∞} k_{q}(t, x, y)dt, x, y ∈ D,

is well defined and is continuous off the diagonal.

Suppose that h > 0 is a positive superharmonic function with respect to X^{D}. Note that by Theorem 2.3 above, we have (see, e.g., page 11 of Dynkin [14]) that

h(x) ≥ E^{x}[h(X_{t}^{D})].

We define

p_{D}^{h}(t, x, y) = h(x)^{−1}p_{D}(t, x, y)h(y), t > 0, x, y ∈ D,

where p_{D} is the transition density function of killed symmetric stable process X^{D} in D. It is easy to check that p_{D}^{h} is a transition density and it determines a Markov process on the state space D_{∂} = D ∪ {∂}. This process is called the h–conditioned symmetric stable process. Similar to Propositions 5.2–5.4 of Chung and Zhao [13], we have the following

Lemma 3.11 For t > 0, if Φ ≥ 0 is an F_{t}–measurable function, then

E_{x}^{h}[Φ; t < τ_{D}] = h(x)^{−1}E_{x}[Φ · h(X_{t}); t < τ_{D}], x ∈ D.

Recall that {F_{t}, t ≥ 0} be the minimal admissible σ-fields generated by X. For any stopping time T of {F_{t}, t ≥ 0}, F_{T +} is the class of subsets Λ of F such that

Λ ∩ {T ≤ t} ∈ F_{t+}, t ≥ 0.

F_{T −} is the σ–field generated by F_{0+} and the class of sets

{t < T } ∩ Λ, t ≥ 0, λ ∈ F_{t}.

Lemma 3.12 For any stopping time T and any F_{T +}–measurable function Φ ≥ 0,

E_{x}^{h}[Φ; T < τ_{D}] = h(x)^{−1}E_{x}[Φ · h(X_{T}); T < τ_{D}].

Lemma 3.13 For any stopping time T, A ∈ F_{T +} and any F_{τ_{D−}}–measurable variable Φ ≥ 0,

E_{x}^{h}[A ∩ (T < τ_{D}); Φ ∘ θ_{T}] = E_{x}^{h}[A ∩ (T < τ_{D}); E_{h}^{X_{T}} (Φ)],

where θ_{t} is the shift operator for process X.

Now let D be a bounded Lipschitz domain. For each z ∈ ∂D, the M_{D}(·, z)–conditioned symmetric stable process will be called the z-symmetric stable process, and the associated probability and expectation will be denoted by P_{x}^{z} and E_{x}^{z}, respectively.

For any y ∈ D, G_{D}(·, y) is harmonic in D \ {y} with respect to X^{D\{y}}. Hence we can define the G_{D}(·, y)–conditioned symmetric stable process on the state space (D \ {y}) ∪ {∂}, with lifetime τ_{D\{y}}. It will be referred to as the y–conditioned symmetric stable process, and the associated probability and expectation will be denoted by P_{x}^{y} and E_{x}^{y} respectively. The following result immediately follows from Theorem 3.10.

Corollary 3.14 (Conditional Lifetime) Suppose that D is a bounded C^{1,1} domain. Then

sup_{x∈D, z∈∂D} E_{x}^{z}[τ_{D}] < ∞.

Theorem 3.15 (Conditional Gauge Theorem). Suppose that D is a bounded C^{1,1} domain and q ∈ K_{n,α}. If (D, q) is gaugeable, then there exists c > 1 such that

c^{−1} ≤ inf_{x∈D,z∈∂D} E_{x}^{z}[e_{q}(τ_{D})] ≤ sup_{x∈D,z∈∂D} E_{x}^{z}[e_{q}(τ_{D})] ≤ c.

Proof. Suppose x, y ∈ D and z ∈ ∂D. For any w ∈ D, by it follows from Lemma 6.5 of Chen and Song [9] that

lim_{y→z} \frac{1}{G_{D}(x, y)}V_{q}(x, w)q(w)G_{D}(w, y) = lim_{y→z} V_{q}(x, w)q(w)\frac{G_{D}(w, y)/G_{D}(x_{0}, y)}{G_{D}(x, y)/G_{D}(x_{0}, y)}
= V_{q}(x, w)q(w)\frac{M_{D}(w, z)}{M_{D}(x, z)}.

