%April 29, 1998 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \documentstyle[12pt,amssymb]{article} %\input amssym.def %\input amssym \oddsidemargin 0in \evensidemargin 0in \topmargin -0.6in \textwidth 6.5in \textheight 8.5in \def\doublespace{\baselineskip=16pt} \def\singlespace{\baselineskip=12pt} \begin{document} \newtheorem{thm}{Theorem}[section] \newtheorem{lemma}[thm]{Lemma} \newtheorem{defn}{Definition}[section] \newtheorem{prop}[thm]{Proposition} \newtheorem{corollary}[thm]{Corollary} \newtheorem{remark}{Remark} \renewcommand{\theequation}{\arabic{section}.\arabic{equation}} \catcode`\@=11 \renewcommand{\section}{ \setcounter{equation}{0} \@startsection {section}{1}{\z@}{-3.5ex plus -1ex minus -.2ex}{2.3ex plus .2ex}{\Large\bf} } \catcode`\@=12 \def\ee{\varepsilon} \def\qed{\hfill\hbox{\vrule height1.5ex width.5em}} \def\RR{{\bf R}} \def\KK{{\bf K}} \def\pf{\noindent{\bf Proof.} } \title{\Large \bf Martin Boundary and Integral Representation for Harmonic Functions of Symmetric Stable Processes} \author{Zhen-Qing Chen \thanks{The research of this author is supported in part by an NSA grant}\\ Departments of Mathematics and Statistics\\ Cornell University \\ Ithaca, NY 14853, USA \\ Email: zchen@math.cornell.edu \smallskip \\ and \smallskip \\ Renming Song\\ Department of Mathematics\\ University of Illinois \\ Urbana, IL 61801, USA \\ Email: rsong@math.uiuc.edu } \date{ } \maketitle \begin{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. \end{abstract} \vspace{.3in} \noindent {\bf Keywords and phrases:} Symmetric stable processes, harmonic functions, conditional stable processes, and Martin boundaries. \vspace{.1in} \noindent {\bf Running Title:} Martin Boundary for Stable Processes \pagebreak \begin{doublespace} \section{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 $\alpha$-stable processes with $0< \alpha <2$, 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 L\'evy processes. A symmetric $\alpha$-stable process $X$ on $\RR^n$ is a L\'evy process whose transition density $p(t, x-y)$ relative to the Lebesgue measure is uniquely determined by its Fourier transform $\int_{\RR^n} e^{i x\cdot \xi} p(t, x) dx =e^{-t |\xi|^\alpha} $. Here $\alpha$ must be in the interval $(0, \, 2]$. When $\alpha=2$, 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 $0<\alpha<2$, 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. \medskip In the sequel, we will use $v^+$ and $v^-$ to denote the positive and negative part of a real-valued Borel measurable function $v$, i.e., $v^+=\max\{ v ,\, 0\}$ and $v^- =\max \{-v ,\, 0\}$. \vspace{.1in} \noindent{\bf Acknowledgement.} After the first draft of this paper was written, we met Krzysztof Bogdan and learned about his paper \cite{bog2} 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 \cite{bog2}. However the approach of this paper and that of \cite{bog2} 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. \section{Definitions and Preliminaries} In sections 2--4 of this paper, we always assume $n\geq 2$. Let $X=(\Omega, {\cal F}, X_t, {\cal F}_t, P_x)$ be a symmetric $\alpha$-stable process on $\RR^n$ with $0 < \alpha < 2$, where $\{ {\cal F}_t, \, t\geq 0\}$ is the minimal admissible $\sigma$-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 $$ G(x, y)= 2^{-\alpha} \pi^{-{n\over2 }}\Gamma \left({n-\alpha\over 2}\right) \Gamma\left({\alpha \over 2}\right)^{-1} |x-y|^{\alpha -n}. $$ For a domain $D$ in $\RR^n$, let $\tau_D =\inf\{ t>0: X_t \not\in D \}$. Adjoin a cemetery point $\partial$ to $D$ and set $$ X^D_t(\omega) = \cases {X_t(\omega) \ & if $t< \tau_D$, \cr \partial \ & if $t \ge \tau_D$.} $$ $X^D$ is a strong Markov process with state space $D_{\partial}= D\cup\{\partial\}$, which is called the subprocess of the symmetric $\alpha$--stable process $X$ killed upon leaving $D$, or simply the symmetric $\alpha$--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. \begin{defn}\label{def:har1} Let $D$ be a domain in $\RR^n$. A locally integrable function $f$ defined on $D$ taking values in $(-\infty, \, \infty]$ and satisfying the condition $\int_{\{|x|>1\}\cap D} |f(x)| |x|^{-(n+\alpha )} dx <\infty$ is said to be \begin{description} \item 1) harmonic with respect to $X^D$ if $f$ is continuous in $D$ and for each $x \in D$ and each ball $B(x, r)$ with $\overline{B(x, r)}\subset D$, $$ f(x)= E_x[f(X_{\tau_{B(x, r)}}); \tau_{B(x, r)}<\tau_D]; $$ \item 2) superharmonic respect to $X^D$ if $f$ is lower semicontinuous in $D$ and for each $x \in D$ and each ball $B(x, r)$ with $\overline{B(x, r)}\subset D$, $$ f(x)\ge E_x[f(X_{\tau_{B(x, r)}}); \tau_{B(x, r)}<\tau_D]. $$ \end{description} \end{defn} The next definition is taken from Landkof \cite{ns:lan}. \begin{defn}\label{def:har2} Let $D$ be a domain in $\RR^n$. A locally integrable function $f$ defined on $\RR^n$ taking