Stopping time convergence for processes associated with Dirichlet forms

TL;DR

This paper proves convergence of solutions to Dirichlet problems involving Markov processes with many small holes, under certain conditions, using stable topology, which strengthens traditional convergence notions.

Contribution

It establishes stable convergence of entrance and hitting times for processes associated with general Dirichlet forms in complex regions with small excluded sets.

Findings

Convergence of solutions in regions with many small holes

Stable topology used for stronger convergence results

Additional results on random center models

Abstract

Convergence is proved for solutions of Dirichlet problems in regions with many small excluded sets (holes), as the holes become smaller and more numerous. The problem is formulated in the context of Markov processes associated with general Dirichlet forms, for random and nonrandom excluded sets. Sufficient conditions are given under which the sequence of entrance times or hitting times of the excluded sets converges in the stable topology. Convergence in the stable topology is a strengthened form of convergence in distribution, introduced by Renyi. Stable convergence of the entrance times implies convergence of the solutions of the corresponding Dirichlet problems. Some additional results are given in a supplement on random center models.

Equations

$$v(x)={\mathbf{E}}_x\left[{\int }_0^{\sigma }{e}^{-\alpha t}f\left(X_t\right)dt\right]+{\mathbf{E}}_x\left[e^{-\alpha \sigma }\phi \left(X_{\sigma }\right)\right].$$

$$u_n(x)={\mathbf{E}}_x\left[{\int }_0^{{\tau }_n\wedge \sigma }{e}^{-\alpha t}f\left(X_t\right)dt\right]+{\mathbf{E}}_x\left[e^{-\alpha {\tau }_n\wedge \sigma }\phi \left(X_{{\tau }_n\wedge \sigma }\right)\right],$$

$$\tau_{n}\,\big{|}_{\displaystyle G\times(0,1)}\Longrightarrow\tau\,\big{|}_{\displaystyle G\times(0,1)}\ \text{in distribution}$$

$$\tau (\omega ,r)=\inf \left\{u:u\in [0,\infty ],S_u^{\tau }(\omega )\le 1-r\right\}.$$

$${\mathcal{E}}_{\alpha }\left(G_{\alpha }\nu ,G_{\alpha }\mu \right)=\int \left(G_{\alpha }\mu \right)d\nu ={\mathcal{E}}_{\alpha }\left(G_{\alpha }\mu ,{\hat{G}}_{\alpha }\nu \right)=\int \left({\hat{G}}_{\alpha }\nu \right)d\mu .$$

$${\delta }_x{R}_{\alpha }<<m\text{ for }m\text{-a.e. }x.$$

$$\underset{n\to \infty }{\mathrm{limsup}}\sum _{i\ne j}\int \left(G_{\alpha }{\bar{\gamma }}_j^n\right)d{\bar{\gamma }}_i^n\le \int \left(G_{\alpha }\eta \right)d\eta ,$$

$$\int {\int }_0^{1}Y\left(\omega ,\tau (\omega ,u)\right)du\mathbf{P}\left(d\omega \right)=\int {\int }_0^{\infty }Y\left(\omega ,s\right){\Phi }_{\omega }^{\tau }\left(ds\right)\mathbf{P}\left(d\omega \right).$$

$$\underset{n\to \infty }{\lim }\int Y_{{\tau }_n}d\mathbf{P}d{\lambda }^{\ast }=\int Y_{\tau }d\mathbf{P}d{\lambda }^{\ast }$$

$$\underset{n\to \infty }{\lim }\int H{\xi }_{{\tau }_n}d\mathbf{P}d{\lambda }^{\ast }=\int H{\xi }_{\tau }d\mathbf{P}d{\lambda }^{\ast }.$$

$$\underset{n\to \infty }{\lim }{\mathbf{E}}_{{\pi }_0}\left[e^{-\alpha {\tau }_n}{\left(1+G_{\alpha }\eta \left(X_{{\tau }_n}\right)-G_{\alpha }{\nu }_n\left(X_{{\tau }_n}\right)\right)}^{+}\right]=0$$

$$\underset{n\to \infty }{\lim }{\mathbf{E}}_{\pi }\left[e^{-\alpha {\tau }_n}{\left(1+G_{\alpha }\eta \left(X_{{\tau }_n}\right)-G_{\alpha }{\mu }_n\left(X_{{\tau }_n}\right)\right)}^{-}\right]=0$$

$$\int H{1}_{\left\{a<\tau \le b\right\}}{e}^{-\alpha \tau }d{\mathbf{P}}_{\pi }d{\lambda }^{\ast }\le \int H{1}_{\left\{a<\tau \right\}}{\int }_{(a,\tau \wedge b]}{e}^{-\alpha r}d{A}_{r}d{\mathbf{P}}_{\pi }d{\lambda }^{\ast }.$$

$$\begin{gathered} W_n+Z_n-Y_n={\mathbf{E}}_{\pi }\left[H{1}_{\left\{a\le {\tau }_n\le b\right\}}{e}^{-\alpha {\tau }_n}\left(1+G_{\alpha }\eta -G_{\alpha }{\nu }_n\right)\left(X_{{\tau }_n}\right)\right] \\ +{\mathbf{E}}_{\pi }\left[H{1}_{\left\{a\le {\tau }_n\right\}}{e}^{-\alpha b}{1}_{\left\{{\tau }_n>b\right\}}\left(G_{\alpha }\eta -G_{\alpha }{\nu }_n\right)\left(X_b\right)\right] \\ \le {\mathbf{E}}_{\pi }\left[e^{-\alpha {\tau }_n}{\left(1+G_{\alpha }\eta -G_{\alpha }{\nu }_n\right)}^{+}\left(X_{{\tau }_n}\right)\right]+\int \left|G_{\alpha }\eta -G_{\alpha }{\nu }_n\right|d\left(\pi p_b\right). \end{gathered}$$

$${\mathbf{P}}_{\pi }\left(t\mapsto M_t\text{ is continuous on }[0,\zeta )\right)=1.$$

$$\begin{gathered} {\mathbf{E}}_{\pi }\left[H\int \int 1_{\left\{(a,\infty )\right\}}(z)1_{(a,z\wedge b]}(u)e^{-\alpha u}d{A}_u{\Phi }^{\sigma }\left(dz\right)\right] \\ ={\mathbf{E}}_{\pi }\left[H\int \int 1_{(a,b]}(u)e^{-\alpha u}{1}_{[u,\infty )}(z){\Phi }^{\sigma }\left(dz\right)d{A}_u\right]={\mathbf{E}}_{\pi }\left[H{\int }_{(a,b]}{e}^{-\alpha u}{S}_u^{\sigma }d{A}_u\right]. \end{gathered}$$

$${\mathbf{E}}_{\pi }\left[H{\int }_{(a,b]}{e}^{-\alpha u}{\Phi }^{\sigma }\left(du\right)\right]\le {\mathbf{E}}_{\pi }\left[H{\int }_{(a,b]}{e}^{-\alpha u}{S}_u^{\sigma }d{A}_u\right]$$

$${\int }_{\left\{D_{E_{\partial }-F}\ge \zeta \right\}}{\Phi }^{\sigma }\left([\zeta ,\infty )\right)d{\mathbf{P}}_{\pi }=0.$$

$${\tilde{\Omega }}^n=\prod _{j=1}^{{\kappa }_n}{\tilde{\Omega }}_j^n,q^n=\prod _{j=1}^{{\kappa }_n}{q}_j^n.$$

$$\int \left(\int fd{\gamma }_j^n\right)d{q}_j^n=\int fd{\bar{\gamma }}_j^n,$$

$$G_{\beta }h(x)=\int g_{\beta }\left(x,y\right)h(y)m(dy)$$

$$\tilde{\mathbf{E}}\left[{\left({\Vert {\gamma }^n\Vert }_{\text{tv}}-{\Vert {\bar{\gamma }}^n\Vert }_{\text{tv}}\right)}^2\right]=\sum _{j=1}^{{\kappa }_n}\tilde{\mathbf{E}}\left[{\left({\Vert {\gamma }_j^n\Vert }_{\text{tv}}-{\Vert {\bar{\gamma }}_j^n\Vert }_{\text{tv}}\right)}^2\right]\le 4\sum _{j=1}^{{\kappa }_n}\tilde{\mathbf{E}}\left[{\Vert {\gamma }_j^n\Vert }_{\text{tv}}^2\right]\le 4{\chi }_n{\Vert {\bar{\gamma }}^n\Vert }_{\text{tv}}\to 0.$$

$$\begin{gathered} \tilde{\mathbf{E}}\left[\int \left(G_{\alpha }{\gamma }_j^n\right)d{\gamma }_i^n\right]=\tilde{\mathbf{E}}\left[\int \left(G_{\alpha }{\gamma }_j^n\right)d{\bar{\gamma }}_i^n\right]=\tilde{\mathbf{E}}\left[\int \left({\hat{G}}_{\alpha }{\bar{\gamma }}_i^n\right)d{\gamma }_j^n\right] \\ =\int \left({\hat{G}}_{\alpha }{\bar{\gamma }}_i^n\right)d{\bar{\gamma }}_j^n=\int \left(G_{\alpha }{\bar{\gamma }}_j^n\right)d{\bar{\gamma }}_i^n. \end{gathered}$$

$$\begin{gathered} {\left(\int \left(G_{\alpha }{\gamma }_j^n\right)d{\gamma }_i^n\right)}^2={\Vert {\gamma }_i^n\Vert }_{\text{tv}}^2{\left(\int \left(G_{\alpha }{\gamma }_j^n\right)d\frac{{\gamma }_i^n}{{\Vert {\gamma }_i^n\Vert }_{\text{tv}}}\right)}^2 \\ \le {\Vert {\gamma }_i^n\Vert }_{\text{tv}}^2\left(\int {\left(G_{\alpha }{\gamma }_j^n\right)}^{2}d\frac{{\gamma }_i^n}{{\Vert {\gamma }_i^n\Vert }_{\text{tv}}}\right)\le {\chi }_n\int \left(G_{\alpha }{\gamma }_j^n\right)d{\gamma }_i^n. \end{gathered}$$

$$\tilde{\mathbf{E}}\left[{\left(\int \left(G_{\alpha }{\gamma }_j^n\right)d{\gamma }_i^n\right)}^2\right]\le {\chi }_n\int \left(G_{\alpha }{\bar{\gamma }}_j^n\right)d{\bar{\gamma }}_i^n.$$

$$Y_{ij}(n)=\int \left(G_{\alpha }{\gamma }_j^n-G_{\alpha }{\bar{\gamma }}_j^n\right)d\left({\gamma }_i^n-{\bar{\gamma }}_i^n\right)=\int \left({\hat{G}}_{\alpha }{\gamma }_i^n-{\hat{G}}_{\alpha }{\bar{\gamma }}_i^n\right)d\left({\gamma }_j^n-{\bar{\gamma }}_j^n\right).$$

$$\begin{gathered} \int Z(n{)}^{2}d{q}^n=\int \left(\sum _{i\ne j}\sum _{k\ne \ell }{Y}_{ij}(n)Y_{k\ell }(n)\right)d{q}^n \\ =\sum _{i\ne j}\int \left(Y_{ij}(n{)}^2+Y_{ij}{Y}_{ji}\right)d{q}^n+\sum _{i\ne j}\sum _{k,\ell }\int Y_{ij}(n)Y_{k\ell }(n)d{q}^n, \end{gathered}$$

$$\int Z(n{)}^{2}d{q}^n=\sum _{i\ne j}\int \left(Y_{ij}(n{)}^2+Y_{ij}{Y}_{ji}\right)d{q}^n.$$

$$\int \left(G_{\alpha }{\gamma }^n\right)d{\gamma }^n=\sum _{i=1}^{{\kappa }_n}{\Vert {\gamma }_i^n\Vert }_{\text{tv}}+\sum _{i\ne j}\int \left(G_{\alpha }{\gamma }_j^n\right)d{\gamma }_i^n\le {\gamma }^n(E)+Z(n)+{\bar{\gamma }}^n(E)+{\gamma }^n(E).$$

$$\widetilde{\mathbf{Var}}\left(V_n\right)=\sum _{j=1}^{{\kappa }_n}\widetilde{\mathbf{Var}}\left(V_{n,j}\right)\le 4\sum _{j=1}^{{\kappa }_n}\tilde{\mathbf{E}}\left[V_{n,j}^2\right]\le 4\sum _{j=1}^{{\kappa }_n}{\Vert v\Vert }_{\text{sup}}^2\tilde{\mathbf{E}}\left[{\Vert {\gamma }_j^n\Vert }_{\text{tv}}^2\right]\le 4{\Vert v\Vert }_{\text{sup}}^2{\chi }_n{\Vert {\bar{\gamma }}^n\Vert }_{\text{tv}}\to 0.$$

Full text

Mathematics classification numbers 60J45, 60K37, 35J25

1 Introduction

Let $X$ be a Markov process and let $U$ be an open subset of its state space $E$. The probabilistic solution $v$ of the Dirichlet problem on $U$, with killing rate $\alpha\in[0,\infty)$, source term $f$ and boundary value function $\varphi$, is given by

[TABLE]

Here $\sigma$ is the exit time of $U$, $f$ is a measurable function on $U$, and $\varphi$ is a measurable function on $E-U$. The solution $v(x)$ is given for those $x\in U$ such that $\mathbf{E}_{x}\left[\int_{0}^{\sigma}e^{-\alpha t}f\left(X_{t}\right)\,dt\right]$ and $\mathbf{E}_{x}\left[e^{-\alpha\sigma}\varphi\left(X_{\sigma}\right)\right]$ exist, and $\varphi$ is defined to be zero at the cemetery point $\partial$ of the process. In some applications it is natural to consider the limit of a sequence of solutions $u_{n}$ for which a subset $\Lambda\left(n\right)$ of the region $U$ is excluded for each $n$, so $u_{n}$ solves a Dirichlet problem on $U-\Lambda\left(n\right)$. Typically $\Lambda\left(n\right)$ is the union of many small sets $\Lambda_{j}\left(n\right)$, which become smaller and more numerous as $n\rightarrow\infty$, and in this case $U-\Lambda\left(n\right)$ is often referred to as a region with many small holes. Define $u_{n}(x)$ on $U$ by