Now from Theorem 1.6 (3G Theorem) of Chen and Song [8] and Theorem 5.2 of Chen and Song [9] we have that

{V_{q}(x, ·)G_{D}(·, y)|q(·)|/G_{D}(x, y) : x, y ∈ D}

is uniformly integrable. Hence it follows from Theorem 5.4 of Chen and Song [9] that

lim_{y→z} E_{x}^{y}[e_{q}(τ_{D\\{y\}})] = 1 + \frac{1}{M_{D}(x, z)} ∫_{D} V_{q}(x, u)q(u)M_{D}(u, z)du.

However, one can show, by using an argument similar to the proof of Theorem 5.4 of Chen and Song [9], that

E_{x}^{z}[e_{q}(τ_{D})] = 1 + \frac{1}{M_{D}(x, z)} ∫_{D} V_{q}(x, u)q(u)M_{D}(u, z)du.

Therefore

lim_{y→z} E_{x}^{y}[e_{q}(τ_{D\{y}})] = E_{x}^{z}[e_{q}(τ_{D})]  (3.9)

and the theorem now follows from Theorem 5.6 of Chen and Song [9]. .

Theorem 3.16 Suppose that D is a bounded C^{1,1} domain and q ∈ K_{n,α}. If (D, q) is gaugeable, then for any fixed point x_{0} ∈ D and z ∈ ∂D,

lim_{D∋y→z} \frac{V_{q}(x, y)}{V_{q}(x_{0}, y)} = \frac{E_{x}^{z}[e_{q}(τ_{D})]}{E_{z}^{x_{0}}[e_{q}(τ_{D})]} M_{D}(x, z).  (3.10)

Furthermore,

lim_{D∋y→z} \frac{V_{q}(x, y)}{δ(y, ∂D)^{α/2}} = E_{x}^{z}[e_{q}(τ_{D})] lim_{D∋y→z∈∂D} \frac{G_{D}(x, y)}{δ(y, ∂D)^{α/2}} .  (3.11)

Proof. Since

E_{x}^{y}[e_{q}(τ_{D\{y}})] = V_{q}(x, y)/G_{D}(x, y)  (3.12)

for x, y ∈ D by Theorem 5.5 of Chen and Song [9] and so (3.10) follows immediately from it. Identity (3.11) follows from (3.12) and from Lemma 6.5 of Chen and Song [8] which asserts that the limit

lim_{D∋y→z∈∂D} \frac{G_{D}(x, y)}{δ(y, ∂D)^{α/2}}

exists and forms a positive and continuous function in (x, z) ∈ D × ∂D.

As a consequence of Theorems 3.15 and 3.16 we get that, for a bounded C^{1,1} domain D and a q ∈ K_{n,α}, if (D, q) is gaugeable, then the Martin kernel of the generalized Schrödinger operator −(−Δ)^{α/2} + q with zero exterior condition on D^{c} is comparable to Martin kernel M_{D}.

Remark 3.1 Recently in [10] we were able to extend the 3G theorem and conditional gauge theorem established for bounded C^{1,1} domains in Chen and Song [8], [9] to bounded Lipschitz domains. Thus Theorems 3.10 and 3.15 and (3.10) in Theorem 3.16 in fact hold on bounded Lipschitz domains as well. Thus under the condition that D is a bounded Lipschitz domain and (D, q) is gaugeable, the Martin kernel of the generalized Schrödinger operator L = −(−Δ)^{α/2} + q with zero exterior condition on D^{c} is comparable to Martin kernel M_{D} and the Martin boundary for L coincides with the Euclidean boundary ∂D of D.

When D is a ball, we can actually get an explicit formula for the Martin kernel of D. This follows easily form the definition of the Martin kernel and the explicit formula for the Green functions of balls (see Corollary 4 of Blumenthal, Getoor and Ray [3]). We record this fact as follows.

Example. If B = B(0, r), then

M_{B}(x, w) = \frac{(r^{2} − |x|^{2})^{α/2}}{|x − w|^{n}} , x ∈ D, w ∈ ∂D.