values in $(-\infty, \, \infty]$ and satisfying the condition $\int_{\{|x|>1\}} |f(x)| |x|^{-(n+\alpha )} dx <\infty$ is said to be \begin{description} \item 1) harmonic in $D$ with respect to $X$ if $f$ is continuous in $D$ and for each $x \in D$ and each ball $B(x, r)$ with $\overline{B(x, r)}\subset D$, $$ f(x)= E_x[f(X_{\tau_{B(x, r)}})]; $$ \item 2) superharmonic in $D$ with respect to $X$ if $f$ is lower semicontinuous in $D$ and for each $x \in D$ and each ball $B(x, r)$ with $\overline{B(x, r)}\subset D$, $$ f(x)\ge E_x[f(X_{\tau_{B(x, r)}})]. $$ \end{description} \end{defn} \noindent{\bf Remark 2.1} (1) If $f$ is a lower semicontinuous function defined on $D$ taking values in $(-\infty , \, \infty]$, 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 \break $\int_{\{|x|>1\}} |f(x)| |x|^{-(n+\alpha )} dx <\infty$, it follows from estimate (\ref{eqn:2.2}) below that $ E_x[f^-(X_{\tau_{B(x, r)}})] <\infty$ for any ball $B(x, r)$ with $\overline{B(x, r)}\subset D$. Therefore the expections in Definitions \ref{def:har1} and \ref{def:har2} are well defined. \smallskip (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 $X$. \smallskip (3) Conversely, if $f$ is a non-negative superharmonic in $D$ with respect to $X$, then clearly it is a superharmonic with respect to $X^D$. \medskip 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 $\RR^n$ is said to be a bounded Lipschitz domain with Lipschitz characteristic constants $(r_0, \, A_0)$ if for every $z \in \partial D$, there is a local coordinate system $(\xi_1,\xi^{(1)}) \in \RR \times \RR^{n-1}$ with origin sitting at $z$ and there is a Lipschitz function $f$ defined on $\RR^{n-1}$ with Lipschitz constant $A_0$ such that $D \cap B(z,r_0) = B(z,r_0) \cap \{\xi = (\xi_1,\xi^{(1)}): \xi_1 > f(\xi^{(1)})\}$. A domain $U$ in $\RR^n$ is said to satisfy the {\it uniform exterior cone condition} if there exist constants $\eta>0$, $r>0$ and a cone ${\cal C} =\{x=(x_1, \dots, x_n)\in \RR^n: 01$ such that for $ x\in U$ and $ z\in {\overline U}^c$, \begin{equation}\label{eqn:2.2} \frac{\delta(x)^{\alpha/2}} {C\, \delta(z)^{\alpha/2}(1+\delta(z))^{\alpha/2}} \ \frac{1}{|x-z|^n} \le K_U(x,z) \le \frac{C \, \delta(x)^{\alpha/2}} {\delta(z)^{\alpha/2}(1+\delta(z))^{\alpha/2}} \ \frac{1}{|x-z|^n}, \end{equation} where $\delta (y)=\hbox{dist}(y, \partial U)$ is the Euclidean distance from point $y$ to the set $\partial U$. Here a domain $U$ is $C^{1, 1}$ means that for every $z\in \partial D$, there exists a $r>0$ such that $B(z, r)\cap\partial D$ is the graph of a function whose first derivatives are Lipschitz. \medskip It is well known that for any domain $D$, there exists an increasing sequence of bounded $C^\infty$-smooth domains $\{D_k \}_{k \geq 1}$ such that $\overline {D_k} \subset D_{k +1}$ for $k\geq 1$ and that $\bigcup_{k=1}^\infty D_k =D$ (see, for example, Lemma 2.4 of \cite{chen}). \medskip \begin{thm}\label{1a} Suppose that $D$ is a bounded domain in $\RR^n$. If $h$ is superharmonic in $D$ with respect to $X$, then for any domain $D_1\subset\overline{D_1}\subset D$, $E_x[ h^-(X_{\tau_{D_1}})] <\infty$ and \[ h(x)\ge E_x[h(X_{\tau_{D_1}})] \quad \mbox{for every } x\in D_1 . \] \end{thm} \pf For a fixed $\epsilon>0$ and each $x\in D_1$ we put \[ r(x)=\frac12\delta(x, \partial D_1)\wedge\epsilon, \,\,\, B(x)=B(x, r(x)), \] where $\delta(x, \partial D_1)$ denotes the Euclidean distance between $x$ and $\partial D_1$. Define a sequence of stopping times $\{T_m, m\ge1\}$ as follows: \[ T_1=\inf\{t>0: X(t)\notin B(X_0)\}, \] and for $m\ge 2$, \[ T_m=\cases{ T_{m-1}+ T_{B(X_{T_{m-1}})}\circ\theta_{T_{m-1}} & if $X_{T_{m-1}}) \in D_1$, \cr \tau_{D_1} & otherwise. \cr} \] The superharmonicity of $h$ and the strong Markov property imply that \[ h(X_{T_{m-1}})\ge E_x[h(X_{T_m})|{\cal F}_{T_{m-1}}]. \] Thus $\{h(X_{T_m}), m\ge1\}$ is a supermartingale under $P_x$. We claim that for each $x\in D_1$, \[ P_x(\lim_{m\to \infty}T_m=\tau_{D_1})=1. \] It is clear that $P_x$--a.\/s., $T_m\uparrow$ and $T_m\le\tau_{D_1}<\infty$ because $D$ is bounded. Let $T_{\infty}=\lim_{m\to\infty}T_m$. Then $X_{T_{\infty}}= \lim_{m\to\infty}X_{T_m}$ by quasi left continuity. On the set $\{T_{\infty}<\tau_{D_1}\}$, we have $X_{T_{\infty}}\in D_1$. Then for all sufficiently large values of $m$, we have $\delta(X_{T_{m-1}}, \partial D_1)>\frac12\delta(X_{T_{\infty}}, \partial D_1)>0$ and $|X_{T_{m-1}}-X_{T_m}|<\frac14 \delta(X_{T_{\infty}},\partial D_1)\wedge\epsilon$. But by the definition of $T_m$, $|X_{T_{m-1}}-X_{T_m}| >\frac12\delta(X_{T_{m-1}}, \partial D_1)\wedge\epsilon$. These inequalities are incompatible. Hence $P_x(T_{\infty}<\tau_{D_1})=0$. Put $A=\{ \tau_{D_1}=T_m \mbox{ for some } t\ge1\}$. Since $h$ is bounded from below on $\overline{D_1}$, we have by Fatou's lemma \[ \liminf_{m\to\infty}E_x[h(X_{T_m}); T_m<\tau_{D_1}] \ge