[TABLE]

where $\tau_{n}$ is the entrance time of $\Lambda\left(n\right)$ and $x$ is such that the expected values exist. We study conditions under which the solutions $u_{n}$ converge to a limit $u$, which could then be considered as an approximation to $u_{n}$ when the holes are small. Analytical formulations of (1.2) can of course be given, as in Lemma 6.2) below, and there are many approaches to this problem. In the classical case of Brownian motion and $\alpha=0$, where $-\Delta u_{n}=f$ holds in $U-\Lambda\left(n\right)$, the size of each small set $\Lambda_{j}\left(n\right)$ as a target for Brownian motion is measured by its capacity. When convergence holds in this setting, the limit $u$ often satisfies the equation $-\Delta u+qu=f$ in $U$, where $q$ is an appropriate limiting density for the capacities of the holes $\Lambda_{j}\left(n\right)$. Convergence problems in regions with many small holes have been considered for many classes of equations, both linear and nonlinear, using a variety of techniques. Early results in this area include [20], [19], [28], and [27]. In the setting of Dirichlet forms, convergence properties for sequences of solutions of Dirichlet problems have been studied using variational methods (cf. [8], [12], [9], [23]), extending a similar approach for elliptic equations (cf. [14], [11], [13]). The present paper uses a probabilistic approach based on [4], applied to the Markov process $X$ which is associated with a Dirichlet form. Theorem 2.1 gives conditions on a sequence of random sets $\Lambda\left(n\right)$ under which the stopping times $\tau_{n}$ in (1.2) converge with respect to the stable topology introduced by Rényi [29]. This implies convergence for the corresponding solutions of the Dirichlet problem (Lemmas 6.1 and 6.2). Theorem 2.1 holds for a wide class of Dirichlet forms, and also strengthens earlier results in the Brownian motion case. Precise statements are given in Section 2. For definitions and results concerning Dirichlet forms we will refer to [22], but some of the facts which are needed are discussed in Section 9. Properties of stable convergence are given in Section 3.

2 Main results

The main theorem is given below in Section 2.4, after some preliminary definitions.

2.1 Stable convergence definitions

Stable convergence was defined by Rényi in [29], as a stronger form of convergence in distribution. A general treatment of stable convergence of random variables is given in [18]. Here we consider stable convergence of randomized stopping times, as in [3], [15], [24]). By definition, a randomized stopping time $\tau$ with respect to a filtration $\left(\mathcal{F}_{t}\right)_{t\geq 0}$ on a probability space $\left(\Omega,\mathcal{F},\mathbf{P}\right)$, is simply a stopping time on $\left(\Omega\times(0,1),\mathcal{F}\times\mathcal{B}^{\ast},\mathbf{P}\times\lambda^{\ast}\right)$, where $\lambda^{\ast}$ is Lebesgue measure on the Borel sets $\mathcal{B}^{\ast}$ of $(0,1)$, and we use the enriched filtration $\left(\mathcal{F}_{t}\times\mathcal{B}^{\ast}\right)_{t\geq 0}$ on the randomized space $\left(\Omega\times(0,1),\mathcal{F}\times\mathcal{B}^{\ast},\mathbf{P}\times\lambda^{\ast}\right)$. An ordinary stopping time on $\left(\Omega,\mathcal{F},\mathbf{P}\right)$ can be regarded as defined on $\Omega\times(0,1)$, and so an ordinary stopping time is a special case of a randomized one. In this setting, stable convergence for randomized stopping times is defined as follows. Let $\mathcal{G}$ be a sub-$\sigma$-field of $\mathcal{F}$. Let $\tau_{n},\tau$ be randomized stopping times, or more generally let $\tau_{n},\tau$ be any $\mathcal{F}\times\mathcal{B}^{\ast}$-measurable maps from $\Omega\times(0,1)$ to $[0,\infty]$. Then $\tau_{n}\rightarrow\tau$, $\mathbf{P},\mathcal{G}$-stably, if for every set $G\in\mathcal{G}$ with $\mathbf{P}\left(G\right)>0$,

[TABLE]

with respect to the conditional probability measure $\mathbf{P}\times\lambda^{\ast}\left(\,\,\cdot\,\,|G\times(0,1)\right)$. Here \tau_{n}\,\big{|}_{\displaystyle G\times(0,1)} denotes the restriction of $\tau_{n}$ to the set $G\times(0,1)$. When $\mathcal{G}$ is known from the context we may simply write $\tau_{n}\rightarrow\tau$, $\mathbf{P}$-stably. The space of randomized stopping times associated with a given filtration is closed and compact with respect to stable convergence ([3], [24]). When stopping times for a Markov process $\left(X_{t}\right)$ are considered, unless otherwise stated $\mathcal{G}$ will be $\sigma\left(\mathcal{F}_{0},X_{t},\,0\leq t<\infty\right)$. A sequence $\tau_{n}$ may converge stably with respect to one probability measure $\mathbf{P}$ and not converge with respect to another probability measure $\mathbf{Q}$, although convergence is preserved if $\mathbf{Q}<<\mathbf{P}$ for enough sets, which can be useful when Girsanov’s theorem is applicable. Other properties of stable convergence are stated in Section 3. If, as in the present paper, $\mathcal{G}$ is countably generated modulo sets of $\mathbf{P}$-measure zero, the topology for stable convergence with respect to $\mathbf{P}$ is metrizable.

For any randomized stopping time $\tau$ with respect to the filtration $\left(\mathcal{F}_{t}\times\mathcal{B}^{\ast}\right)_{t\geq 0}$ on $\Omega\times(0,1)$, let $F^{\tau}_{t}=\mathbf{E}\left[\left.\mathbf{1}_{\left\{\tau\leq t\right\}}\,\,\right|\mathcal{F}\times\left\{\emptyset,(0,1)\right\}\right]$ for $t\in[0,\infty]$, where here the conditional expectation is with respect to the probability measure $\mathbf{P}\times\lambda^{\ast}$. A version of $F^{\tau}$ will always be chosen such that $t\mapsto F^{\tau}_{t}(\omega)$ nondecreasing and right continuous for each $\omega$ and such that $F^{\tau}_{t}$ is $\mathcal{F}_{t}$-measurable. The randomized stopping time $\tau$ can be chosen so that $\tau(\omega,\cdot)$ is left-continuous and nondecreasing on $(0,1)$, and we will always use such a version. For convenience in stating formulas, we also define $S^{\tau}_{t}=1-F^{\tau}_{t}$. $S^{\tau}$ and $F^{\tau}$ describe the observable properties of $\tau$. For $\omega\in\Omega$, $S^{\tau}_{t}(\omega)$ can be thought of as the fraction of $\omega$ which is not yet stopped at time $t$. In the case of an ordinary stopping time, $S^{\tau}_{t}$ is either zero or one for each $t$. It is easy to check that for any randomized stopping time $\tau$,

[TABLE]

Also, $\tau$ is an $\left(\mathcal{F}_{t}\times\mathcal{B}^{\ast}\right)$-stopping time if and only if $\tau(\cdot,u)$ is a $\left(\mathcal{F}_{t}\right)$-stopping time for each $u\in(0,1)$.

2.2 Rate measures for randomized stopping times

We are interested in stopping times in the setting of a Markov process. Let $E$ be a separable metric space with Borel $\sigma$-algebra $\mathcal{B}$. Let $\left(\Omega,\mathcal{F},\left(\mathcal{F}_{t}\right)_{t\geq 0},\left(X_{t}\right)_{t\in[0,\infty]},\mathbf{P}_{z}\right)$, $z\in E_{\partial}$, be a Hunt process with state space $E$, cemetery point $\partial$ and lifetime $\zeta$. Unless otherwise stated $\mathcal{F}_{t}$ is the natural filtration for $X$, that is, an appropriate closure of the filtration generated by $X$, and $\mathcal{F}=\mathcal{F}_{\infty}$. Let $\Lambda\left(n\right)$ be a sequence of closed subsets of $E$, and let $\tau_{n}$ be the entrance time $D_{\Lambda\left(n\right)}$ of $\Lambda\left(n\right)$ or the hitting time $T_{\Lambda\left(n\right)}$ of $\Lambda\left(n\right)$. Sufficient conditions will be given under which stable convergence holds for the stopping time sequence $\tau_{n}$. In the cases studied here, convergence will be proved for situations in which the sets $\Lambda\left(n\right)$ are sparse enough that $\tau_{n}$ can converge to a randomized stopping time $\tau$ which is associated with a rate of stopping. The rate of stopping is expressed by $S^{\tau}_{t}=e^{-A_{t}}$ for $t\in[0,\infty)$, where $A$ is a positive continuous additive functional. For a path $\omega$ and $t\in[0,\infty)$, the probability that the path has not yet stopped by time $t$ is equal to $e^{-A_{t}(\omega)}$. If there is a nonnegative Borel function $h$ on the state space $E$ such that $A_{t}=\int_{0}^{t}h\left(X_{s}\right)\,ds$, then one can say that the stopping time $\tau$ results from stopping at a rate $h(x)$ when the process $X$ is near the point $x$. More generally, let $m$ be a fixed $\sigma$-finite excessive measure on $E$. Any positive continuous additive functional $A$ for $X$ is associated with a Revuz measure $\mu_{A}$ on $E$ with respect to $m$ (cf. Theorem A.3.5 in [10]). We will refer to the Revuz measure $\mu_{A}$ as the rate measure for the stopping time.

2.3 Assumptions on $X$

Unless otherwise stated, from now on it is assumed that the Hunt process $X$ is properly associated with a quasi-regular Dirichlet form $\left(\mathcal{E},D\left(\mathcal{E}\right)\right)$ on $L^{2}\left(m\right)$, not necessarily symmetric, as defined in IV.1.13 and IV.2.5 of [22]. Here $m$ is a $\sigma$-finite measure on $E$. $E$ is assumed to be a metrizable Lusin space, which is defined to be the continuous one-to-one image of a Polish space, or, equivalently, a space which is homeomorphic to a Borel subset of a compact metric space. Unless otherwise stated, $\mathcal{E}$ is assumed to have the local property, so that $\mathbf{P}_{x}\left(t\mapsto X_{t}\text{ is continuous on }[0,\zeta)\right)=1$ for $\mathcal{E}$-q.e. $x$. The special case in which $\left(\mathcal{E},D\left(\mathcal{E}\right)\right)$ is a regular Dirichlet form on a locally compact separable metric space (cf. IV.4.3(a) in [22]) will be referred to as the regular case. Our main interest is in the regular case, and the transfer method (Chapter VI in [22]) allows one to obtain proofs in the quasi-regular case from results in the regular case. However, the proofs here for the regular case do not seem significantly easier, so direct proofs will be given under the general quasi-regular conditions.

Let $\hat{\mathcal{E}}$ be the form defined by by $\hat{\mathcal{E}}\left(x,y\right)=\mathcal{E}\left(y,x\right)$. There is a special standard Markov process $\hat{X}$ properly associated with $\hat{\mathcal{E}}$. For any $\alpha>0$ the form $\mathcal{E}_{\alpha}$ is defined by $\mathcal{E}_{\alpha}\left(u,v\right)=\mathcal{E}\left(u,v\right)+\alpha\left(u,v\right)$, where $\left(,\right)$ is the usual inner product on $L^{2}\left(m\right)$. The set $D\left(\mathcal{E}\right)$ with the symmetric inner product $\widetilde{\mathcal{E}}_{\alpha}=(1/2)\left(\mathcal{E}_{\alpha}+\hat{\mathcal{E}}_{\alpha}\right)$ is a Hilbert space. Let $\left\lVert\cdot\right\rVert_{\mathcal{E},\alpha}$ be the norm on this Hilbert space, so that $\left\lVert u\right\rVert_{\mathcal{E},\alpha}=\widetilde{\mathcal{E}}_{\alpha}\left(u,u\right)^{1/2}=\mathcal{E}_{\alpha}\left(u,u\right)^{1/2}$. Clearly all the norms $\left\lVert\cdot\right\rVert_{\mathcal{E},\alpha}$, $\alpha>0$, are equivalent. Let $K_{\alpha}$ denote a continuity constant for the weak sector condition (equation I.2.(2.3) in [22]), so that $\left\lvert\mathcal{E}_{\alpha}\left(u,v\right)\right\rvert\ \leq K_{\alpha}\left\lVert u\right\rVert_{\mathcal{E},\alpha}\left\lVert v\right\rVert_{\mathcal{E},\alpha}$ for all $u,v\in D\left(\mathcal{E}\right)$. We will say that a sequence $u_{n}\in D\left(\mathcal{E}\right)$ converges $\mathcal{E}$-weakly to $u$ if it converges weakly in the Hilbert space $D\left(\mathcal{E}\right)$ with inner product $\widetilde{\mathcal{E}}_{\alpha}$. The arguments used in proving I.2.12 in [22] show that a sequence $u_{n}$ converges $\mathcal{E}$-weakly to $u$ if and only if $\mathcal{E}_{\alpha}\left(u_{n},v\right)\rightarrow\mathcal{E}_{\alpha}\left(u,v\right)$ for every $v\in D\left(\mathcal{E}\right)$.

The bounded and nonnegative $\mathcal{B}$-measurable functions on $E$ will be denoted by $b\mathcal{B}$ and $\mathcal{B}^{+}$, respectively. Let $R_{\alpha}$, $p_{t}$ and $\hat{R}_{\alpha}$, $\hat{p}_{t}$ be the resolvents and Markov operators associated with $X$ and $\hat{X}$. $\mathcal{E}$-exceptional sets, $\mathcal{E}$-quasi-everywhere properties, and $\mathcal{E}$-quasi-continuity are defined in III.2.1 and III.3.2 of [22].