From the formula above, we know that

h(x) = ∫_{∂B} \frac{(r^{2} − |x|^{2})^{α/2}}{|x − w|^{n}} dw, x ∈ D

is a positive harmonic function with respect to X^{B}. From Theorem 2.5 we know that h can not be a bounded function on B. In fact one can check directly in this case that for each z ∈ ∂B

lim_{B∋x→z} h(x) = ∞.

In the Brownian motion case, the Martin boundary can be approached along Brownian paths. While for a symmetric stable process, we know from Lemma 6 of Bogdan [5] that, with probability 1, it will not hit ∂D upon first exiting from a bounded domain D satisfying the uniform exterior cone condition. Our next theorem gives the relationship between the Martin boundary and the (conditioned) stable paths.

Theorem 3.17 Suppose D is a bounded Lipschitz domain. Then for every x ∈ D and z ∈ ∂D,

P_{x}^{z}{τ_{D} < ∞} = 1;
P_{x}^{z}{lim_{t↑τ_{D}} X(t) = z} = 1.

Proof. Let z ∈ ∂D, r_{m} ↓ 0, B_{m} = (z, r_{m}), D_{m} = D \ \overline{B_{m}} and set

T_{m} = inf{t > 0 : X_{t} ∈ B_{m}}, R_{m} = τ_{B_{m}∩D}.

We may assume that x ∈ D_{m} for n ≥ 1. By Lemma 3.3, M_{D}(·, z) can be continuously extended onto \overline{D_{m}} by setting M_{D}(w, z) = 0 for w ∈ ∂D \ B_{m}. Since M_{D}(·, z) is harmonic in D_{m} with respect to X^{D}, we have by Theorem 2.4 and Lemma 3.12

M_{D}(x, z) = E_{x} [M_{D}(X_{τ_{Dm}}, z)]
= E_{x} [M_{D}(X_{T_{m}}, z); T_{m} < τ_{D}]
= M_{D}(x, z)P_{x}^{z} (T_{m} < τ_{D}) .

It follows that for all n ≥ 1 we have

P_{x}^{z}{T_{m} < τ_{D}} = 1.  (3.13)

Note that for each fixed z ∈ ∂D, M_{D}(x, z) is bounded in x ∈ B_{k}^{c} ∩ D by continuity. Let C_{k} denote its bound. Applying Lemmas 3.12 and 3.13 twice, we have for all k < m:

P_{x}^{z}{T_{m} < τ_{D}, R_{k} ∘ θ_{T_{m}} < τ_{D}} = E_{x}^{z} [P_{z}^{X_{Tm}}[R_{k} < τ_{D}]; T_{m} < τ_{D}]

= \frac{1}{M_{D}(x, z)}E_{x} [M(X_{T_{m}}, z)P_{z}^{X_{Tm}}[R_{k} < τ_{D}]; T_{m} < τ_{D}]
= \frac{1}{M_{D}(x, z)}E_{x} [E^{X_{Tm}}[M_{D}(X_{R_{k}}, z); R_{k} < τ_{D}]; T_{m} < τ_{D}]
≤ \frac{C_{k}}{M_{D}(x, z)}P_{x} (T_{m} < τ_{D}) .  (3.14)

By the definition of T_{m} and the quasi left continuity of the unconditioned process X, we have

lim_{m→∞} P_{x}{T_{m} < τ_{D}} ≤ P_{x}{ lim_{m→∞} T_{m} ≤ τ_{D}}
≤ P_{x}{T_{{z}} ≤ τ_{D}} = 0

because z ∈ ∂D and by (2.1) with D in place of U that P_{x}(X_{τ_{D}} ∈ ∂D) = 0. It follows from (3.14) that the left hand side there converges to zero as m → ∞ for each k. Therefore there exists a subsequence {m_{j}} such that

∑_{j=1}^{∞} P_{x}^{z}{T_{m_{j}} < τ_{D}; R_{k} ∘ θ_{T_{m_{j}}} < τ_{D}} < ∞,

and consequently by Borel–Cantelli lemma we have

P_{x}^{z}{[T_{m_{j}} < τ_{D}; R_{k} ∘ θ_{T_{m_{j}}} < τ_{D}] infinitely often } = 0.