E_x[\liminf_{m\to\infty}h(X_{T_m}); A^c]\\ \ge E_x[h(X_{\tau_{D_1}}); A^c]. \] Take two smooth domains $D_2$ and $D_3$ such that $\overline{D_1}\subset D_2\subset\overline{D_2}\subset D_3\subset\overline{D_3} \subset D$, then $h$ is bounded from below on $\overline{D_3}$. Since \begin{eqnarray*} h(X_{\tau_{D_1} })&=& h(X_{\tau_D})1_{\{ X_{\tau_{D_1}}\in D_3 \} } + h(X_{\tau_{D_1} }) 1_{ \{X_{\tau_{D_1} }\notin D_3 \} }\\ &= & h (X_{\tau_{D_1}}) 1_{\{ X_{\tau_{D_1} }\in D_3 \} } + h (X_{\tau_{D_2}}) 1_{\{ X_{\tau_{D_1}}\notin D_3 \}}, \end{eqnarray*} we have by estimate (\ref{eqn:2.2}) with $D_2$ in place of $U$ and the integrability assumption about $h$ in Definition \ref{def:har2} that \[ E_x[ h^- (X_{\tau_{D_1} })]<\infty. \] Thus by Fatou's Lemma \begin{eqnarray*} h(x)&\geq &\liminf_{m\to\infty}E_x[h(X_{T_m})]\\ &\ge&\liminf_{m\to\infty}E_x[h(X_{\tau_{D_1}}) , T_m=\tau_{D_1}] +\liminf_{m\to\infty}E_x[h(X_{T_m }), T_m<\tau_{D_1}]\\ &\ge&E_x[h(X_{\tau_{D_1}}); A]+E_x[h(X_{\tau_{D_1}}); A^c]\\ &= & E_x[h(X_{\tau_{D_1}})]. \end{eqnarray*} This completes the proof. \qed \vspace{.1in} \begin{thm}\label{1b} Suppose that $D$ is a bounded domain in $\RR^n$. If $h$ is a harmonic in $D$ with respect to $X$, then for any domain $D_1\subset\overline{D_1}\subset D$, $h(X_{\tau_{D_1}})$ is $P_x$-integrable and \begin{equation}\label{eqn:har} h(x)= E_x[h(X_{\tau_{D_1}})] \quad \mbox{for every }x\in D_1 . \end{equation} \end{thm} \pf We can always take a smooth domain $D_2$ such that $D_1\subset\overline{D_1}\subset D_2\subset\overline{D_2}\subset D$. If we could prove that for any $x\in D_2$, $h(X_{\tau_{D_2}})$ is $P_x$-integrable and \[ h(x)= E_x[h(X_{\tau_{D_2}})]. \] then by strong Markov property we immediately get $h(X_{\tau_{D_1}})$ is $P_x$-integrable and \[ h(x)= E_x[h(X_{\tau_{D_1}})],\qquad x\in D_1. \] 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 $\{h(X_{T_m}), m\ge1\}$ is a martingale under $P_x$ for any $x\in D_1$. By (\ref{eqn:2.1}) with $D_1$ in place of $U$, we have $P_x(\tau_{D_1}=T_m\, \mbox{ for some $m\ge1$})=1$. Since $h$ is bounded on $D_1$, we have \[ |E_x[h(X_{T_m}), T_m<\tau_{D_1}]|\le C P_x(T_m<\tau_{D_1})\rightarrow 0. \] Take a domain $D_2$ such that $\overline{D_1}\subset D_2 \subset\overline{D_2}\subset D$, then $h$ is continuous and therefore bounded on $\overline{D_2}$. By the estimate (\ref{eqn:2.2}) with $D_1$ in place of $U$ and the integrability assumption about $h$ in Definition 2.2 we have $E_x[|h|(X_{\tau_{D_1}})] <\infty$. Thus by the dominated convergence theorem \[ \lim_{m\to\infty}E_x[h(X_{\tau_{D_1}}) , T_m=\tau_{D_1}]=E_x[h(X_{\tau_{D_1}})]. \] Therefore \begin{eqnarray*} h(x)& =& \lim_{m\to\infty}E_x[h(X_{T_m})] \\ &=& \lim_{m\to\infty}E_x[h(X_{\tau_{D_1}}) , T_m=\tau_{D_1}] + \lim_{m\to\infty} E_x[h(X_{T_m}), T_m<\tau_{D_1}]\\ & = & E_x[h(X_{\tau_{D_1}})]. \end{eqnarray*} \qed \vspace{.1in} Similarly, we have the following result for functions harmonic with respect to $X^D$. \begin{thm}\label{1c} Suppose that $D$ is a bounded domain in $\RR^n$. If $h$ is superharmonic in $D$ with respect to $X^D$, then for any domain $D_1\subset\overline{D_1}\subset D$, $E_x [h^- (X^D_{\tau_{D_1}})] <\infty$ and \[ h(x)\ge E_x[h(X^D_{\tau_{D_1}})] \quad \mbox{for every } x\in D_1 . \] If $h$ is harmonic in $D$ with respect to $X^D$, then for any domain $D_1\subset\overline{D_1}\subset D$, $h(X^D_{\tau_{D_1}})$ is $P_x$-integrable and \[ h(x)= E_x[h(X^D_{\tau_{D_1}})] \quad \mbox{for every } x\in D_1. \] \end{thm} \begin{thm}\label{1d} Suppose that $D$ is a bounded domain and $h$ is harmonic in $D$ with respect to $X$ and continuous on $\overline{D}$, then $h(X_{\tau_D})$ is $P_x$-integrable and \[ h(x)=E_x[h(X_{\tau_D})], \qquad \hbox{for each } x\in D. \] \end{thm} \pf Take an increasing sequence of smooth domains $\{D_m\}_{m\geq 1}$ such that $ \overline{D_m}\subset D_{m+1}$ and $\bigcup_{m=1}^\infty D_m =D$. Set $\tau_m=\tau_{D_m}$. Then $\tau_m\uparrow \tau_D$ and $\lim_{m\to\infty}X_{\tau_m}=X_{\tau_D}$ by quasi-left continuity of $X$. Set \begin{equation}\label{eqn:2.3} A=\{\tau_m=\tau_D \mbox{ for some } m\ge1\}. \end{equation} From Theorem \ref{1b} we know that for any $m\ge1$, \[ h(x)=E_x[h(X_{\tau_m})], \qquad x\in D_m. \] Since $h$ is continuous on $\overline{D}$, we have by dominated convergence theorem that \begin{equation}\label{eqn:2.4} \lim_{m\to\infty}E_x[h(X_{\tau_m}), \tau_m<\tau_D] =E_x[h(X_{\tau_D}), A^c]. \end{equation} Since $h$ is continuous on $\overline{D}$, we can find two smooth domains $U_1$ and $U_2$ such that $\overline{D}\subset U_1\subset \overline{U_1}\subset U_2$ and that $h$ is bounded on $\overline{U_2}$. Since \begin{eqnarray*} |h|(X_{\tau_D})&=& |h|(X_{\tau_D})1_{\{ X_{\tau_D}\in U_2 \} } + |h|(X_{\tau_D}) 1_{ \{X_{\tau_D}\notin U_2 \} }\\ & = & |h| (X_{\tau_D}) 1_{\{ X_{\tau_D}\in U_2\} } + |h| (X_{\tau_{U_1}}) 1_{\{ X_{\tau_{U_1}}\notin U_2 \}}, \end{eqnarray*} we have by estimate (\ref{eqn:2.2}) with $U_1$ in place of $U$ that \[ E_x[|h|(X_{\tau_D})]<\infty. \] Thus by the dominated convergence theorem \begin{eqnarray*} h(x)&=&\lim_{m\to\infty}E_x[h(X_{\tau_m})]\\ &=&\lim_{m\to\infty}E_x[h(X_{\tau_D}) , \tau_m=\tau_{D}] +\lim_{m\to\infty}E_x[h(X_{\tau_m}), \tau_m<\tau_D]\\ &=& E_x[h(X_{\tau_D})]. \end{eqnarray*} \qed \medskip \noindent{\bf Remark 2.2} If $D$ is a bounded domain satisfying the uniform exterior cone condition, then the conclusion of Theorem \ref{1d} 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 (\ref{eqn:2.1}) $P_x (A) =1$ for $x\in D$, where $A$ is the set defined in (\ref{eqn:2.3}) and the term in (\ref{eqn:2.4}) vanishes. The rest of the argument goes through without the continuous assumption on $h$ up to the boundary $\partial D$. \vspace{.1in} 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. \begin{thm}\label{1e} Suppose that $D$ is a bounded domain in $\RR^n$ 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. \end{thm} \pf Take an increasing sequence of smooth domains $D_m$ such that $D_m\subset \overline{D_m}\subset D_{m+1}\subset\overline{D_{m+1}} \subset D$ and set $\tau_m=\tau_{D_m}$. Then $\tau_m\uparrow \tau_D$. By (\ref{eqn:2.1}), we know that $P_x(\tau_D=\tau_m\, \mbox{ for some $m\ge 1$}) =1$ for $x\in D$. From Theorem \ref{1b} we get that \begin{eqnarray*} |h(x)|&=&|E_x[h(X^D_{\tau_m})]|\\ &\le& C\, P_x(\tau_m<\tau_D)\rightarrow0. \end{eqnarray*} The proof is now complete. \qed \vspace{.1in} \section{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 \cite{wa:kun}). 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 $\partial D$ 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 $\partial D$ coincide. Our proof of the identification between the Martin boundary and the Euclidean boundary is similar to the argument of Bass--Burdzy \cite{bu:bas} in the Brownian motion case. Fix $x_0\in D$ and set \[ M_D(x, y)=\frac{G_D(x, y)}{G_D(x_0, y)}, \qquad x, y\in D. \] The Martin boundary is the set $\partial _MD=D^*\setminus D$, where $D^*$ is the smallest compact set for which $M_D(x, y)$ is continuous in $y$ in the extended sense. \begin{lemma}\label{2A} Suppose that $D$ is a bounded Lipschitz domain. Then for any $\gamma>0$, \[ \lim_{x\to\partial D}G_D(x, y)=0 \] uniformly on $D_{\gamma}=\{y\in D: \delta(y, \partial D)\ge\gamma\}$. \end{lemma} \pf Suppose that ${\cal C} =\{x=(x_1, \dots, x_n)\in \RR^n: 00$, set \[ {\cal C}_r={\cal C}\cap B(O, r) \] and \[ T_{{\cal C}_r}=\inf\{t>0, X_t\in\overline{{\cal C}_r}\}. \] One can easily show (similar to the proof of Proposition 1.19 of \cite{kl:chu}) that for any $t>0$, the function \[ x\mapsto P_x(t0$, we have \[ \limsup_{x\to O} P_x(T_{{\cal C}_r}>s)\le P^O(T_{{\cal C}_r}>s)=0 \] since $P^O(T_{{\cal C}_r}=0)=1$. Now use the fact that $D$ satisfies the uniform exterior cone condition we can easily see that \[ \lim_{x\to z\in \partial D}P_x(\tau_D>s)=0 \] uniformly in $z\in\partial D$, i.\/e., for any $\epsilon>0$ there exists $\delta'>0$ such that \begin{equation}\label{eqn:21} P_x(\tau_D>s)<\epsilon, \qquad \mbox{ if } \delta(x, \partial D)<\delta'. \end{equation} We know that \[ G_D(x, y)=G(x, y)-E_x[G(X_{\tau_D}, y)] \] and that $G(x, y)$ is bounded $D^c\times D_{\gamma}$. Now use the fact (\ref{eqn:21}) and argue along the line of the proof of Theorem 1.23 of \cite{kl:chu} we easily arrive at our conclusion. \qed \vspace{.1in} Take a positive number $\epsilon<\delta(x_0, \partial D)/4$. \begin{lemma}\label{2a} Suppose $x\in D$ with $|x-x_0|>4\epsilon$. There exists a constant $c_1=c_1(\epsilon, D, x, x_0)$ such that \[ M_D(x, y)\le c_1 \quad \mbox{ for } y\in D\setminus \left( \overline{B(x_0, \epsilon)}\cup\overline{B(x, \epsilon)} \right). \] \end{lemma} \pf Pick $y_0\in \partial B(x_0, 2\epsilon)$. By the explicit formula for the Green function of balls (see \cite{g1:blu} for instance) we know that \[ G_D(x_0, y_0)\ge G_{B(x_0, 3\epsilon)}(x_0, y_0)\ge \delta(\epsilon)>0. \] On the other hand, we know that $G_D(x, y_0)\le 2^{-\alpha}\pi^{-n/2}\Gamma( (n-\alpha)/2)\Gamma(\alpha/2)^{-1}|x-y_0|^{\alpha-n}$. Therefore $G_D(x, y_0)$ is bounded above by a constant depending on $\epsilon$ in $x\in D$ with $|x-x_0|>4 \epsilon $. Thus $M_D(x, y_0)$ is bounded above in $x\in D$ with $|x-x_0|>4 \epsilon $. But from the boundary Harnack principle (see \cite{kr:bog}) we get that $M_D(x, y)$ is comparable to $M_D(x, y_0)$ for all points $y$ in $D\setminus(B(x, \epsilon) \cup B(x_0, \epsilon))$. The lemma follows. \qed \vspace{.1in} \begin{lemma}\label{2b} Let $x$, $x_0$, $\epsilon$ be as above. Then $M_D(x, y)$ is a H\"older continuous function of $y$ in $D\setminus \left(\overline{B(x_0, \epsilon)}\cup\overline{B(x, \epsilon)} \right)$ with H\"{o}lder exponent and coefficient depending only on $x$, $x_0$, $\epsilon$ and $D$. \end{lemma} \pf For any set $A$, we define that \[ \mbox{Osc}_A\,f=\sup_{y\in A} f(y)-\inf_{y\in A}f(y). \] Let $f(y)=M_D(x, y)$. Let $y_0\in D_{\epsilon}= D\setminus \left(\overline{B(x_0, \epsilon)}\cup\overline{B(x, \epsilon)} \right)$. Since by Lemma \ref{2a} $f$ is bounded by $c_1$ on $D_{\epsilon}$, $\mbox{Osc}_{D_{\epsilon}}\, f \, \le c_1$. So it suffices to show that there exists $\rho=\rho(\epsilon, D, x, x_0)<1$ such that \begin{equation} \mbox{Osc}_{D\cap B(y_0, r)}\,f\le \rho \, \mbox{Osc}_ {D\cap B(y_0, 2r)}\, f \quad \mbox{ for }r<\epsilon/4. \end{equation} Suppose $r<\epsilon/4$, and let $g$ be the ratio of any two positive harmonic functions on $D_{\epsilon/4}$ vanishing continuously on $D^c$. By considering $ag+b$ for suitable $a$ and $b$, we may assume \[ \sup_{D\cap B(y_0, 2r)}g=1,\,\,\,\, \inf_{D\cap B(y_0, 2r)}g=0. \] If $\sup_{D\cap B(y_0, r)}g\le 1/2$, then since $\inf_{D\cap B(y_0, 2r)}g\ge 0$, we have \[ \mbox{Osc}_{D\cap B(y_0, r)}\,g\le\frac12 =\frac12 \mbox{Osc}_{D\cap B(y_0, 2r)}\,g . \] If $\sup_{D\cap B(y_0, r)}g\ge 1/2$, there exists a point $y_1$ in $D\cap B(y_0, r)$ with $g(y_1)\ge 1/2$. But then by the boundary Harnack principle with $V=\{x: \delta(x, D)<\epsilon\}\setminus(\overline{B} (x_0, r)\cup\overline{B}(x, r))$ and $K=\overline{D}\setminus(\overline{B} (x_0, r)\cup\overline{B}(x, r))$ , there exists a constant $c_2=c_2(\epsilon, D, x, x_0)\in (0, 1)$ such that \[ \inf_{D\cap B(y_0, r)}g\ge c_2g(y_1). \] Since $\sup_{D\cap B(y_0, r)}g\le 1$, in this case we have \[ \mbox{Osc}_{D\cap B(y_0, r)}\,g\le 1-\frac{c_2}2 = \left( 1-\frac{c_2}2 \right) \mbox{Osc}_{D\cap B(y_0, 2r)}\, g . \] So we have (3.1) with $\rho=\max \left\{\frac12, 1-\frac{c_2}2 \right\} $. Therefore $M_D(x, y)$ is a (globlly) H\"older continuous in $y \in D\setminus \left(\overline{B(x_0, \epsilon)}\cup\overline{B(x, \epsilon)} \right)$. \qed \vspace{.1in} A direct consequence of Lemma \ref{2b} is that $M_D(x, y)=G_D(x, y)/G_D(x_0, y)$ converges when $y\to z\in \partial D$. Let the limit be denoted as $M_D(x, z)$. This implies that the Martin boundary of $D$ can identified with a subset of $\partial D$. It is also well known that for a bounded Lipschitz domain $D$ with Lipschitz characteristic constants $(r_0, \, A_0)$, there exists $\kappa=\kappa(A_0)\in (0, 1)$ such that for every $\epsilon \in(0, r_0)$ and $z\in\partial D$, there is a point $A_\epsilon (z)\in D\cap B(z, r)$ such that $B(A_\epsilon (z), \kappa r) \subset D\cap B(z, r)$. It is not difficult to show the following (cf. Lemma 6 of Bogdan \cite{bog2}). \begin{lemma}\label{2B} For any $z\in \partial D$, $M_D(\cdot, z)$ is harmonic with respect to $X^D$. \end{lemma} \pf Clearly any fixed $x\in D$ and and $r<\delta(x, \partial D)$, \[ M_D(x, y)= E_x \left[ M_D(X_{\tau_{B(x, r)}}, y); \, \tau_{B(x, r)} < \tau_D \right] \quad \hbox{for } y\in D\setminus \overline{B(x, r)}. \] In particular, \begin{equation} M_D(x, A_{\epsilon}(z)) = E_x \left[ M_D(X_{\tau_{B(x, r)}}, A_{\epsilon}(z)); \, \tau_{B(x, r)} < \tau_D \right]\label{BB} \end{equation} for any $0<\epsilon <\min\{r, \, r_0 \}$. By Fatou's lemma, $$ M_D(x, z)\ge E_x \left[ M_D(X_{\tau_{B(x, r)}}, z); \, \tau_{B(x, r)} < \tau_D \right]. $$ Therefore $M_D(X_{\tau_{B(x, r)}}, z)$ is $P_x$-integrable. Put \[ \epsilon_0=\min\left\{\frac{\delta(x_0, \partial D)}4, \frac{r_0}2, \frac{r}4\right\}, \] Then by Lemma 13 of \cite{kr:bog} we get that there exists $C_1=C_1(D)>0$ such that for any $y\in D\cap B(z, \epsilon_0)$ and $\epsilon\in(0, \epsilon_0)$, \[ M_D(w, A_{\epsilon}(z))\le C_1 M_D(w, y) \quad \hbox{for } w\in D\setminus B(z, 2\epsilon). \] Letting $y\to z$ we get that for any $\epsilon\in(0, \epsilon_0)$, \begin{equation} M_D(w, A_{\epsilon}(z))\le C_1 M_D(w, z) \quad \hbox{for } w\in D\setminus B(z, 2\epsilon).\label{1} \end{equation} For $w\in D\cap B(z, 2\epsilon)$, $|w-x|>3r/2$ 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 $K_{B(x, r)}(x, w)\le C_2$ for $w\in D\cap B(z, 2\epsilon)$. Hence for any $w\in D\cap B(z, 2\epsilon)$, \begin{eqnarray*} &&E_x \left[ M_D(X_{B(x, r)}, \, A_{\epsilon}(z)); \, X_{B(x, r)} \in D\cap B(z, 2\epsilon) \right] \\ &\le&\frac{C_2}{G_D(x_0, A_{\epsilon}(z))}\int_{D\cap B(z, 2\epsilon)} G(w, A_{\epsilon}(z))dw\\ &\le&\frac{C_3}{G_D(x_0, A_{\epsilon}(z))}\int_{D\cap B(z, 2\epsilon)} |w-A_{\epsilon}(z)|^{\alpha-n}dw\\ &\le&\frac{C_4}{G_D(x_0, A_{\epsilon}(z))}\epsilon^{\alpha}. \end{eqnarray*} From Lemma 5 of \cite{kr:bog} we know that there exists a constant $C_5=C_5(D, x_0)>0$ and positive number $\gamma=\gamma(D)<\alpha$ such that $ G_D(x_0, A_{\epsilon}(z))\ge C_5 \epsilon^{\gamma} $. Therefore \begin{equation} E_x \left[ M_D(X_{B(x, r)}, \, A_{\epsilon}(z)); \, X_{B(x, r)} \in D\cap B(z, 2\epsilon) \right] \le C_4 C_5^{-1}\epsilon^{\alpha-\gamma}.