A finite measure is said to be smooth if it does not charge $\mathcal{E}$-exceptional sets. Conversely, one can show that a set is $\mathcal{E}$-exceptional if every smooth measure gives it measure zero. General smooth measures are defined in VI.2.3 of [22], and by VI.2.4 in [22], for any smooth measure $\mu$ there is a unique positive continuous additive functional $A$ such that $\mu$ is the Revuz measure for $A$. For a smooth probability measure $\mu$, any martingale with respect to the natural filtration of $X$ has continuous paths $\mathbf{P}_{\mu}$-almost surely (Proposition 4.4). This property is used in proving convergence of stopping times.

The potentials $G_{\alpha}$ and $\hat{G}_{\alpha}$ for the forms $\mathcal{E}$ and $\hat{\mathcal{E}}$ are defined in I.2.8 of [22] and satisfy $R_{\alpha}\,f=G_{\alpha}\,f$, $\hat{R}_{\alpha}\,f=\hat{G}_{\alpha}\,f$, $\mathcal{E}$-q.e., for all $f\in L^{2}\left(m\right)$, by IV.2.9 and IV.3.3 of [22]. $\mathcal{E}$-quasi-continuity is defined in II.3 of [22]. Each $u\in D\left(\mathcal{E}\right)$ has $\mathcal{E}$-quasi-continuous versions, any of which is denoted by $\tilde{u}$. A $\sigma$-finite measure $\mu$ on $\mathcal{B}$ will be said to have finite energy if $\mu$ does not charge $\mathcal{E}$-exceptional sets and the map $u\mapsto\int\tilde{u}\,d\mu$ is bounded on $D\left(\mathcal{E}\right)$ with respect to $\left\lVert\cdot\right\rVert_{\mathcal{E},\alpha}$-norm for some (and hence all) $\alpha>0$. This condition only depends on the symmetric part of $\mathcal{E}$. By I.2.7 in [22], there exist unique elements $v,w\in D\left(\mathcal{E}\right)$ such that $\mathcal{E}_{\alpha}\left(v,u\right)=\int\tilde{u}\,d\mu=\mathcal{E}_{\alpha}\left(u,w\right)$ for all $u\in D\left(\mathcal{E}\right)$. $G_{\alpha}\,\mu,\hat{G}_{\alpha}\,\mu$ are defined to be $v,w$ respectively. The new definition for potentials is consistent with the old, in the sense that if a measure $\mu$ has a density $f\in L^{2}\left(m\right)$ with respect to $m$, then $\mu$ has finite energy, and $G_{\alpha}\,\mu=G_{\alpha}\,f$. We will choose $\mathcal{E}$-quasi-continuous versions of $G_{\alpha}\,\mu,\hat{G}_{\alpha}\,\mu$ whenever pointwise values are needed. By VI.2.1 in [22], the measure $\mu$ is uniquely determined by $G_{\alpha}\,\mu$. When $\mu$ and $\nu$ are measures with finite energy,

[TABLE]

The definitions imply that the resolvent equation holds for potentials of measures, so that in particular $G_{\alpha}$ and $G_{\beta}$ commute.

The $\alpha$-equilibrium measure for a closed set $B$ is defined as the unique measure $\gamma$ such that $G_{\alpha}\,\gamma=1$ holds $\mathcal{E}$-q.e. on $B$ and $\gamma\left(B^{c}\right)=0$, when such a measure exists. The $\alpha$-equilibrium measure for $B$ will exist if there is an $\mathcal{E}$-quasi-continuous $v\in D\left(\mathcal{E}\right)$ such that $v\geq 1$ holds $\mathcal{E}$-q.e. on $B$ (see Section 9). Define the $\alpha$-capacity of $B$ by $\mathsf{Cap}_{\alpha}\left(B\right)=\gamma(B)=\mathcal{E}_{\alpha}\left(G_{\alpha}\,\gamma,G_{\alpha}\,{\gamma}\right)=\int\left(G_{\alpha}\,\gamma\right)\,d\gamma$. The collection of closed sets which have $\alpha$-equilibrium measures will be denoted by $\mathsf{C}$. When $\mathcal{E}$ is a regular Dirichlet form all compact sets are in $\mathsf{C}$.

2.4 A convergence theorem for random holes

The main convergence result, Theorem 2.1, deals with random sets of the following sort. Let $\kappa_{n}$, $n=1,2,\ldots$ be a sequence of positive integers with $\kappa_{n}\nearrow\infty$, and for each $n$ let $\Lambda_{j}\left(n\right)$, $j=1,\ldots,\kappa_{n}$ be an independent sequence of random variables (not necessarily identically distributed), whose values are compact sets in $\mathsf{C}$. Since identical distributions are not assumed, nonrandom sets are included as a special case.

The $\Lambda_{j}\left(n\right)$ are assumed to be measurable as maps into the space $F(E)$ of compact subsets of $E$, equipped with the Hausdorff metric and its Borel $\sigma$-algebra. Let $\Lambda\left(n\right)=\Lambda_{1}\left(n\right)\cup\cdots\cup\Lambda_{\kappa_{n}}\left(n\right)$. Each random set $\Lambda\left(n\right)$ provides a random environment for the Markov process $X$. Let $\tilde{\mathbf{P}}_{n}$ and $\tilde{\mathbf{E}}_{n}$ denote probability and expectation for the probability space $\tilde{\Omega}_{n}$ on which $\Lambda\left(n\right)$ is defined. The probability space for the environment can depend on $n$, but for convenience in stating results, we will usually assume that the $\Lambda\left(n\right)$ are all defined on the same space $\tilde{\Omega}$, and write $\tilde{\mathbf{P}}_{n}$ and $\tilde{\mathbf{E}}_{n}$ as $\tilde{\mathbf{P}}$ and $\tilde{\mathbf{E}}$. For any probability measure $\pi$ on $E$, it will be assumed that the entrance time $D_{\Lambda\left(n\right)}$ and hitting time $T_{\Lambda\left(n\right)}$ are measurable as maps from the sample space to the space of randomized stopping times, when the space of randomized stopping times is given the topology of stable convergence with respect to $\mathbf{P}_{\pi}$. This measurability will hold automatically in the regular case, since in the regular case $B\mapsto D_{B}$ is a pointwise limit of certain maps $G_{k}$, where each $G_{k}$ is constant on the sets of a measurable partition of $F(E)$.

Fix $\alpha\in(0,\infty)$. For each $n$ and each $j$, let $\gamma^{n}_{j}$ be the $\alpha$-equilibrium measure for $\Lambda_{j}\left(n\right)$, as defined in Section 9. It is assumed that $\gamma^{n}_{j}(A)$ is a measurable function of the random environment for each $A\in\mathcal{B}$, and that $\int\left(G_{\alpha}\,\gamma^{n}_{j}\right)\left(G_{\alpha}\,\gamma^{n}_{i}\right)\,dm$ is measurable for all $i,j$. In the regular case this is automatically true. For each $n,j$, define the averaged measure $\bar{\gamma}^{n}_{j}$ by $\bar{\gamma}^{n}_{j}(A)=\tilde{\mathbf{E}}\left[\gamma^{n}_{j}(A)\right]$ for each $A\in\mathcal{B}$. Then $\tilde{\mathbf{E}}\left[\int f\,d\gamma^{n}_{j}\right]=\int f\,d\bar{\gamma}^{n}_{j}$, for all $n,j$ and $f\in b\mathcal{B}\cup\mathcal{B}^{+}$. Let $\gamma^{n}=\sum_{j}\gamma^{n}_{j}$ and $\bar{\gamma}^{n}=\sum_{j}\bar{\gamma}^{n}_{j}$.

It is assumed that $\left\lVert\gamma^{n}_{j}\right\rVert_{\text{tv}}\leq\chi_{n}\text{ for all }j=1,\ldots,\kappa_{n}$, where $\chi_{n}$ is a deterministic sequence of numbers with $\chi_{n}\rightarrow 0$, and that $\sup_{n}\left\lVert\bar{\gamma}^{n}\right\rVert_{\text{tv}}<\infty$, where $\left\lVert\bar{\gamma}^{n}\right\rVert_{\text{tv}}=\bar{\gamma}^{n}\left(E\right)$ is the total variation norm of $\bar{\gamma}^{n}$. Since $\left\lVert G_{\alpha}\,\gamma^{n}_{j}\right\rVert_{\mathcal{E},\alpha}^{2}=\int\left(G_{\alpha}\,\gamma^{n}_{j}\right)\,d\gamma^{n}_{j}=\left\lVert\gamma^{n}_{j}\right\rVert_{\text{tv}}\leq\chi_{n}$, $\left\lvert\int v\,d\bar{\gamma}^{n}_{j}\right\rvert=\left\lvert\tilde{\mathbf{E}}\left[\int v\,d\gamma^{n}_{j}\right]\right\rvert\leq K_{\alpha}\left\lVert v\right\rVert_{\mathcal{E},\alpha}\sqrt{\chi_{n}}$ for any $v\in D\left(\mathcal{E}\right)$. Hence $\bar{\gamma}^{n}_{j}$ has finite energy.

Theorem 2.1

Suppose that for $\alpha\in(0,\infty)$,

[TABLE]

Let $\eta$ be a finite measure with finite energy, such that $G_{\alpha}\,\bar{\gamma}^{n}\rightarrow G_{\alpha}\,\eta$, $\mathcal{E}$-weakly, and let $A$ be the positive continuous additive functional with Revuz measure $\eta$. Let $\tau_{n}$ be the entrance time $D_{\Lambda\left(n\right)}$ or the hitting time $T_{\Lambda\left(n\right)}$ for $\Lambda\left(n\right)$. If

[TABLE]

then for any smooth probability $\pi$, the sequence $\tau_{n}\wedge\zeta$ converges $\mathbf{P}_{\pi}$-stably to $\tau\wedge\zeta$ in $\tilde{\mathbf{P}}$-probability, where $\tau$ denotes the randomized stopping time with $S^{\tau}_{t}=e^{-A_{t}}$. If with $\tilde{\mathbf{P}}$-probability one all the sets $\Lambda\left(n\right)$ are contained in a single compact subset of $E$, $\tau_{n}$ converges $\mathbf{P}_{\pi}$-stably to $\tau$ in $\tilde{\mathbf{P}}$-probability.

In the statement of the theorem, stable convergence in $\tilde{\mathbf{P}}$-probability means convergence in $\tilde{\mathbf{P}}$-probability with respect to any metric for $\mathbf{P}_{\pi}$-stable convergence. Theorem 2.1 is proved in Section 5.

Stable convergence of $\tau_{n}\wedge\zeta$ implies convergence of the solutions of the corresponding Dirichlet problems. This is discussed in more detail in Section 6.

It follows from the resolvent equation that (2.4) implies that $\delta_{x}\,R_{\alpha}<<m$ for $\mathcal{E}$-q.e. $x$, and thus is similar to condition $\text{(AC)}^{\prime}$ of Definition A.2.16 in [10].

Condition (2.5) was introduced in [27] for deterministic sets $\Lambda_{i}\left(n\right)$, in the Brownian motion case. This condition ensures that the sets $\Lambda_{j}\left(n\right)$ are rather evenly distributed. The proof of Theorem 2.1 shows that even when (2.5) is not satisfied, $\left(\tau_{n}\wedge\tau\right)\wedge\zeta$ converges $\mathbf{P}_{\pi}$-stably to $\tau\wedge\zeta$ in $\tilde{\mathbf{P}}$-probability, and $\tau_{n}\wedge\tau\rightarrow\tau$ when all the sets $\Lambda\left(n\right)$ are contained in a compact set. Thus asymptotically $\tau_{n}\geq\tau$, verifying the physical intuition that when the sets $\Lambda_{j}\left(n\right)$ are allowed to clump together, some of their capacity may be wasted. When $\bar{\gamma}^{n}\left(E\right)\rightarrow 0$, so that $A=0$, the same limit shows that the bodies $\Lambda\left(n\right)$ have a negligible stopping effect for large $n$.

In the regular case, the assumption that $G_{\alpha}\,\bar{\gamma}^{n}\rightarrow G_{\alpha}\,\eta$, $\mathcal{E}$-weakly, will hold whenever $\sup_{n}\left\lVert G_{\alpha}\,\bar{\gamma}^{n}\right\rVert_{\mathcal{E},\alpha}<\infty$ and $\bar{\gamma}^{n}$ converges vaguely to $\eta$ as a sequence of measures (Lemma 9.6).

Most of the assumptions on $\gamma^{n}$ in Theorem 2.1 only involve the average measure $\bar{\gamma}^{n}$. This simplifies applications, especially in the iid case.

Corollary 2.2

Suppose that (2.4) holds, and that the sequences $\Lambda_{j}\left(n\right)$, $j=1,\ldots,\kappa_{n}$, are iid for each $n$. Assume as above that $\left\lVert\gamma^{n}_{j}\right\rVert_{\text{tv}}\leq\chi_{n}$ and $\sup_{n}\left\lVert\bar{\gamma}^{n}\right\rVert_{\text{tv}}<\infty$. Let $\eta$ be a finite measure with finite energy such that for some $\alpha$, $\lim_{n\rightarrow\infty}\left\lVert G_{\alpha}\,\bar{\gamma}^{n}-G_{\alpha}\,\eta\right\rVert_{\mathcal{E},\alpha}=0$. Then the hypotheses of Theorem 2.1 are satisfied.