Together with (3.13) this implies that for k ≥ 1 and P_{x}^{z}–a.e. ω, there exists an integer N(ω) < ∞ such that

X_{t}(ω) ∈ B(z, r_{k}) for all t ∈ [T_{N(ω)}(ω), τ_{D}(ω)).

For each k let N(k) be the smallest N for which the above is true. Then T_{N(k)} ↑ τ_{D}; otherwise, we would have X_{t} = z for all t ∈ [lim_{k→∞} T_{N(k)}, τ_{D}), which is impossible since z ∉ D_{∂}. This proves that X_{t} → z as t ↑ τ_{D}.

Functions harmonic in D with respect to X^{D} do not come from solving Dirichlet exterior problems. Therefore the usual probabilistic interpretation of harmonic functions as solutions of Dirichlet problems is not true anymore. The following result, which follows easily from Theorem 3.17, provides some probabilistic interpretation to these kind of harmonic functions.

Theorem 3.18 Suppose that D is a bounded Lipschitz domain and μ is a finite measure on ∂D. Define

h(x) = ∫_{∂D} M_{D}(x, z)μ(dz).

Then for any Borel measurable subset A ⊂ ∂D,

P_{x}^{h}(lim_{t↑τ_{D}} X(t) ∈ A) = \frac{1}{h(x)} ∫_{A} M_{D}(x, z)μ(dz).

In particular, when D = B(O, r) and h(x) = ∫_{∂D} M_{D}(x, z)σ(dz), where σ is the surface measure, lim_{t↑τ_{D}} X(t) is distributed uniformly on ∂D under P_{0}^{h}.

4 Integral Representations of Positive Harmonic Functions

Functions which are (super)harmonic in D with respect to X are studied in Landkof [17] and Bogdan [5]. However, it seems that no one has studied the integral representations of this kind of (super)harmonic functions. We intend to establish such a representation. To prove the uniqueness of such a representation theorem we need the following result:

Lemma 4.1 Suppose that D is a bounded Lipschitz domain. If a function f satisfies the following

∫_{D^{c}} \frac{f(z)}{|y − z|^{n+α}} dz = 0,  ∀y ∈ D,  (4.1)

Then f = 0 almost everywhere on D^{c}.

Proof. Without loss of generality we can assume that the origin O is in D.

We claim that for all 0 ≤ m ≤ k,

∫_{D^{c}} ∏_{j=1}^{m}(x_{i_{j}} − y_{i_{j}})|x − y|^{−(n+α+2k)}f(y)dy = 0,  x ∈ D,  (4.2)

where for each j, 1 ≤ i_{j} ≤ n. We are going to prove the claim by induction on k.

The case of k = 0, m = 0 follows by the assumption. Take the partial derivative of (4.1) with respect to x_{j} we get

0 = \frac{∂}{∂x_{j}} ∫_{D^{c}} |x − y|^{−n−α}f(y)dy = −(n + α) ∫_{D^{c}}(x_{j} − y_{j})|x − y|^{−n−α−2}f(y)dy,

Thus

∫_{D^{c}}(x_{j} − y_{j})|x − y|^{−n−α−2}f(y)dy = 0,  x ∈ D,  (4.3)

that is, the claim is true for k = m = 1. Now take the partial derivative of (4.3) with respect to x_{j} we get

0 = \frac{∂}{∂x_{j}} ∫_{D^{c}}(x_{j} − y_{j})|x − y|^{−n−α−2}f(y)dy
= ∫_{D^{c}} |x − y|^{−n−α−2}f(y)dy − (n + α + 2) ∫_{D^{c}}(x_{j} − y_{j})^{2}|x − y|^{−n−α−4}f(y)dy

Summing the above from j = 1 to j = n we get

(α + 2) ∫_{D^{c}} |x − y|^{−n−α−2}f(y)dy = 0,

which implies that the claim is true for the case of k = 1, m = 0. Therefore the claim is true for k = 1.