\label{2} \end{equation} Now combine (\ref{1}) and (\ref{2}) we see that the family of functions $\{M_D(X_{B(x, r)}), \, A_{\epsilon}(z) ): \, 0<\epsilon < \epsilon_0 \}$ is $P_x$-uniformly integrable. Letting $\epsilon\to 0$ in (\ref{BB}) yields \[ M_D(x, z)= E_x \left[ M_D(X_{\tau_{B(x, r)}}, z); \, \tau_{B(x, r)} < \tau_D \right] \quad \hbox{ for any } r < \delta ( x , \partial D). \] Thus $M_D(\cdot, z)$ is harmonic with respect to $X^D$. \qed \vspace{.1in} The next result tells that each Euclidean boundary point corresponds to a different non-negative harmonic function. Hence the Martin boundary can not be identified with a proper subset of the Euclidean boundary. \begin{lemma}\label{2c} If $M_D(\cdot, z_1) \equiv M_D(\cdot, z_2)$ for $z_1$, $z_2\in \partial D$, then $z_1=z_2$. \end{lemma} \pf Let $\epsilon>0$ be such that \[ \epsilon<\min\left\{ \frac{\delta(x_0, \, \partial D)}4, \, \frac{r_0}2\right\}, \] where $r_0$ comes from the Lipschitz characteristic constants $(r_0, A_0)$ of $D$. First we are going to show that $M_D(x, w)\to 0$ uniformly in $w\in\partial D$ as $\delta(x, \partial D\setminus B(w, 3\epsilon))\to 0$. In fact, for any given $\eta>0$, by Lemma \ref{2A} there is a $\beta=\beta (\eta, A_0, \epsilon)>0$ such that $G_D(x, A_{\epsilon}(w))<\eta$ for $ w\in\partial D$ and $x\in D$ with $\delta(x, \partial D)< \beta$. Let $D_0$ be a smooth domain such that \[ \left\{x\in D; \delta(x, \partial D)\ge\frac{\kappa \epsilon}2 \right\}\subset D_0\subset\overline{D_0}\subset D. \] Then for all $w\in \partial D$, $\delta(A_{\epsilon}(w), \partial D_0)>\kappa\epsilon/2$ and so by Theorem 1.1 of \cite{s1:che} \begin{eqnarray*} G_D(x_0, A_{\epsilon}(w))&\ge& G_{D_0}(x_0, A_{\epsilon}(w))\\ &\ge&C(D_0)\min\left\{\frac1{|x_0-A_{\epsilon}(w)|^{n-\alpha}}, \, \frac{ \delta(x_0, \partial D_0)^{\alpha/2}\delta(A_{\epsilon}(w), \partial D_0)^{\alpha/2}}{|x_0-A_{\epsilon}(w)|^{n}}\right\}\\ &\ge&C(D_0)\min\left\{\frac1{d_D^{n-\alpha}}, \, \frac{(3\epsilon)^{\alpha/2} (\kappa\epsilon/2)^{\alpha/2}}{d_D^n}\right\}\\ & := &C_1>0, \end{eqnarray*} where $d_D$ is the diameter of $D$. Therefore \[ M_D(x, A_{\epsilon}(w))<\eta/C_1, \qquad \forall w\in\partial D, \] whenever $\delta(x, \partial D)< \beta$. Fix $w\in\partial D$. Clearly $|x-w|>2\epsilon$ for any $x\in D$ with $\delta(x, \partial D\setminus B(w, 3\epsilon)) <\epsilon$. Now by Lemma 13 of \cite{kr:bog} with $r=\epsilon$, we get that there is a $C_2=C_2(A_0)>0$ such that \[ M_D(x, y)\le C_2M_D(x, A_{\epsilon}(w)) \quad \hbox{for } y\in D\cap B(w, \epsilon) \] whenever $x\in D$ satisfies $\delta(x, \partial D\setminus B(w, 3\epsilon)) <\epsilon$. Therefore \[ M_D(x, y)\le \frac{C_2}{C_1}\eta \quad \hbox{for } y\in D\cap B(w, \epsilon) \] whenever $x\in D$ satisfies $\delta(x, \partial D\setminus B(w, 3\epsilon)) <\epsilon\wedge\beta$. Letting $y\to w$ we get that \[ M_D(x, w)\le \frac{C_2}{C_1}\eta \] for $x\in D$ with $\delta(x, \partial D\setminus B(w, 3\epsilon)) <\epsilon\wedge\beta$. Therefore \begin{equation}\label{eqn:unif} M_D(x, w)\to 0 \hbox{ uniformly in } w\in\partial D \hbox{ as } \delta(x, \partial D\setminus B(w, 3\epsilon))\to 0. \end{equation} Suppose that $M_D(\cdot, w)=M_D(\cdot, z)$ for some $w$, $z\in \partial D$, $w\neq z$, and let $\epsilon<|w-z|/8$. By the above argument, $M_D(x, w)\to 0$ uniformly when $\delta(x, \partial D\setminus B(w, 2\epsilon))\to 0$ or when $\delta(x, \partial D\setminus B(z, 2\epsilon))\to 0$. Therefore $M_D(x, w)\to 0$ uniformly as $\delta(x, \partial D)\to 0$. Since $M_D(\cdot, w)$ is a non-negative harmonic function with respect to $X^D$ which continuously vanishes on $\partial D$, it must be identically zero by Theorem \ref{1e}. This contradicts the fact that $M_D(x_0, w)=1$. The proof is now complete. \qed \vspace{.1in} Combining the lemmas above we get the following result. \begin{thm}\label{2d} The Martin boundary of $D$ can be identified with its Euclidean boundary $\partial D$. \end{thm} \begin{thm}\label{2e} For each $z\in \partial D$, $M_D(x, z)$ is minimal harmonic with respect to $X^D$. \end{thm} \pf Fix $z\in\partial D$ and suppose $h\le M_D(\cdot, z)$, where $h$ is a positive harmonic function with respect to $X^D$. By Theorem \ref{2d} we know that there is a measure $\mu$ on $\partial D$ such that \[ h(\cdot)=\int_{\partial D}M_D(\cdot, w)\mu(dw). \] If $\mu$ is not a multiple of the point mass at $z$, then there is a finite measure $\nu\le \mu$ such that $\delta(z, \mbox{supp}(\nu)) >0$. Let \[ u(\cdot)=\int_{\partial D}M_D(\cdot, w)\nu(dw). \] Then $u$ is a positive harmonic function with respect to $X^D$ bounded by $M_D(\cdot, z)$. Recall from (\ref{eqn:unif} in the proof of Lemma \ref{2c} that $M_D(x, z) \to 0$ uniformly as $\delta(x, \partial D\setminus