Proof   Clearly $G_{\alpha}\,\bar{\gamma}^{n}\rightarrow G_{\alpha}\,\eta$, $\mathcal{E}$-weakly, and $\bar{\gamma}^{n}_{j}=(1/\kappa_{n})\bar{\gamma}^{n}$. Thus $\sum_{i\neq j}\int\left(G_{\alpha}\,\bar{\gamma}^{n}_{j}\right)\,d\bar{\gamma}^{n}_{i}=(1-1/\kappa_{n})\left\lVert G_{\alpha}\,\bar{\gamma}^{n}\right\rVert_{\mathcal{E},\alpha}^{2}\rightarrow\left\lVert G_{\alpha}\,\eta\right\rVert_{\mathcal{E},\alpha}^{2}$, so (2.5) holds.

In the setting of the classical Dirichlet problem with iid random holes in a subset of ${\mathbb{R}}^{d}$, Corollary 2.2 gives a more general form of Theorem 4.2 of [1] (Section 8), and similarly extends Theorem 4.2 of [2], on the Dirichlet problem for the Laplace-Beltrami operator on a compact Riemannian manifold.

The proof of Theorem 2.1 is given in Section 5, based on a more general convergence criterion, Theorem 4.2. Details of the proofs for Proposition 4.4 and the examples in Section 8 are given in [7]. Transformations which simplify applying Theorem 2.1 are discussed in Section 7. In particular one can use Girsanov’s theorem to deal with a drift term.

2.5 Relaxed Dirichlet problems

In the setting of Dirichlet forms, the solution $u$ of a Dirichlet problem can be defined by specifying the boundary values for $u$ and requiring that the equation $\mathcal{E}_{\alpha}\left(u,v\right)=\int vf\,dm$ must hold for all $v$, where $f$ represents the source term in the equation, and $v$ lies in a suitable class of test functions. More generally, $u$ is said to solve a relaxed Dirichlet problem, with penalty measure $\eta$, if $\mathcal{E}_{\alpha}\left(u,v\right)+\int uv\,d\eta=\int vf\,dm$ for suitable $v$, where the penalty measure $\eta$ is a measure which does not charge sets of capacity zero, but may be infinite on some sets. Any ordinary Dirichlet problem on a region $U$ can be represented as a relaxed Dirichlet problem using a suitable infinite penalty measure $\eta$, so that convergence for solutions of ordinary Dirichlet problems can be formulated as a special case of convergence of solutions of relaxed Dirichlet problems. General properties of relaxed Dirichlet problems in the setting of Dirichlet forms have been studied in a number of papers, including [11], [8], [12], [9], [23]. These papers deal with a class of Dirichlet forms which have certain extra regularity properties. A Dirichlet form $\mathcal{E}$ in this class is regular and strongly local, and satisfies some additional assumptions, in particular that a Poincaré inequality holds for $\mathcal{E}$ and $m$ has a doubling property. [23] proves some results for nonsymmetric Dirichlet forms, while the other references study symmetric forms. In the present paper the extra regularity assumptions are replaced by (2.4). For symmetric forms satisfying the extra regularity assumptions, (2.4) holds, since it follows from equation (1.10) of [8], using the arguments of Theorem 4.2.7 in [16]. We do not know if equation (2.4) always holds for nonsymmetric forms under the assumptions in [23].

The results in [11], [8], [12], [9], [23] include some necessary and sufficient conditions for convergence of solutions of general nonrandom relaxed Dirichlet problems. Thus these results are relevant to the problems considered here. However, further arguments would be needed in order to verify the hypotheses of these results and obtain convergence in that way. Such an approach was used in the papers [1] and [2] mentioned earlier, dealing with the Laplacian operator and the Laplace-Beltrami operator.

3 Stable convergence facts

Let $\mathcal{B}_{\infty}$ denote the Borel sets on $[0,\infty]$, and for any randomized stopping time $\tau$, let $\Phi^{\tau}=\Phi^{\tau}_{\omega}$ be the random measure on $\mathcal{B}_{\infty}$ such that $\Phi^{\tau}_{\omega}\left((t,\infty]\right)=S^{\tau}_{t}(\omega)$. The Functional Monotone Class Theorem shows that for any bounded $\mathcal{F}\times\mathcal{B}_{\infty}$-measurable $Y$,

[TABLE]

Lemma 3.1

Let $\left(\mathcal{G}_{t}\right)_{t\geq 0}$ be a filtration on a probability space $\left(\Omega,\mathcal{G},\mathbf{P}\right)$, with $\mathcal{G}_{t}\subset\mathcal{G}$. Let $\tau_{n},\tau$ be randomized $\left(\mathcal{G}_{t}\right)$-stopping times on $\Omega\times(0,1)$, such that $\tau_{n}\rightarrow\tau$, $\mathbf{P},\mathcal{G}$-stably. Let $\lambda^{\ast}$ denote Lebesgue measure on $(0,1)$.

(i) Let $Y$ be a real-valued process on $\Omega\times[0,\infty]$ which is $\mathcal{G}\times\mathcal{B}_{\infty}$-measurable, where $\mathcal{B}_{\infty}$ is the collection of Borel subsets of $[0,\infty]$. Let $Z$ be a nonnegative random variable with $\mathbf{E}\left[Z\right]<\infty$, and such that for $\mathbf{P}$-a.e. $\omega$, $\left\lvert Y\left(\omega,t\right)\right\rvert\leq Z(\omega)$ for all $t$. Let $A\in\mathcal{G}\times\mathcal{B}_{\infty}$, such that $A$ contains all pairs $(\omega,u)$ for which the map: $t\mapsto Y(\omega,t)$ is discontinuous at $u$. If $\mathbf{P}\times\lambda^{\ast}\left(\left\{(\omega,s):\ (\omega,\tau(\omega,s))\in A\right\}\right)=0$, then

[TABLE]

(ii) Let $\xi:\Omega\times[0,\infty]\rightarrow[-\infty,\infty]$ be $\mathcal{G}\times\mathcal{B}_{\infty}$-measurable, càdlàg and quasi-left continuous with respect to $\mathbf{P}$ on the closed interval $[0,\infty]$. Let $H\in L^{1}\left(\mathcal{G},\mathbf{P}\right)$ be such that $\mathbf{E}\left[\left\lvert H\right\rvert\sup_{t}\left\lvert\xi_{t}\right\rvert\right]<\infty$. Then

[TABLE]

Statement (i) of Lemma 3.1 follows from the Corollary to Theorem 7 in [24]. Statement (ii) follows from Theorem (1.10) in [3] or Theorem 5 in [24]. Statement (i) is the main tool in applying stable convergence. The next two lemmas are simple observations based on the definitions.

Lemma 3.2

Let $\left(\Omega,\mathcal{G},\mathbf{P}\right)$ be a probability space. Let $\tau_{n},\tau,\sigma$ be randomized times, meaning $\mathcal{F}\times\mathcal{B}^{\ast}$-measurable maps from $\Omega\times(0,1)$ to $[0,\infty]$.

(a) Suppose that $\tau_{n}\rightarrow\tau$, $\mathbf{P},\mathcal{G}$-stably. Then $\tau_{n}\wedge\sigma\rightarrow\tau\wedge\sigma$, $\mathbf{P},\mathcal{G}$-stably.

(b) Let $\sigma_{k}$ be a sequence of randomized times such that $\sigma_{k}\nearrow\sigma$ and $\tau_{n}\wedge\sigma_{k}\rightarrow\tau\wedge\sigma_{k}$, $\mathbf{P},\mathcal{G}$-stably as $n\rightarrow\infty$, for each $k$. Then $\tau_{n}\wedge\sigma\rightarrow\tau\wedge\sigma$, $\mathbf{P},\mathcal{G}$-stably.

Proof   (i) follows at once from Lemma 3.1 and the definitions (see the corollary to Lemma 3.1 in [5]). To prove (ii), by choosing a subsequence we may assume that $\tau_{n}$ converges stably to some limit $\psi$. By (i), $\tau_{n}\wedge\sigma$ converges stably to $\psi\wedge\sigma$, and also $\tau_{n}\wedge\sigma_{k}$ converges stably to $\psi\wedge\sigma_{k}$ for each $k$. Hence $\psi\wedge\sigma_{k}=\tau\wedge\sigma_{k}$ for all $k$, and so $\psi\wedge\sigma=\tau\wedge\sigma$, independent of the choice of subsequence.

It is usually sufficient to have convergence for $\tau_{n}\wedge\zeta$ rather than for $\tau_{n}$, as in Lemma 6.1. The two forms of convergence are sometimes equivalent, as the next result shows.

Lemma 3.3

Let $\tau_{n},\tau$ be randomized $\left(\mathcal{G}_{t}\right)$-stopping times. Suppose that for any $\mathbf{P},\mathcal{G}$-stable limit point $\tau$ of the sequence $\tau_{n}$, $\mathbf{P}\times\lambda^{\ast}\left(\left\{\zeta\leq\tau<\infty\right\}\right)=0$. Suppose also that there is a randomized stopping time $\sigma$ such that $\tau_{n}\wedge\zeta\rightarrow\sigma\wedge\zeta$, $\mathbf{P},\mathcal{G}$-stably. Then $\tau_{n}$ converges $\mathbf{P},\mathcal{G}$-stably to the randomized stopping time $\hat{\tau}$ defined by $\hat{\tau}=\sigma$ if $\sigma<\zeta$, $\hat{\tau}=\infty$ otherwise.

Proof   Suppose $\tau_{n_{k}}\rightarrow\tau$, $\mathbf{P},\mathcal{G}$-stably. By Lemma 3.2 (a), $\tau_{n_{k}}\wedge\zeta\rightarrow\tau\wedge\zeta$, so $\tau\wedge\zeta=\sigma\wedge\zeta$ modulo $\mathbf{P}\times\lambda^{\ast}$. Hence modulo $\mathbf{P}\times\lambda^{\ast}$ we have $\tau=\sigma$ if $\tau\wedge\zeta<\zeta$, and if $\tau\wedge\zeta\geq\zeta$, then $\tau\geq\zeta$, so $\tau=\infty$ modulo $\mathbf{P}\times\lambda^{\ast}$. Thus $\tau=\hat{\tau}$ modulo $\mathbf{P}\times\lambda^{\ast}$.

The final lemma in this section is only used in proving analytical consequences of stable convergence (Lemma 6.1).

Lemma 3.4

Let $X$ be a Markov process satisfying the assumptions of Section 2.3. Let $\mathcal{G}$ be a $\sigma$-algebra with $\mathcal{F}_{t}\subset\mathcal{G}$ for all $t$. Let $A_{t}$ be a positive continuous additive functional, and let $\tau$ be the randomized stopping time with $S^{\tau}_{t}=e^{-A_{t}}$. Let $\sigma$ be a terminal time, and let $\tau_{n}$ be a sequence of terminal times, such that $\tau_{n}\wedge\sigma\rightarrow\tau\wedge\sigma$, $\mathbf{P}_{\pi},\mathcal{G}$-stably for some probability measure $\pi$. Suppose that for $\pi$-a.e. $x$, $\tau_{n}\rightarrow\tau$, $\mathbf{P}_{\mu},\mathcal{G}$-stably for $\mu$ with $\mu<<\delta_{x}\,R_{\alpha}$-stably. (This last condition will be true, for example, if $\pi$ is smooth and $\tau_{n}\wedge\sigma\rightarrow\tau\wedge\sigma$, $\mathbf{P}_{\mu},\mathcal{G}$-stably whenever $\mu<<m$.) Then there exists a subsequence $n_{k}$ such that for $\pi$-a.e $x$, $\tau_{n_{k}}\wedge\sigma\rightarrow\tau\wedge\sigma$, $\mathbf{P}_{x},\mathcal{G}$-stably.

This follows from the proof of Theorem 7.1 in [25], and holds for the general case of stopping times such that $S^{\tau_{n}}_{t}$ and $\tau$ are multiplicative functionals. For Brownian motion a different proof was given in Theorem 1.3 of [6].

4 A convergence criterion

Theorem 2.1 will be derived from a more general convergence result, given below as Theorem 4.2. The conditions for convergence in this theorem are motivated by the intuitive idea that the limiting rate of hitting $\Lambda\left(n\right)$ within a neighborhood of a point should be determined by the size (i.e. capacity) of that part of the $\Lambda\left(n\right)$ which lies the neighborhood (cf. [4], [6] for the Brownian motion case). Expressing this picture in terms of potentials leads to the following.

Definition 4.1

Let $\eta$ be a measure with finite energy. Let $\alpha\in(0,\infty)$. For any sequence $\gamma_{n}$ of measures with finite energy, we will write $\gamma_{n}\overset{\mathcal{E}}{\sim}\eta$ if $G_{\alpha}\,\gamma_{n}\rightarrow G_{\alpha}\,\eta$, $\mathcal{E}$-weakly, and also $\lim_{n\rightarrow\infty}\int h\left\lvert G_{\alpha}\,\gamma_{n}-G_{\alpha}\,\eta\right\rvert\,dm=0$ for all $h\in L^{2}\left(m\right)$.

Let $\Lambda\left(n\right)$ be a closed set, $n=1,2,\ldots$. Let $\tau_{n}=D_{\Lambda\left(n\right)}$, the entrance time of $\Lambda\left(n\right)$. Let $\nu_{n}$ be a sequence of measures with finite energy, such that $\sup_{n}\nu_{n}\left(E\right)<\infty$, and $\nu_{n}\overset{\mathcal{E}}{\sim}\eta$. If

[TABLE]

for every smooth probability measure $\pi$, then we will say that $\eta$ is $\alpha$-bounded from above, for the sequence $\left(\Lambda\left(n\right)\right)$, and $\nu_{n}$ will be called an $\alpha$-upper sequence for $\Lambda\left(n\right),\eta$.

If there exists a sequence of measures $\mu_{n}$ with finite energy, with $\sup_{n}\mu_{n}\left(E\right)<\infty$ and $\mu_{n}\left(\Lambda\left(n\right)^{c}\right)=0$ for all $n$, such that $\mu_{n}\overset{\mathcal{E}}{\sim}\eta$ and

[TABLE]

for every smooth probability measure $\pi$, then $\eta$ will be said to be $\alpha$-bounded from below for the sequence $\left(\Lambda\left(n\right)\right)$, and $\mu_{n}$ will be called an $\alpha$-lower sequence for $\Lambda\left(n\right),\eta$.