Now we assume that the claim is true for all 0 ≤ m ≤ k ≤ N. Take the partial derivative of (4.2) with respect to x_{i_{m+1}} we get

0 = \frac{∂}{∂x_{i_{m+1}}} ∫_{D^{c}} ∏_{j=1}^{m}(x_{i_{j}} − y_{i_{j}})|x − y|^{−(n+α+2k)}f(y)dy

= ∫_{D^{c}} \frac{∂}{∂x_{i_{m+1}}} (∏_{j=1}^{m}(x_{i_{j}} − y_{i_{j}})|x − y|^{−(n+α+2k)}) f(y)dy

= −(n + α + 2k) ∫_{D^{c}} ∏_{j=1}^{m+1}(x_{i_{j}} − y_{i_{j}})|x − y|^{−(n+α+2(k+1))}f(y)dy

+ ∫_{D^{c}} \frac{∂}{∂x_{i_{m+1}}} (∏_{j=1}^{m}(x_{i_{j}} − y_{i_{j}})) |x − y|^{−(n+α+2k)}f(y)dy

= −(n + α + 2k) ∫_{D^{c}} ∏_{j=1}^{m+1}(x_{i_{j}} − y_{i_{j}})|x − y|^{−(n+α+2(k+1))}f(y)dy,

where in the last equality we used the induction assumption. Therefore the claim is true for all 0 ≤ m ≤ k ≤ N + 1, and hence the claim is always true.

Evaluate (4.2) at x = O we get that for any non-negative integer k, and any multi–index β = (β_{1}, ⋯, β_{n}) with |β| = β_{1} + ⋯ + β_{n} ≤ k,

∫_{D^{c}} y^{β}|y|^{−2k} \frac{f(y)}{|y|^{n+α}} dy = 0

where y^{β} = y_{1}^{β_{1}} ⋯ y_{n}^{β_{n}}. Since the linear span of the set {y^{β}|y|^{−2k} : |β| ≤ k} is an algebra of real–valued continuous functions on D^{c} which separates points in D^{c} and vanishes at infinity, by the Stone–Weierstrass Theorem the linear span of {y^{β}|y|^{−2k} : |β| ≤ k} is dense in C_{∞}(D^{c}) with respect to the uniform topology. Here C_{∞}(D^{c}) is the space of continuous functions on D^{c} which vanishes at infinity. Thus for all ϕ ∈ C_{∞}(D^{c}),

∫_{D^{c}} ϕ(y)\frac{f(y)}{|y|^{n+α}} dy = 0

which implies that f(y)|y|^{−(n+α)} = 0 almost everywhere on D^{c}. Therefore f = 0 almost everywhere on D^{c}.

Theorem 4.2 Suppose that D is a bounded domain in R^{n}. If h and f are both harmonic in D with respect to X with h = f in D, then h = f in R^{n}.

Proof. Take x_{0} ∈ D and B(x_{0}, r) ⊂ \overline{B(x_{0}, r)} ⊂ D, then it follows from Theorem 2.2 that for any x ∈ B(x_{0}, r),

E_{x}[h(X_{τ_{B(x_{0},r)}})] = h(x) = f(x) = E_{x}[f(X_{τ_{B(x_{0},r)}})].

Therefore we have

E_{x}[(h − f)(X_{τ_{B(x_{0},r)}})] = 0, x ∈ B(x_{0}, r).

By Theorem 1.4 of Chen and Song [8] we know that for all x ∈ B(x_{0}, r),

E_{x}[(h − f)(X_{τ_{B(x_{0},r)}})] = ∫_{B(x_{0},r)^{c}} K_{B(x_{0},r)}(x, z)(h − f)(z)dz
= A(n, α) ∫_{B(x_{0},r)^{c}} (∫_{B(x_{0},r)} G_{B(x_{0},r)}(x, y)\frac{1}{|y − z|^{n+α}} dy) (h − f)(z)dz
= A(n, α) ∫_{B(x_{0},r)} G_{B(x_{0},r)}(x, y) (∫_{B(x_{0},r)^{c}} \frac{(h − f)(z)}{|y − z|^{n+α}}dz) dy.