B(z, \epsilon) \to 0$. So the same is true of $u$. But for each $w\in\mbox{supp}(\nu)$, we can see that $M_D(x, w)\to 0$ uniformly as $\delta(x, \partial D\cap B(z, 2\epsilon) \to 0$ provided $2\epsilon<\delta(z, \mbox{supp}(\nu))$. So it follows by the dominated convergence theorem that $u(x)\to 0$ as $\delta(x,\partial D\cap B(z, 2\epsilon) \to 0$. But then $u$ is a positive harmonic function of $X^D$ which continuously vanishes on $\partial D$. This implies that $\nu$ is 0, or that $\mu=c\delta_z$ for some $c$. \qed \vspace{.1in} From Theorem \ref{2d} and the general theory of Martin representation (cf. \cite{wa:kun}), we have \begin{thm}\label{2f} If $D$ is a bounded Lipschitz domain in $\RR^n$, then the restriction to $D$ of any positive superharmonic function $f$ with respect to $X^D$ can be written uniquely as \begin{equation}\label{eqn:martin} f(x)=\int_DG_D(x, y)\nu(dy)+\int_{\partial D}M_D(x, z)\mu(dz), \end{equation} where $\nu$ and $\mu$ are finite measures on $D$ and $\partial D$ respectively. \end{thm} When $D$ is a bounded $C^{1, 1}$ domain, we can say more about the Martin kernel of $D$. \begin{thm}\label{2g} Suppose that $D$ is a bounded $C^{1, 1}$ domain, then $M_D(\cdot, \cdot)$ is continuous function on $D\times \partial D$. Furthermore, there exists a constant $C=C(D, \alpha)>0$ such that for any $x\in D$ and $z\in \partial D$, \[ \frac{\delta(x, \partial D)^{\alpha/2}}{C|x-z|^n}\le M_D(x, z)\le \frac{C\delta(x, \partial D)^{\alpha/2}}{|x-z|^n}. \] \end{thm} \pf The joint continuity of $M_D$ follows from the definition of Martin kernel and Lemma \ref{2b}. The estimates on $M_D$ follows easily from Theorems 1.1 and 1.2 of Chen and Song \cite{s1:che}. \qed \vspace{.1in} From Theorems 1.1--1.2 of Chen and Song \cite{s1:che} and Theorem \ref{2g} above we have \begin{thm}\label{2n} (3G Theorem). Suppose that $D$ is a bounded $C^{1, 1}$ domain, then there exists $C=C(D, \alpha)>0$ such that \[ \frac{G_D(x, y)M_D(y, z)}{M_D(x, z)}\le C\frac{|x-z|^{n-\alpha}} {|x-y|^{n-\alpha}|y-z|^{n-\alpha}}, \qquad x, y\in D, z\in\partial D. \] \end{thm} \pf The proof is the same as the proof of Theorem 1.6 of Chen and Song \cite{s1:che} and we omit it here. \qed \vspace{.1in} Using Theorems \ref{2g} and \ref{2n}, we can prove a conditional gauge theorem, which complements the two conditional gauge theorems established in Chen and Song \cite{s2:che}. Before we state and prove the conditional gauge theorem, we need to do some preparations first. \begin{defn}\label{def:kato} A Borel measurable function $q$ on $\RR^n$ is said to be in the Kato class $\KK_{n, \alpha}$ if \begin{equation}\label{eqn:kato} \lim_{r\downarrow 0}\sup_{x\in \RR^n}\int_{|x-y|\le r} \frac {|q(y)|}{|x-y|^{n-\alpha}}dy=0. \end{equation} \end{defn} \smallskip For $q\in\KK_{n, \alpha}$, set \[ e_q(t)=\exp \left(\int^t_0q(X_s)ds \right). \] From Chen and Song \cite{s2:che}, we know that the following semigroup \[ T_tf(x)=E_x[e_q(t)f(X_t); \, t<\tau_D], \qquad x\in D, \] admits an integral kernel $k_q(t, x, y)$. The function \[ g(x) := E_x[e_q(\tau_D)] \] is called the gauge function of $(D, q)$. It is shown in \cite{r2:chu} that either $g$ is identically identically infinite or $g$ is bounded on $D$. In the latter case, $(D, q)$ is said to be {\it gaugeable}. When $(D, q)$ is gaugeable, it can be showb (see \cite{s2:che}) that \[ V_q(x, y)=\int^{\infty}_0k_q(t, x, y)dt, \qquad x, y\in D, \] 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 \ref{1c} above, we have (see, e.g., page 11 of Dynkin \cite{eb:dyn}) that $$ h(x) \geq E^x [h(X^D_t)]. $$ We define \[ p_D^h(t, x, y)=h(x)^{-1}p_D(t, x, y)h(y), \qquad t>0, x, y\in D, \] 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 $D_{\partial}=D\cup \{\partial\}$. This process is called the $h$--conditioned symmetric stable process. Similar to Propositions 5.2--5.4 of Chung and Zhao \cite{zh:chu}, we have the following \begin{lemma}\label{2i} For $t>0$, if $\Phi\ge 0$ is an ${\cal F}_t$--measurable function, then \[ E_x^h[\Phi; t<\tau_D]=h(x)^{-1}E_x[\Phi\cdot h(X_t); t<\tau_D], \qquad x\in D. \] \end{lemma} Recall that $\{ {\cal F}_t , \, t \geq 0\}$ be the minimal admissible $\sigma$-fields generated by $X$. For any stopping time $T$ of $\{ {\cal F}_t , \, t \geq 0\}$, ${\cal F}_{T+}$ is the class of subsets $\Lambda$ of ${\cal F}$ such that \[ \Lambda\cap\{T\le t\}\in{\cal F}_{t+}, \qquad t\ge0. \] ${\cal F}_{T-}$ is the $\sigma$--field generated by ${\cal F}_{0+}$ and the class of sets \[ \{t1$ such that \[ c^{-1}\le\inf_{x\in D, z\in\partial D}E_x^z[e_q(\tau_D)] \le \sup_{x\in D, z\in\partial D}E_x^z[e_q(\tau_D)]\le c. \] \end{thm} \pf Suppose $x, y\in D$ and $z\in\partial D$. For any $w\in D$, by it follows from Lemma 6.5 of Chen and Song \cite{s2:che} that \begin{eqnarray*} \lim_{y\to z}\frac1{G_D(x, y)}V_q(x, w)q(w)G_D(w, y) &=&\lim_{y\to 