If (2.4) holds, Lemma 9.7 shows that $\gamma_{n}\overset{\mathcal{E}}{\sim}\eta$ automatically holds whenever $G_{\alpha}\,\gamma_{n}\rightarrow G_{\alpha}\,\eta$, $\mathcal{E}$-weakly.

The proof of Theorem 2.1 in Section 5 will show that in the setting of that theorem the conditions in Definition 4.1 are satisfied in the nonrandom case, with $\Lambda\left(n\right)$ equal to the union of many small bodies $\Lambda_{i}\left(n\right)$, and $\nu_{n},\mu_{n}$ each approximately equal to the sum of the $\alpha$-equilibrium measures of the $\Lambda_{i}\left(n\right)$. The same is true in the general case with probability one for a subsequence.

Theorem 4.2

Let $\Lambda\left(n\right)$ be a closed set, $n=1,2,\ldots$. Let $\alpha\in(0,\infty)$, and let $\eta$ be $\alpha$-bounded from above for $\left(\Lambda\left(n\right)\right)$. Let $\tau$ be the randomized stopping time with rate measure $\eta$. Let $\tau_{n}=D_{\Lambda\left(n\right)}$ or $\tau_{n}=T_{\Lambda\left(n\right)}$, and let $\pi$ be a smooth probability measure. Then $\tau_{n}\wedge\tau\wedge\zeta\rightarrow\tau\wedge\zeta$, $\mathbf{P}_{\pi}$-stably. In the case that all the sets $\Lambda\left(n\right)$ are contained in a single compact subset of $E$, $\tau_{n}\wedge\tau\rightarrow\tau$, $\mathbf{P}_{\pi}$-stably.

Suppose $\eta$ is also $\alpha$-bounded from below for $\left(\Lambda\left(n\right)\right)$. Then $\tau_{n}\wedge\zeta\rightarrow\tau\wedge\zeta$, $\mathbf{P}_{\pi}$-stably. If the sets $\Lambda\left(n\right)$ are contained in a single compact subset of $E$, $\tau_{n}\rightarrow\tau$, $\mathbf{P}_{\pi}$-stably.

Although Theorem 4.2 does not require the absolute continuity condition (2.4), it may not be easy to show that the hypotheses are satisfied without using condition (2.4). Before proving the theorem a few auxiliary facts are needed.

Lemma 4.3

Let $\Lambda\left(n\right)$ be a closed set, $n=1,2,\ldots$, and let $\tau_{n}=D_{\Lambda\left(n\right)}$. Let $\alpha\in(0,\infty)$, and let $\pi$ be a smooth probability measure. Let $\eta$ be $\alpha$-bounded from above for $\left(\Lambda\left(n\right)\right)$. Let $A$ be the positive continuous additive functional with Revuz measure $\eta$. Let $\tau$ be a randomized stopping time on $\Omega\times(0,1)$ which is a $\mathbf{P}_{\pi}$-stable limit point of the sequence $\left(\tau_{n}\right)$. Then $\mathbf{P}_{\pi}\left(\tau=0\right)=0$. Furthermore, for any $a,b\in[0,\infty)$ with $a<b$, and any bounded $\mathcal{G}_{a}$-measurable function $H$,

[TABLE]

If $\eta$ is $\alpha$-bounded from below for $\left(\Lambda\left(n\right)\right)$, then equality holds in (4.3).

Proof   By approximating $\pi$, we may assume that $\pi$ has finite energy.

Let $\nu_{n}$ be a $\pi,\alpha$-upper sequence for $\eta,\Lambda\left(n\right)$.

We may assume that the random variable $H$ in (4.3) satisfies $0\leq H\leq 1$, and also, by passing to a subsequence and relabelling, that $\tau_{n}\rightarrow\tau$, $\mathbf{P}_{\pi}$-stably. For $t\in[0,\infty)$, let us say that $t$ is good if $\mathbf{P}_{\pi}\times\lambda^{\ast}\left(\tau=t\right)=0$ and also $\lim_{n\rightarrow\infty}\int\left\lvert G_{\alpha}\,\nu_{n}-G_{\alpha}\,\eta\right\rvert\,d\left(\pi\,p_{t}\right)=0$. Since $\pi$ has finite energy it is easy to show from the definitions that $\pi\,R_{\alpha}$ has a density with respect to $m$ which is in $L^{2}\left(m\right)$. Hence $\lim_{n\rightarrow\infty}\int\left\lvert G_{\alpha}\,\nu_{n}-G_{\alpha}\,\eta\right\rvert\,d\left(\pi\,R_{\alpha}\right)=0$. Thus $\lim_{n\rightarrow\infty}\int_{0}^{\infty}e^{-\alpha t}\int\left\lvert G_{\alpha}\,\nu_{n}-G_{\alpha}\,\eta\right\rvert\,d\left(\pi\,p_{t}\right)\,\lambda(dt)=0$, where $\lambda$ be Lebesgue measure on $\mathbb{R}$. Replacing $\nu_{n}$ by a suitable subsequence, we may assume that $\lambda$-a.e. $t\in[0,\infty)$ is good.

Let $a,b$ be given. For the moment assume that $b$ is good and that either $a$ is good or $a=0$.

Let $Y_{n}=\mathbf{E}_{\pi}\left[H\mathbf{1}_{\left\{a\leq\tau_{n}\right\}}e^{-\alpha\tau_{n}\wedge b}G_{\alpha}\,\nu_{n}\left(X_{\tau_{n}\wedge b}\right)\right]$, $K_{n}=\mathbf{E}_{\pi}\left[H\mathbf{1}_{\left\{a\leq\tau_{n}\right\}}e^{-\alpha a}G_{\alpha}\,\nu_{n}\left(X_{a}\right)\right]$ for $n=1,2,\ldots$. By 9.2, $e^{-\alpha t}G_{\alpha}\,\nu_{n}\left(X_{t}\right)$ is a supermartingale, and so $Y_{n}\leq K_{n}$.

Let $V_{n}=\mathbf{E}_{\pi}\left[H\mathbf{1}_{\left\{a\leq\tau_{n}\right\}}e^{-\alpha a}G_{\alpha}\,\eta\left(X_{a}\right)\right]$, for $n=1,2,\ldots$, $V_{\infty}=\int H\mathbf{1}_{\left\{a\leq\tau\right\}}e^{-\alpha a}G_{\alpha}\,\eta\left(X_{a}\right)\,d\mathbf{P}_{\pi}\,d\lambda^{\ast}$. If $a$ is good, so that $\mathbf{P}_{\pi}\times\lambda^{\ast}\left(\tau=a\right)=0$, $\lim_{n\rightarrow\infty}V_{n}=V_{\infty}$, by Lemma 3.1 (i). If $a=0$, $V_{n}=V_{\infty}$ for all $n$.

Let $\gamma$ be the distribution of $X_{0}$ with respect to $H\mathbf{P}_{\pi}$, so that for $a=0$ we have $K_{n}=\int\left(G_{\alpha}\,\nu_{n}\right)\,d\gamma=\mathcal{E}_{\alpha}\left(G_{\alpha}\,\nu_{n},\hat{G}_{\alpha}\,\gamma\right)$ and $V_{n}=\int\left(G_{\alpha}\,\eta\right)\,d\gamma=\mathcal{E}_{\alpha}\left(G_{\alpha}\,\eta,\hat{G}_{\alpha}\,\gamma\right)$. Then $\lim_{n\rightarrow\infty}\left(K_{n}-V_{n}\right)=0$ if $a=0$ since $\nu_{n}\overset{\mathcal{E}}{\sim}\eta$, and the same limit holds by the definition of good if $a$ is good. By (9.1) and the additive functional property, $V_{\infty}=\int H\mathbf{1}_{\left\{a\leq\tau\right\}}e^{-\alpha a}\mathbf{E}_{X_{a}}\left[\int_{0}^{\infty}e^{-\alpha s}dA_{s}\right]\,d\mathbf{P}_{\pi}\,d\lambda^{\ast}=\int H\mathbf{1}_{\left\{a\leq\tau\right\}}\int_{a}^{\infty}e^{-\alpha r}dA_{r}\,d\mathbf{P}_{\pi}\,d\lambda^{\ast}$.

For $n=1,2,\ldots$, let $Z_{n}=\mathbf{E}_{\pi}\left[H\mathbf{1}_{\left\{a\leq\tau_{n}\right\}}e^{-\alpha\tau_{n}\wedge b}G_{\alpha}\,\eta\left(X_{\tau_{n}\wedge b}\right)\right]=\mathbf{E}_{\pi}\left[H\mathbf{1}_{\left\{a\leq\tau_{n}\right\}}\int_{\tau_{n}\wedge b}^{\infty}e^{-\alpha r}\,dA_{r}\right]$, where the second equality follows from (9.1) and the additive functional property. We have $\int_{\tau_{n}\wedge b}^{\infty}e^{-\alpha v}\,dA_{v}\leq\int_{0}^{\infty}e^{-\alpha v}\,dA_{v}$ for all $n$, and $\mathbf{E}_{\pi}\left[\int_{0}^{\infty}e^{-\alpha r}\,dA_{r}\right]=\int\left(G_{\alpha}\,\eta\right)\,d\pi=\mathcal{E}_{\alpha}\left(G_{\alpha}\,\eta,\hat{G}_{\alpha}\,\pi\right)<\infty$. Thus Lemma 3.1 (i) is applicable and $\lim_{n\rightarrow\infty}Z_{n}=\int H\mathbf{1}_{\left\{a\leq\tau\right\}}\int_{\tau\wedge b}^{\infty}e^{-\alpha r}\,dA_{r}\,d\mathbf{P}_{\pi}\,d\lambda^{\ast}\equiv Z_{\infty}$.

Let $W_{n}=\mathbf{E}_{\pi}\left[H\mathbf{1}_{\left\{a\leq\tau_{n}\leq b\right\}}e^{-\alpha\tau_{n}}\right]$, for $n=1,2,\ldots$, $W_{\infty}=\int H\mathbf{1}_{\left\{a\leq\tau\leq b\right\}}e^{-\alpha\tau}\,d\mathbf{P}_{\pi}\,d\lambda^{\ast}$. By Lemma 3.1 (i), $\lim_{n\rightarrow\infty}W_{n}=W_{\infty}$. For $n=1,2,\ldots$ we have

[TABLE]

By (4.1) and the fact that $b$ is good, $\limsup_{n\rightarrow\infty}\left(W_{n}+Z_{n}-Y_{n}\right)\leq 0$. Hence $\limsup_{n\rightarrow\infty}\left(W_{n}+Z_{n}-K_{n}\right)\leq 0$, so $W_{\infty}\leq V_{\infty}-Z_{\infty}$. Taking $a=0$ and $H=1$ gives $\int\mathbf{1}_{\left\{\tau\leq b\right\}}e^{-\alpha\tau}\,d\mathbf{P}_{\pi}\,d\lambda^{\ast}\leq\int\int_{0}^{\tau\wedge b}e^{-\alpha r}dA_{r}\,d\mathbf{P}_{\pi}\,d\lambda^{\ast}\leq\mathbf{E}_{\pi}\left[\int_{0}^{b}e^{-\alpha r}dA_{r}\right]$. Letting $b\searrow 0$ gives $\mathbf{P}_{\pi}\times\lambda^{\ast}\left(\tau=0\right)=0$. For general $H$ and good $a,b$, $W_{\infty}\leq V_{\infty}-Z_{\infty}$ gives (4.3). Both sides of (4.3) are right continuous in $a$ and $b$, so (4.3) holds for all $a,b$.

If $\eta$ is $\alpha$-bounded from below, let $\mu_{n}$ be an $\alpha$-lower sequence. Define quantities $Y_{n},K_{n},V_{n},Z_{n},W_{n}$ as above, except with $\nu_{n}$ replaced by $\mu_{n}$. By Lemma 9.3, $e^{-\alpha t\wedge\tau_{n}}G_{\alpha}\,\mu_{n}\left(X_{t\wedge\tau_{n}}\right)$ is a martingale, so now $Y_{n}=K_{n}$. All the remaining arguments work as before, with inequalities reversed. This gives  (4.3) with the inequality reversed, so the equality case of (4.3) holds when $\eta$ is also $\alpha$-bounded from below for $\left(\Lambda\left(n\right)\right)$.

Proposition 4.4

Let $\mathcal{G}_{t}=\sigma\left(X_{s},\,s\leq t\right)$. There exists an exceptional set $K\in\mathcal{B}$ such that for any probability measure $\pi$ on $E$ with $\pi\left(K\right)=0$, if $M=\left(M_{t}\right)_{t\in[0,\infty)}$ is a right continuous $\left(\mathcal{G}_{t}\right)$-martingale with respect to $\mathbf{P}_{\pi}$, then

[TABLE]

The proof of Proposition 4.4 is similar to the proof for the Brownian motion case. Let $V$ denote the set of all $Y\in L^{1}\left(\pi\right)$, such that there exists a right continuous $\left(\mathcal{F}_{t}\right)$-martingale $N^{Y}_{t}$ with $N^{Y}_{t}=\mathbf{E}_{\pi}\left[\left.Y\,\,\right|\mathcal{F}_{t}\right]$ for each $t\in[0,\infty)$, and such that with $\mathbf{P}_{\pi}$-probability one, $t\mapsto N^{Y}_{t}$ is continuous on $[0,\zeta)$. One can show that $V$ is closed in $L^{1}\left(\pi\right)$ and contains a dense class, consisting of all $Y=R_{\alpha}\,{h}\left(X_{u}\right)$, where $\alpha,u\in{\mathbb{R}}$ with $\alpha>0$, $u\geq 0$. Thus $V=L^{1}(\pi)$, and this implies the result. Details of the proof are given in [7].