Therefore by general potential theory (see Section 5.2 of [11], for instance) we know that the function

y ↦ ∫_{B(x_{0},r)^{c}} \frac{(h − f)(z)}{|y − z|^{n+α}}dz

is zero almost everywhere on B(x_{0}, r). Since the function above is continuous in B(x_{0}, r), we have

∫_{B(x_{0},r)^{c}} \frac{(h − f)(z)}{|x − z|^{n+α}}dz = 0, ∀x ∈ B(x_{0}, r).

It follows from Lemma 4.1 we know that u − f = 0 almost everywhere on B(x_{0}, r)^{c}, and the proof is finished..

Remark 4.1 In fact the proof actually shows that for a function h harmonic in a domain D, the values of h in any ball B(x_{0}, r) ⊂ \overline{B(x_{0}, r)} ⊂ D determine h uniquely.

Theorem 4.3 Suppose D is a bounded Lipschitz domain in R^{n}. If f is a non-negative harmonic function in D with respect to X, then there exists a unique finite measure μ on ∂D such that the restriction of f to D can be written as

f(x) = ∫_{D^{c}} K_{D}(x, z)f(z)dz + ∫_{∂D} M_{D}(x, z)μ(dz),  ∀x ∈ D.  (4.4)

Proof. First we are going to show that the difference

f(x) − ∫_{D^{c}} K_{D}(x, z)f(z)dz

is non-negative. Take a sequence of domains D_{m} such that D_{m} ⊂ \overline{D_{m}} ⊂ D_{m+1} ⊂ \overline{D_{m+1}} ⊂ D and set τ_{m} = τ_{D_{m}}. Then τ_{m} ↑ τ_{D}. For any x ∈ D, since P_{x}(X_{τ_{D−}} ≠ X_{τ_{D}}) = 1, we know that P_{x}(τ_{D} = τ_{m} for some m ≥ 1) = 1. From Theorem 2.2 we know that

f(x) = E_{x}f(X_{τ_{m}})
= E_{x}[f(X_{τ_{D}}); τ_{m} = τ_{D}] + E_{x}[f(X_{τ_{m}}); τ_{m} < τ_{D}]
≥ E_{x}[f(X_{τ_{D}}); τ_{m} = τ_{D}].

Therefore

f(x) ≥ E_{x}[f(X_{τ_{D}})].

Therefore by Theorem 3.6 we know that there exists a unique finite measure μ on ∂D such that

f(x) − E_{x}[f(X_{τ_{D}})] = ∫_{∂D} M_{D}(x, z)μ(dz), ∀x ∈ D.

From the above theorem we can easily get the following

Theorem 4.4 If D is a bounded Lipschitz domain, then the restriction to D of any nonnegative function f which is superharmonic in D with respect to X can be written as

f(x) = ∫_{D^{c}} K_{D}(x, z)f(z)dz + ∫_{D} G_{D}(x, y)ν(dy) + ∫_{∂D} M_{D}(x, z)μ(dz),

where ν and μ are finite measures on D and ∂D respectively.

Proof. Similar to the first part of the proof of Theorem 4.3, we have that the function

f(x) − E_{x}[f(X_{τ_{D}})]

is a non-negative function which vanishes outside D and is superharmonic in D with respect to X. Hence it is a non-negative function which is harmonic in D with respect to X^{D}. Now our claim follows from Theorem 3.8.

However, the above decomposition is not unique anymore. This non-uniqueness is due to the following relation between the Green function G_{D} and Poisson kernel K_{D} established in Theorem 1.4 of Chen and Song [8]:

K_{D}(x, z) = A(n, α) ∫_{D} \frac{G_{D}(x, y)}{|y − z|^{n+α}} dy.

Using this relation, we can absorb the first term on the right hand side above, or part of it, into the second term.

5 Extensions

The results of this paper can be extended to dimension n = 1, by noting that Green functions and Poisson kernels on bounded open intervals for symmetric stable processes are explicitly known (see, for example, [3]). In particular (2.1)–(2.2) hold for bounded open interval U in R. Note that the Kato class K_{1,α} is defined by (3.8) for 0 < α < 2 (cf. Corollary 4 of [3] and [19]).

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