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)}. \end{eqnarray*} Now from Theorem 1.6 (3G Theorem) of Chen and Song \cite{s1:che} and Theorem 5.2 of Chen and Song \cite{s2:che} we have that \[ \{V_q(x, \cdot)G_D(\cdot, y)|q(\cdot)|/G_D(x, y): x, y\in D\} \] is uniformly integrable. Hence it follows from Theorem 5.4 of Chen and Song \cite{s2:che} that \[ \lim_{y\to z}E_x^y[e_q(\tau_{D\setminus\{y\}})]= 1+\frac1{M_D(x, z)}\int_DV_q(x, u)q(u)M_D(u, z)du. \] However, one can show, by using an argument similar to the proof of Theorem 5.4 of Chen and Song \cite{s2:che}, that \[ E_x^z[e_q(\tau_D)]=1+\frac1{M_D(x, z)}\int_DV_q(x, u)q(u)M_D(u, z)du. \] Therefore \begin{equation} \lim_{y\to z}E_x^y[e_q(\tau_{D\setminus\{y\}})]= E_x^z[e_q(\tau_D)] \end{equation} and the theorem now follows from Theorem 5.6 of Chen and Song \cite{s2:che}. \qed. \medskip \begin{thm}\label{2p} Suppose that $D$ is a bounded $C^{1, 1}$ domain and $q\in\KK_{n, \alpha}$. If $(D, q)$ is gaugeable, then for any fixed point $x_0 \in D$ and $z\in \partial D$, \begin{equation}\label{eqn:3.3} \lim_{D\ni y\to z}\frac{V_q(x, y)}{V_q(x_0, y)}= \frac{E_x^z[e_q(\tau_D)]}{E^{x_0}_z[e_q(\tau_D)]} \, M_D(x, z). \end{equation} Furthermore, \begin{equation}\label{eqn:3.4} \lim_{D\ni y\to z}\frac{V_q(x, y)}{\delta(y, \partial D) ^{\alpha/2}}=E_x^z[e_q(\tau_D)] \, \lim_{D\ni y\to z\in\partial D}\frac{G_D(x, y)}{\delta(y, \partial D) ^{\alpha/2}}. \end{equation} \end{thm} \pf Since \begin{equation}\label{eqn:3.5} E_x^y[e_q(\tau_{D\setminus\{y\}})]=V_q(x, y)/ G_D(x, y) \end{equation} for $x, y \in D$ by Theorem 5.5 of Chen and Song \cite{s2:che} and so (\ref{eqn:3.3}) follows immediately from it. Identity (\ref{eqn:3.4}) follows from (\ref{eqn:3.5}) and from Lemma 6.5 of Chen and Song \cite{s1:che} which asserts that the limit $$ \lim_{D\ni y\to z\in\partial D}\frac{G_D(x, y)}{\delta(y, \partial D) ^{\alpha/2}} $$ exists and forms a positive and continuous function in $(x, z)\in D\times \partial D$. \qed \vspace{.1in} As a consequence of Theorems \ref{2o} and \ref{2p} we get that, for a bounded $C^{1, 1}$ domain $D$ and a $q\in\KK_{n, \alpha}$, if $(D, q)$ is gaugeable, then the Martin kernel of the generalized Schr\"odinger operator $-(-\Delta)^{\alpha/2}+q$ with zero exterior condition on $D^c$ is comparable to Martin kernel $M_D$. \medskip \noindent{\bf Remark 3.1} Recently in \cite{s3:che} we were able to extend the 3G theorem and conditional gauge theorem established for bounded $C^{1,1}$ domains in Chen and Song \cite{s1:che}, \cite{s2:che} to bounded Lipschitz domains. Thus Theorems \ref{2n} and \ref{2o} and (\ref{eqn:3.3}) in Theorem \ref{2p} 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 Schr\"odinger operator $L= -(-\Delta)^{\alpha/2}+q$ 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 $\partial D$ of $D$. \medskip 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 \cite{g1:blu}). We record this fact as follows. \vspace{.1in} \noindent {\bf Example}. If $B=B(0, r)$, then \[ M_B(x, w)=\frac{(r^2-|x|^2)^{\alpha/2}}{|x-w|^n}, \qquad x\in D, w\in \partial D. \] \vspace{.1in} From the formula above, we know that \[ h(x)=\int_{\partial B}\frac{(r^2-|x|^2)^{\alpha/2}}{|x-w|^n}dw, \qquad x\in D \] is a positive harmonic function with respect to $X^B$. From Theorem \ref{1e} we know that $h$ can not be a bounded function on $B$. In fact one can check directly in this case that for each $z\in\partial B$ \[ \lim_{B\ni x\to z }h(x)=\infty. \] 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 \cite{kr:bog} that, with probability 1, it will not hit $\partial D$ 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. \begin{thm}\label{2l} Suppose $D$ is a bounded Lipschitz domain. Then for every $x\in D$ and $z\in\partial D$, \begin{eqnarray*} P_x^z\{\tau_D<\infty\}&=&1;\\ P_x^z\{\lim_{t\uparrow\tau_D}X(t)=z\}&=&1. \end{eqnarray*} \end{thm} \pf Let $z\in\partial D$, $r_m\downarrow 0$, $B_m=(z, r_m), D_m=D\setminus \overline{B_m}$ and set \[ T_m=\inf\{t>0: X_t\in B_m\}, \qquad R_m=\tau_{B_m\cap D}. \] We may assume that $x\in D_m$ for $n\ge1$. By Lemma \ref{2b}, $M_D(\cdot, z)$ can be continuously extended onto $\overline{D_m}$ by setting $M_D(w, z)=0$ for $w\in \partial D\setminus B_m$. Since $M_D(\cdot, z)$ is harmonic in $D_m$ with respect to $X^D$, we have by Theorem \ref{1d} and Lemma \ref{2j} \begin{eqnarray*} M_D(x, z)&=&E_x\left[ M_D(X_{\tau_{D_m}}, z)\right] \\ &=& E_x\left[ M_D(X_{T_m}, z); T_m<\tau_D\right] \\ &=& M_D(x, z)P_x^z\left( T_m<\tau_D\right) . \end{eqnarray*} It follows that for all $n\ge1$ we have \begin{equation}\label{eqn:3.7} P_x^z\{T_m<\tau_D\}=1. \end{equation} Note that for each fixed $z\in \partial D$, $M_D(x, z)$ is bounded in $x\in B^c_k\cap D$ by continuity. Let $C_k$ denote its bound. Applying Lemmas \ref{2j} and \ref{2k} twice, we have for all $k