Lemma 4.5

Let $\left(\Omega,\mathcal{G},\mathbf{P}\right)$ be a probability space and let $\left(\mathcal{G}_{t}\right)_{t\geq 0}$ be a filtration with $\mathcal{G}_{t}\subset\mathcal{G}$ for all $t$. Let $E$ be a set and let $\partial$ be a point not in $E$, and assume that $E\cup\left\{\partial\right\}$ has a metrizable topology. Let $\left(X_{t}\right)_{t\in[0,\infty]}$ be any $\mathcal{G}\times\mathcal{B}_{\infty}$-measurable process with $X_{t}\in E$ for $t\in[0,\infty)$ and $X_{\infty}=\partial$. Assume that $X$ is càdlàg and quasi-left continuous on $[0,\infty)$ with respect to $\mathbf{P}$. Let $\tau_{n},\tau$ be randomized $\left(\mathcal{G}_{t}\right)$-stopping times. Let $F\subset E$ be closed in $E_{\partial}$ and let $B_{F}\in\mathcal{G}$ be such that $\mathbf{P}\times\lambda^{\ast}\left(\left\{X_{\tau_{n}}\notin F,\tau_{n}<\infty\right\}\cap\left(B_{F}\times(0,1)\right)\right)=0$ for each $n$, where $\lambda^{\ast}$ is Lebesgue measure on $(0,1)$. Then for any $\mathbf{P},\mathcal{G}$-stable limit point $\tau$ of the sequence $\tau_{n}$, $\mathbf{P}\times\lambda^{\ast}\left(\left\{X_{\tau}\notin F,\tau<\infty\right\}\cap\left(B_{F}\times(0,1)\right)\right)=0$.

Proof   $\tau_{n_{k}}\rightarrow\tau$, $\mathbf{P},\mathcal{G}$-stably, for some $n_{k}$. Let $s\in(0,\infty)$. Let $\varphi$ be a continuous function on $E_{\partial}$ such that $0\leq\varphi\leq\mathbf{1}_{E_{\partial}-F}$, and let $\psi$ be a continuous function on $[0,\infty]$ such that $0\leq\psi\leq\mathbf{1}_{[0,s]}$. Let $\xi_{t}=\varphi\left(X_{t}\right)\psi(t)$ on $B_{F}$, $\xi=0$ otherwise. Since $\xi$ is càdlàg and quasi-left continuous on $[0,\infty]$, $\lim_{k\rightarrow\infty}\int\mathbf{1}_{B_{F}}\varphi\left(X_{\tau_{n_{k}}}\right)\psi\left(\tau_{n_{k}}\right)\,d\mathbf{P}\,d\lambda^{\ast}=\int\mathbf{1}_{B_{F}}\varphi\left(X_{\tau}\right)\psi\left(\tau\right)\,d\mathbf{P}\,d\lambda^{\ast}$ by Lemma 3.1 (ii). $\int\mathbf{1}_{B_{F}}\varphi\left(X_{\tau_{n_{k}}}\right)\psi\left(\tau_{n_{k}}\right)\,d\mathbf{P}\,d\lambda^{\ast}=0$ for all $k$, so $\int\mathbf{1}_{B_{F}}\varphi\left(X_{\tau}\right)\psi\left(\tau\right)\,d\mathbf{P}\,d\lambda^{\ast}=0$ for all $\varphi,\psi$, which gives $\mathbf{P}\times\lambda^{\ast}\left(\left\{X_{\tau}\notin F,\tau<\infty\right\}\cap\left(B_{F}\times(0,1)\right)\right)=0$.

Proof of Theorem 4.2

Suppose $\tau_{n}=D_{\Lambda\left(n\right)}$. Let $\alpha\in(0,\infty)$. Let $\eta$ be $\alpha$-bounded from above for for $\left(\Lambda\left(n\right)\right)$.

Let $\sigma$ be any randomized stopping time which is a $\mathbf{P}_{\pi}$-stable limit point of a subsequence $\tau_{n_{i}}$. Let $\pi$ be a smooth probability measure. We will show that with $\mathbf{P}_{\pi}$-probability one, $S^{\sigma}_{t}\geq e^{-A_{t}}$ for $t\in[0,\zeta)$. By relabelling we may assume that the full sequence $\tau_{n}$ converges $\mathbf{P}_{\pi}$-stably to $\sigma$.

Applying Lemma 4.3 shows that (4.3) holds. Let $\Phi^{\sigma}$ be the random measure on $[0,\infty]$ such that $\Phi^{\sigma}{\left((t,\infty]\right)}=S^{\sigma}_{t}$. The left side of (4.3) is $\mathbf{E}_{\pi}\left[H\int_{(a,b]}e^{-\alpha u}\,\Phi^{\sigma}{\left(du\right)}\right]$. The right side of (4.3) is:

[TABLE]

Hence by (4.3), for all $a,b\in[0,\infty)$ with $a\leq b$,

[TABLE]

for any bounded $\mathcal{G}_{a}$-measurable function $H$. Then $N_{t}\equiv\int_{(0,t]}e^{-\alpha u}\,\left(\Phi^{\sigma}{\left(du\right)}-S^{\sigma}_{u}dA_{u}\right)$ is a $\mathcal{G}_{t}$-supermartingale.

Let $F$ be a compact subset of $E$, and as usual let $D_{E_{\partial}-F}$ be the entrance time of $E_{\partial}-F$. Then by Lemma 4.5, with $B_{F}=\left\{D_{E_{\partial}-F}\geq\zeta\right\}$,

[TABLE]

Let $N^{F}_{t}=N_{t\wedge D_{E_{\partial}-F}}$. It is easy to check using equation (9.1) that $N^{F}$ is of class (DL), so by Doob-Meyer, $N^{F}_{t}=M_{t}-G_{t}$, where $M$ is a right continuous martingale and $G$ is a natural right-continuous increasing process, and $M$ and $G$ are unique. Thus $M_{t}=M_{t\wedge D_{E_{\partial}-F}}$ and $G_{t}=G_{t\wedge D_{E_{\partial}-F}}$. Also $M=N^{F}+G$. By Proposition 4.4, $t\mapsto M_{t}$ is continuous on $[0,\zeta)$ with probability one.

The formula for $N$ shows that the only discontinuities in the paths of $N$ and $N^{F}$ occur in the form of positive jumps. The same is true for $G$, so if a positive jump for $N_{\cdot}^{F}(\omega)$ occurs at time $t$ then a positive jump for $M_{\cdot}(\omega)$ must also occur at time $t$. With probability one, since $M$ has continuous paths on $[0,\zeta)$, $N^{F}$ must also have continuous paths on $[0,\zeta)$. Hence $t\mapsto N^{F}_{t}$ is continuous on $[0,\infty)$ for $\mathbf{P}_{\pi}$-a.e. path in $\left\{\zeta=\infty\right\}\cup\left\{D_{E_{\partial}-F}<\zeta\right\}$, and also for $\mathbf{P}_{\pi}$-a.e. path in $\left\{D_{E_{\partial}-F}\geq\zeta\right\}\cap\left\{\zeta<\infty\right\}$, by (4.5). Hence $t\mapsto N^{F}_{t}$ is continuous on $[0,\infty)$ with probability one. Since $N^{F}$ is continuous, $M_{\zeta-}-M_{\zeta}=G_{\zeta}-G_{\zeta-}\geq 0$. Since $G$ is natural, $\mathbf{E}_{\pi}\left[\left(M_{\zeta-}-M_{\zeta}\right)\left(G_{\zeta}-G_{\zeta-}\right)\right]=0$, so $t\mapsto M_{t}$ is continuous at $\zeta$, and $M$ is continuous on $[0,\infty)$. Since the paths of $M$ have finite variation, they are constant on $[0,\infty)$ with probability one. Hence $t\mapsto N^{F}_{t}$ is continuous and monotonic decreasing with probability one. Thus $t\mapsto N_{t}$ is continuous and monotonic decreasing on $[0,D_{E_{\partial}-F})$ with probability one.

Let $\left(F_{k}\right)$ be an $\mathcal{E}$-nest, with $F_{k}$ compact for each $k=1,2,\ldots$. By IV.5.30 in [22], $\mathbf{P}_{\pi}\left(\lim_{k\rightarrow\infty}D_{E_{\partial}-F_{k}}<\zeta\right)=0$. It follows that with probability one, $t\mapsto N_{t}$ is continuous and monotonic decreasing on $[0,\zeta)$, i.e.  $\Phi^{\sigma}{\left(dt\right)}-S^{\sigma}_{t}d\,A_{t}\leq 0$ on $[0,\zeta)$. By Lemma 4.3, $\mathbf{P}_{\pi}\left(S^{\sigma}_{0}=1\right)=1$. Multiplying by $e^{A_{t}}$ shows $d\left(e^{A_{t}}S^{\sigma}_{t}\right)\geq 0$ on $[0,\zeta)$, and so with $\mathbf{P}_{\pi}$-probability one, $S^{\sigma}_{t}\geq e^{-A_{t}}$ for $t\in[0,\zeta)$. By (2.2), if $\sigma(\omega,r)<c<\zeta(\omega)$ then there exists $u$ such that $S^{\sigma}_{u}\leq 1-r$. But then $e^{A_{u}}\leq 1-r$, so $\tau(\omega,r)<c$ by (2.2). It follows that $\sigma\wedge\zeta\geq\tau\wedge\zeta$ with $\mathbf{P}_{\pi}$-probability one, and hence $\sigma\wedge\tau\wedge\zeta=\tau\wedge\zeta$. Since this is true for any stable limit point $\sigma$, and compactness holds for stable convergence, we have proved that $\tau_{n}\wedge\tau\wedge\zeta\rightarrow\tau\wedge\zeta$, $\mathbf{P}_{\pi}$-stably.

In the special case that for some compact set $F$ we have $\Lambda\left(n\right)\subset F$ for all $n$, $\mathbf{P}\times\lambda^{\ast}\left(\left\{X_{\sigma}\notin F,\sigma<\infty\right\}\right)=0$ by Lemma 4.5 with $B_{F}=\Omega$. Obviously $\mathbf{P}\times\lambda^{\ast}\left(\left\{X_{\sigma}\in F,\zeta\leq\sigma<\infty\right\}\right)=0$. Hence $\int_{E}\Phi^{\sigma}{\left([\zeta,\infty)\right)}\,d\mathbf{P}_{\pi}=0$, so with probability one $t\mapsto S^{\sigma}_{t}$ is constant on $[\zeta,\infty)$ and has no discontinuity at $t=\zeta$. Hence with $\mathbf{P}_{\pi}$-probability one, $S^{\sigma}_{t}\geq e^{-A_{t}}$ for $t\in[0,\infty)$. Using (2.2) now shows that $\sigma\geq\tau$, so $\sigma\wedge\tau=\tau$, and hence $\tau_{n}\wedge\tau\rightarrow\tau$, $\mathbf{P}_{\pi}$-stably.

Up to this point we have assumed that $\eta$ is $\alpha$-bounded from above. Now let $\eta$ be $\alpha$-bounded from below for $\left(\Lambda\left(n\right)\right)$ as well, and let $\sigma$ be a $\mathbf{P}_{\pi}$-stable limit point. Lemma 4.3 now gives the equality case of (4.3), and we can repeat the same arguments as before, replacing inequalities by equalities at each step. Thus the process $N$ is now a martingale. As before, but omitting the Doob-Meyer decomposition step, one obtains $S^{\sigma}_{t}=e^{-A_{t}}\text{ for all }t\in[0,\zeta)$, or $S^{\sigma}_{t}=e^{-A_{t}}\text{ for all }t\in[0,\infty)$ if all the sets $\Lambda\left(n\right)$ are contained in some compact set. By (2.2), $\tau\wedge\zeta=\sigma\wedge\zeta$, and $\tau=\sigma$ if all the sets $\Lambda\left(n\right)$ are contained in some compact set. Since $\sigma$ was any limit point, it follows that the full sequences $\tau_{n}\wedge\zeta$ and $\tau_{n}$ converge as claimed. This completes the proof of the convergence statements in the case that $\tau_{n}=D_{\Lambda\left(n\right)}$. We know by right continuity of the process that $\mathbf{P}_{\pi}\left(\zeta=0\right)=0$. Thus $\limsup_{n\rightarrow\infty}\pi\left(\Lambda\left(n\right)\right)\leq\limsup_{n\rightarrow\infty}\mathbf{P}_{\pi}\left(\tau_{n}=0\right)\leq\limsup_{n\rightarrow\infty}\mathbf{P}_{\pi}\left(\tau_{n}\wedge\zeta=0\right)\leq\mathbf{P}_{\pi}\times\lambda^{\ast}\left(\tau\wedge\zeta=0\right)=0$, by what has already been proved. Consequently $\mathbf{P}_{\pi}\left(D_{\Lambda\left(n\right)}\neq T_{\Lambda\left(n\right)}\right)\rightarrow 0$, so the conclusion of the theorem remains true when $\tau_{n}=T_{\Lambda\left(n\right)}$.

5 Proof of

Theorem 2.1

Let all the assumptions for Theorem 2.1 other than (2.5) hold.

We can choose our sample space for the random variables $\Lambda_{j}\left(n\right)$ to be a product space. That is, for each $n,j$ let $\Lambda_{j}\left(n\right)$ be defined as a random variable on some probability space $\left(\tilde{\Omega}^{n}_{j},q^{n}_{j}\right)$. Let

[TABLE]

For convenience, also let $\tilde{\Omega}=\prod_{n=1}^{\infty}\tilde{\Omega}^{n}$, $\tilde{\mathbf{P}}=\prod_{n=1}^{\infty}q^{n}$. We will regard the random variables $\Lambda_{j}\left(n\right)$ as defined on either $\tilde{\Omega}^{n}$ or $\tilde{\Omega}$ when it seems helpful. By definition $\bar{\gamma}^{n}_{j}(A)=\int\gamma^{n}_{j}(A)\,dq^{n}_{j}$ for $A\in\mathcal{B}$. It is straightforward to show that $\int f\,d\gamma^{n}_{j}$ is measurable for $f\in b\mathcal{B}\cup\mathcal{B}^{+}$. It is straightforward to show that $\bar{\gamma}^{n}_{j}$ is a measure, that

[TABLE]

for all $n,j$ and each $f\in b\mathcal{B}\cup\mathcal{B}^{+}$. For nonnegative bounded $f\in L^{1}\left(m\right)$, $\tilde{\mathbf{E}}\left[\int f\left(G_{\alpha}\,\gamma^{n}_{j}\right)\,dm\right]=\tilde{\mathbf{E}}\left[\int\left(\hat{G}_{\alpha}\,f\right)\,d\gamma^{n}_{j}\right]=\int\left(\hat{G}_{\alpha}\,f\right)\,d\bar{\gamma}^{n}_{j}=\int f\left(G_{\alpha}\,\bar{\gamma}^{n}_{j}\right)\,dm$, so $\int f\left(G_{\alpha}\,\bar{\gamma}^{n}_{j}\right)\,dm\leq\int f\,dm$. and it follows that $G_{\alpha}\,\bar{\gamma}^{n}_{j}\leq 1$, $m$-a.e. Hence $G_{\alpha}\,\bar{\gamma}^{n}_{j}\leq 1$, $\mathcal{E}$-q.e., by IV.3.3 of [22], and $\left\lVert G_{\alpha}\,\bar{\gamma}^{n}_{j}\right\rVert_{\mathcal{E},\alpha}^{2}=\int\left(G_{\alpha}\,{\bar{\gamma}^{n}_{j}}\right)\,d\bar{\gamma}^{n}_{j}\leq\chi_{n}$.

For any $x$ such that $\delta_{x}\,R_{\alpha}<<m$, and for each $\beta>0$, let $g_{\beta}\left(x,\cdot\right)$ be a density for $\delta_{x}\,R_{\alpha}$ with respect to $m$. Otherwise let $g_{\alpha}\left(x,\cdot\right)=0$. Since (2.4) is assumed to hold, we can choose $g_{\beta}\left(x,\cdot\right)$ (for example, via the martingale theorem) so that $g_{\beta}\left(x,y\right)$ is jointly measurable in $x$ and $y$. For any $h\in L^{2}\left(m\right)$,

[TABLE]

for $m$-a.e. $x$. It follows that also $\hat{G}_{\beta}\,h(y)=\int g_{\beta}\left(x,y\right)h(x)\,m(dx)$ for $m$-a.e. $y$.

Lemma 5.1

Let the assumptions for Theorem 2.1 other than (2.5) hold. Let a subsequence $\left(n_{i}\right)$ be given. Then a further subsequence $(n_{i_{\ell}})$ can be chosen, such that with $\tilde{\mathbf{P}}$-probability one, $\gamma^{n_{i_{\ell}}}\overset{\mathcal{E}}{\sim}\eta$ in the sense of Definition 4.1.

Proof   Since $\left\lVert\gamma^{n}_{j}\right\rVert_{\text{tv}}\leq\chi_{n}$, $\sum_{j}\left\lVert\gamma^{n}_{j}\right\rVert_{\text{tv}}^{2}\leq\chi_{n}\sum_{j}\left\lVert\gamma^{n}_{j}\right\rVert_{\text{tv}}$. Using independence,

[TABLE]

Let the subsequence $\left(n_{i}\right)$ be given. Using Borel-Cantelli we can choose $n_{i_{\ell}}$ so that $\left(\left\lVert\gamma^{n_{i_{\ell}}}\right\rVert_{\text{tv}}-\left\lVert\bar{\gamma}^{n_{i_{\ell}}}\right\rVert_{\text{tv}}\right)\rightarrow 0$, $\tilde{\mathbf{P}}$-a.e. Since $\sup_{n}\left\lVert\bar{\gamma}^{n}\right\rVert_{\text{tv}}<\infty$, $\left\lVert\gamma^{n_{i_{\ell}}}\right\rVert_{\text{tv}}$ is bounded in $\ell$, $\tilde{\mathbf{P}}$-a.e.

By independence, for $i\neq j$ we have

[TABLE]

If $\gamma^{n}_{i}\neq 0$, we have, using Jensen and the fact that $G_{\alpha}\,\gamma^{n}_{j}\leq 1$, $\mathcal{E}$-q.e., that

[TABLE]

When $i\neq j$, (5.3) gives

[TABLE]

Using Jensen, the same bound holds if $\gamma^{n}_{i}$ is replaced by $\bar{\gamma}^{n}_{i}$ and/or $\gamma^{n}_{j}$ is replaced by $\bar{\gamma}^{n}_{j}$.

For each $n$ and each $i,j\in\left\{1,\ldots,\kappa_{n}\right\}$, let

[TABLE]

Using (5.1), $\int Y_{ij}(n)\,dq^{n}_{j}=0=\int Y_{ij}(n)\,dq^{n}_{i}$. Let $Z(n)=\sum_{i\neq j}Y_{ij}(n)$. Then

[TABLE]

where $\sum^{\ast}_{k,\ell}$ is the sum over all $k\neq\ell$ such that either $k$ is different from $i$ and $j$ or $\ell$ is different from $i$ and $j$. Consider a term $\int Y_{ij}(n)Y_{k\ell}(n)\,dq^{n}$ where $k$ is different from $i$ and $j$. Evaluating the integral as an iterated integral, and integrating with respect to $q^{n}_{k}$ first, the value is zero. Thus

[TABLE]

Using (5.4), $\sum_{i\neq j}\int Y_{ij}(n)^{2}\,dq^{n}\leq 4\chi_{n}\int G_{\alpha}\,\bar{\gamma}^{n}\,d\bar{\gamma}^{n}$. Since $\int\left(G_{\alpha}\,\bar{\gamma}^{n}\right)\,d\bar{\gamma}^{n}$ is bounded in $n$, it follows that $\int Z(n)^{2}\,dq^{n}\rightarrow 0$ as $n\rightarrow\infty$. We have $\int\left(G_{\alpha}\,\gamma^{n}_{j}\right)\,d\gamma^{n}_{i}\leq Y_{ij}+\int\left(G_{\alpha}\,\gamma^{n}_{j}\right)\,d\bar{\gamma}^{n}_{i}+\int\left(G_{\alpha}\,\bar{\gamma}^{n}_{j}\right)\,d\gamma^{n}_{i}\leq Y_{ij}+\left\lVert\bar{\gamma}^{n}_{i}\right\rVert_{\text{tv}}+\left\lVert\gamma^{n}_{i}\right\rVert_{\text{tv}}$.

[TABLE]

Hence we can refine the subsequence $n_{i_{\ell}}$ to ensure that with $\tilde{\mathbf{P}}$-probability one, $\int\left(G_{\alpha}\,\gamma^{n_{i_{\ell}}}\right)\,d\gamma^{n_{i_{\ell}}}$ is bounded in $\ell$.

By IV.3.3 in [22], there exists a countable dense subset $\Gamma$ of $D\left(\mathcal{E}\right)$. By I.4.17 of [22] , we may choose the functions in $\Gamma$ to be bounded, and we will also choose them to be $\mathcal{E}$-quasi continuous. For any fixed $v\in\Gamma$, let $V_{n}=\int v\,d\gamma^{n}$, so that $\tilde{\mathbf{E}}\left[V_{n}\right]=\int v\,d\bar{\gamma}^{n}=\mathcal{E}_{\alpha}\left(G_{\alpha}\,\bar{\gamma}^{n},v\right)\rightarrow\mathcal{E}_{\alpha}\left(G_{\alpha}\,\eta,v\right)=\int v\,d\eta$. Let $V_{n,j}=\int v\,d\gamma^{n}_{j}$. Using independence,

[TABLE]

Hence we can refine the subsequence $n_{i_{\ell}}$ so that with $\tilde{\mathbf{P}}$-probability one, $\int v\,d\gamma^{n_{i_{\ell}}}\rightarrow\int v\,d\eta$ for each $v\in\Gamma$. Then with $\tilde{\mathbf{P}}$-probability one, $G_{\alpha}\,{\gamma^{n_{i_{\ell}}}}\rightarrow G_{\alpha}\,\eta$, $\mathcal{E}$-weakly. By Lemma 9.6, $\bar{\gamma}^{n_{i_{\ell}}}\overset{\mathcal{E}}{\sim}\eta$.

Lemma 5.2

Let the assumptions for Theorem 2.1 other than (2.5) hold. Let a subsequence $\left(n_{i_{\ell}}\right)$ be given. Let a particular environment $\tilde{\omega}\in\tilde{\Omega}$ be such that the corresponding sequence of measures $\gamma^{n_{i_{\ell}}}$ has the properties stated in Lemma 5.1. Then $\eta$ is $\alpha$-bounded from above for $\left(\Lambda\left(n_{i_{\ell}}\right)(\tilde{\omega})\right)$, in the sense of Definition 4.1.

Proof   By assumption $\gamma^{n_{i_{\ell}}}\overset{\mathcal{E}}{\sim}\eta$. Let $\pi$ be a smooth probability measure. By approximating $\pi$ we may assume that $\pi$ has finite energy. We must verify (4.1).

Let $\varepsilon>0$ be given. For any $\beta\in(0,\infty)$, let $H(\beta)=\left\{\beta G_{\beta+\alpha}\,G_{\alpha}\,{\eta}+\varepsilon\geq G_{\alpha}\,\eta\right\}$. Since $\int\left(\hat{G}_{\alpha}\,\pi\right)\,d\eta<\infty$ and $\int\left(\hat{G}_{\alpha}\,\eta\right)\,d\eta<\infty$, (9.2) tells us that for all sufficiently large $\beta$ we have $\int_{H(\beta)^{c}}\left(\hat{G}_{\alpha}\,{\pi}\right)\,d\eta<\varepsilon$ and $\int_{H(\beta)^{c}}\left(\hat{G}_{\alpha}\,{\eta}\right)\,d\eta<\varepsilon$. Choose such a $\beta$, and let $\psi^{1}_{\varepsilon}=\mathbf{1}_{H(\beta)^{c}}\eta$. Then $\int\left(\hat{G}_{\alpha}\,{\pi}\right)\,d\psi_{\varepsilon}^{1}<\varepsilon$ and $\int\left(\hat{G}_{\alpha}\,{\eta}\right)\,d\psi_{\varepsilon}^{1}<\varepsilon$. By the domination principle ( Lemma 9.4), $\beta G_{\beta+\alpha}\,G_{\alpha}\,{\eta}+\varepsilon\geq G_{\alpha}\,\left(\eta-\psi^{1}_{\varepsilon}\right)$ holds $\mathcal{E}$-q.e. on $E$. Thus $\beta G_{\beta+\alpha}\,G_{\alpha}\,{\eta}+\varepsilon+G_{\alpha}\,\psi^{1}_{\varepsilon}\geq G_{\alpha}\,\eta$, $\mathcal{E}$-q.e.

Since $\int\left(\hat{G}_{\alpha}\,\pi\right)\left(G_{\alpha}\,\eta\right)\,dm<\infty$, by (9.2) we have $\lim_{\lambda\rightarrow\infty}\int\left(\hat{G}_{\alpha}\,\pi\right)\left(G_{\alpha}\,\eta-\lambda G_{\lambda+\alpha}\,G_{\alpha}\,\eta\right)\,dm=0$. Choose $\lambda\geq\alpha$ such that $\beta\int\left(\hat{G}_{\alpha}\,\pi\right)\left(G_{\alpha}\,\eta-\lambda G_{\lambda+\alpha}\,G_{\alpha}\,\eta\right)\,dm<\varepsilon$ and $\beta\int\left(\hat{G}_{\alpha}\,\eta\right)\left(G_{\alpha}\,\eta-\lambda G_{\lambda+\alpha}\,G_{\alpha}\,\eta\right)\,dm<\varepsilon$. Let $\psi^{2}_{\varepsilon}=\beta\left(G_{\alpha}\,\eta-\lambda G_{\lambda+\alpha}\,G_{\alpha}\,\eta\right)\,m$. Then $\int\left(\hat{G}_{\alpha}\,\pi\right)\,d\psi^{2}_{\varepsilon}<\varepsilon$ and $\int\left(\hat{G}_{\alpha}\,\eta\right)\,d\psi^{2}_{\varepsilon}<\varepsilon$. Also $G_{\beta+\alpha}\,{\psi^{2}_{\varepsilon}}=\beta G_{\beta+\alpha}\,G_{\alpha}\,\eta-\beta\lambda G_{\beta+\alpha}\,{G_{\lambda+\alpha}\,{G_{\alpha}\,\eta}}$, so $\beta G_{\beta+\alpha}\,G_{\alpha}\,\eta=\beta\lambda G_{\beta+\alpha}\,{G_{\lambda+\alpha}\,{G_{\alpha}\,\eta}}+G_{\beta+\alpha}\,\psi^{2}_{\varepsilon}$. Hence $\beta\lambda G_{\beta+\alpha}\,{G_{\lambda+\alpha}\,G_{\alpha}\,\eta}+\varepsilon+G_{\alpha}\,{\left(\psi^{1}_{\varepsilon}+\psi^{2}_{\varepsilon}\right)}\geq G_{\alpha}\,\eta$, $\mathcal{E}$-q.e.

For any $A\in\mathcal{B}$ with $m\left(A\right)<\infty$, and any number $c>0$, for any $h\in L^{2}\left(m\right)$ let

[TABLE]

As $A\nearrow E$ and $c\nearrow\infty$, $M_{A,c}G_{\alpha}\,\eta\nearrow\lambda G_{\lambda+\alpha}\,G_{\alpha}\,\eta\leq G_{\alpha}\,\eta$, $m$-a.e. We can choose $A=A_{\varepsilon}$ and $c=c_{\varepsilon}$ such that $\beta\int\left(\hat{G}_{\alpha}\,{\pi}\right)\left(\lambda G_{\lambda+\alpha}\,G_{\alpha}\,\eta-M_{A,c}G_{\alpha}\,\eta\right)\,dm<\varepsilon$ and $\beta\int\left(\hat{G}_{\alpha}\,\eta\right)\left(\lambda G_{\lambda+\alpha}\,G_{\alpha}\,\eta-M_{A,c}G_{\alpha}\,\eta\right)\,dm<\varepsilon$.

Let $\psi^{3}_{\varepsilon}=\beta\left(\lambda G_{\lambda+\alpha}\,G_{\alpha}\,\eta-M_{A,c}G_{\alpha}\,\eta\right)\,m$. Then $\int\left(\hat{G}_{\alpha}\,\pi\right)\,d\psi^{3}_{\varepsilon}<\varepsilon$ and $\int\left(\hat{G}_{\alpha}\,\eta\right)\,d\psi^{3}_{\varepsilon}<\varepsilon$. Also $G_{\beta+\alpha}\,\psi^{3}_{\varepsilon}=\beta\lambda G_{\beta+\alpha}\,G_{\lambda+\alpha}\,G_{\alpha}\,\eta-\beta G_{\beta+\alpha}\,M_{A,c}G_{\alpha}\,\eta$, so $\beta\lambda G_{\beta+\alpha}\,G_{\lambda+\alpha}\,G_{\alpha}\,\eta=\beta G_{\beta+\alpha}\,M_{A,c}G_{\alpha}\,\eta+G_{\beta+\alpha}\,\psi^{3}_{\varepsilon}$. Hence $\beta G_{\beta+\alpha}\,M_{A,c}G_{\alpha}\,\eta+\varepsilon+G_{\alpha}\,{\left(\psi^{1}_{\varepsilon}+\psi^{2}_{\varepsilon}+\psi^{3}_{\varepsilon}\right)}\geq G_{\alpha}\,\eta$, $\mathcal{E}$-q.e. Let $\psi_{\varepsilon}=\psi^{1}_{\varepsilon}+\psi^{2}_{\varepsilon}+\psi^{3}_{\varepsilon}$, so that $\beta G_{\beta+\alpha}\,M_{A,c}G_{\alpha}\,\eta+\varepsilon+G_{\alpha}\,{\psi_{\varepsilon}}\geq G_{\alpha}\,\eta$, $\mathcal{E}$-q.e., where

[TABLE]

Since $\left\lVert M_{A,c}\left(G_{\alpha}\,\gamma^{n}-G_{\alpha}\,\eta\right)\right\rVert_{\text{sup}}\leq c\int_{A}\left\lvert G_{\alpha}\,\gamma^{n}-G_{\alpha}\,\eta\right\rvert\,dm$, we have $\left\lVert\beta G_{\beta+\alpha}\,{M_{A,c}\left(G_{\alpha}\,\gamma^{n}-G_{\alpha}\,\eta\right)}\right\rVert_{\text{sup}}=\left\lVert\beta R_{\beta+\alpha}\,{M_{A,c}\left(G_{\alpha}\,\gamma^{n}-G_{\alpha}\,\eta\right)}\right\rVert_{\text{sup}}\leq c\int_{A}\left\lvert G_{\alpha}\,\gamma^{n}-G_{\alpha}\,\eta\right\rvert\,dm$, $\mathcal{E}$-q.e. Hence

[TABLE]

For any $j=1,\ldots,\kappa_{n}$,

[TABLE]

Thus $M_{A,c}G_{\alpha}\,{\gamma^{n}_{j}}\leq c\chi_{n}/\alpha$ everywhere and hence $\beta G_{\beta+\alpha}\,M_{A,c}G_{\alpha}\,\gamma^{n}_{j}=\beta R_{\beta+\alpha}\,M_{A,c}G_{\alpha}\,\gamma^{n}_{j}\leq c\chi_{n}/\alpha$, $\mathcal{E}$-q.e.

For $j=1,\ldots,\kappa_{n}$, let $\breve{\gamma}^{n}_{j}=\sum_{r\neq j}\gamma^{n}_{r}$. Then $G_{\alpha}\,\breve{\gamma}^{n}_{j}+\tfrac{c\chi_{n}}{\alpha}\geq\beta\lambda G_{\beta+\alpha}\,G_{\lambda+\alpha}\,{G_{\alpha}\,{\breve{\gamma}^{n}_{j}}}+\tfrac{c\chi_{n}}{\alpha}\geq\beta G_{\beta+\alpha}\,M_{A,c}G_{\alpha}\,{\breve{\gamma}^{n}_{j}}+\tfrac{c\chi_{n}}{\alpha}\geq\beta G_{\beta+\alpha}\,M_{A,c}G_{\alpha}\,{\gamma^{n}}$, $\mathcal{E}$-q.e. It follows that

[TABLE]

On $\Lambda_{j}\left(n\right)$, $G_{\alpha}\,\gamma^{n}=G_{\alpha}\,\breve{\gamma}^{n}_{j}+1$, $\mathcal{E}$-q.e., and so on every $\Lambda_{j}\left(n\right)$,

[TABLE]

By (5.7), $\left(1+G_{\alpha}\,\eta-G_{\alpha}\,\gamma^{n_{i_{\ell}}}\right)^{+}\leq 2\varepsilon+G_{\alpha}\,\psi_{\varepsilon}$, $\mathcal{E}$-q.e. on $\Lambda\left(n\right)$, for large $\ell$.

It is not hard to show that $e^{-\alpha t}G_{\alpha}\,\psi_{\varepsilon}\left(X_{t}\right)$ is a supermartingale with respect to $\mathbf{P}_{\pi}$. Hence $\mathbf{E}_{\pi}\left[G_{\alpha}\,\psi_{\varepsilon}\left(X_{0}\right)\right]\geq\mathbf{E}_{\pi}\left[e^{-\alpha\tau_{n_{i_{\ell}}}}G_{\alpha}\,\psi_{\varepsilon}\left(X_{\tau_{n_{i_{\ell}}}}\right)\right]$, so $\mathbf{E}_{\pi}\left[e^{-\alpha\tau_{n_{i_{\ell}}}}G_{\alpha}\,{\psi_{\varepsilon}}\left(X_{\tau_{n_{i_{\ell}}}}\right)\right]\leq\int\left(G_{\alpha}\,\psi_{\varepsilon}\right)\,d\pi<3\varepsilon$ by (5.5). It follows that (4.1) holds with $n$ replaced by $n_{i_{\ell}}$ and $\nu_{n}$ replaced by $\gamma^{n_{i_{\ell}}}$, so $\eta$ is a $\alpha$-bounded from above for the sets $\Lambda\left(n_{i_{\ell}}\right)$.

Proof of Theorem 2.1

Lemma 5.2 and Theorem 4.2 prove the statements of Theorem 2.1 for which (2.5) is not assumed to hold. Suppose now that (2.5) holds, along with the other assumptions for Theorem 2.1.

Let a subsequence $\left(n_{i}\right)$ be given. By Lemma 5.1 we can choose $n_{i_{\ell}}$ so that with $\tilde{\mathbf{P}}$-probability one, $\sup_{\ell}\left\lVert\gamma^{n_{i_{\ell}}}\right\rVert_{\text{tv}}<\infty$ and $\gamma^{n_{i_{\ell}}}\overset{\mathcal{E}}{\sim}\eta$. In particular, $\int\left(G_{\alpha}\,\eta\right)\,d\bar{\gamma}^{n}\rightarrow\int\left(G_{\alpha}\,\eta\right)\,d\eta$. Thus by (2.5),

[TABLE]

We have $\tilde{\mathbf{E}}\left[\int\left(G_{\alpha}\,\eta\right)\,d\gamma^{n}\right]=\int\left(G_{\alpha}\,\eta\right)\,d\bar{\gamma}^{n}$. Using that and (5.3),

[TABLE]

Let $\varepsilon>0$ be given. Let $\psi_{\varepsilon}$ be the measure defined in the proof of Lemma 5.2, so that (5.6) holds. Let $A_{\epsilon},c_{\varepsilon}$ be the quantities $A,c$ appearing in (5.6). Let $e_{\epsilon}(n)=\tfrac{c_{\varepsilon}\chi_{n}}{\alpha}+c_{\varepsilon}\int_{A_{\varepsilon}}\left\lvert G_{\alpha}\,\gamma^{n}-G_{\alpha}\,\eta\right\rvert\,dm+\varepsilon$. Then $G_{\alpha}\,\breve{\gamma}^{n}_{j}+e_{\varepsilon}\left(n_{i_{\ell}}\right)+G_{\alpha}\,\psi_{\varepsilon}\geq G_{\alpha}\,\eta$, $\mathcal{E}$-q.e., so

[TABLE]

Since $\limsup_{\ell\rightarrow\infty}e_{\varepsilon}\left(n_{i_{\ell}}\right)\leq\varepsilon$, and $\int\left(G_{\alpha}\,\psi_{\varepsilon}\right)\,d\bar{\gamma}^{n}\rightarrow\int\left(G_{\alpha}\,\psi_{\varepsilon}\right)\,d\eta$, (5.5) gives

[TABLE]

Since $\varepsilon>0$ is arbitrary, $\lim_{\ell\rightarrow\infty}\tilde{\mathbf{E}}\left[\sum_{j}\int\left(G_{\alpha}\,\breve{\gamma}^{n_{i_{\ell}}}_{j}-G_{\alpha}\,\eta\right)^{+}\,d\gamma^{n_{i_{\ell}}}_{j}\right]=0$. That is, $\lim_{\ell\rightarrow\infty}\tilde{\mathbf{E}}\left[\int\left(G_{\alpha}\,\gamma^{n_{i_{\ell}}}-1-G_{\alpha}\,\eta\right)^{+}\,d\gamma^{n_{i_{\ell}}}\right]=0$. By Borel-Cantelli, we can refine the subsequence $n_{i_{\ell}}$ to ensure that $\int\left(G_{\alpha}\,\gamma^{n_{i_{\ell}}}-1-G_{\alpha}\,\eta\right)^{+}\,d\gamma^{n_{i_{\ell}}}\rightarrow 0$ holds with $\tilde{\mathbf{P}}$-probability one. From now on we deal with an environment $\tilde{\omega}$ such that $\int\left(G_{\alpha}\,\gamma^{n_{i_{\ell}}}-1-G_{\alpha}\,\eta\right)^{+}\,d\gamma^{n_{i_{\ell}}}\rightarrow 0$. We will show that $\eta$ is $\alpha$-bounded from below for $\Lambda\left(n_{i_{\ell}}\right)$.

Let $H_{\ell}=\left\{\left(G_{\alpha}\,\gamma^{n_{i_{\ell}}}-1-G_{\alpha}\,\eta\right)^{+}\leq\delta_{\ell}\right\}$, where $\delta_{\ell}\rightarrow 0$ is chosen so that $\gamma^{n_{i_{\ell}}}\left(H_{\ell}^{c}\right)\rightarrow 0$. Let $\mu_{\ell}^{\prime}=\mathbf{1}_{H_{\ell}}\gamma^{n_{i_{\ell}}}$, so that we have $G_{\alpha}\,\mu_{\ell}^{\prime}\leq 1+G_{\alpha}\,\eta+\delta_{\ell}$, $\mu_{\ell}^{\prime}$-a.e. By the domination principle, $G_{\alpha}\,\mu_{\ell}^{\prime}\leq 1+G_{\alpha}\,\eta+\delta_{\ell}$ holds $\mathcal{E}$-q.e. on $E$. Since $\gamma^{n_{i_{\ell}}}\overset{\mathcal{E}}{\sim}\eta$ and $\left\lVert\gamma^{n_{i_{\ell}}}-\mu_{\ell}^{\prime}\right\rVert_{\text{tv}}\rightarrow 0$, it is easy to check from the definition that $\mu_{\ell}^{\prime}\overset{\mathcal{E}}{\sim}\eta$.

Let $\tau_{n}=D_{\Lambda\left(n\right)}$. Let $\pi$ be any smooth probability measure. Since

[TABLE]

(4.2) holds, with $\mu_{n}$ in that equation replaced by $\mu_{\ell}^{\prime}$. Thus with $\tilde{\mathbf{P}}$-probability one, $\eta$ is $\alpha$-bounded from below for the sequence $\Lambda\left(n_{i_{\ell}}\right)$. By Lemma 5.2, we already know that $\eta$ is a $\alpha$-bounded from above for $\Lambda\left(n_{i_{\ell}}\right)(\tilde{\omega})$ with $\tilde{\mathbf{P}}$-probability one, so Theorem 4.2 applies.

6 Dirichlet problems

Let $X$ be a right Markov process with state space $E$, cemetery point $\partial$ and lifetime $\zeta$. As in Section 1, let $U$ be an open subset of $E$ and $\sigma=D_{E_{\partial}-U}$, the entrance time of $E_{\partial}-U$. Let $\alpha\in[0,\infty)$, let $\varphi$ a $\mathcal{B}$-measurable function on $E$ and $f$ a $\mathcal{B}$-measurable function on $U$. Let $\Lambda\left(n\right)$ be a closed subset of $E$ for $n=1,2,\ldots$. If $\alpha=0$ and $\varphi\neq 0$, assume that $\Lambda\left(n\right)$ is such that $\mathbf{P}_{\pi}\left(\sigma=\infty\right)=0$. Let $\pi$ be a probability measure on $E$ such that $\mathbf{E}_{\pi}\left[\int_{0}^{\sigma}e^{-\alpha t}\left\lvert f\right\rvert\left(X_{t}\right)\,dt\right]<\infty$ and $\mathbf{E}_{\pi}\left[e^{-\alpha\tau_{n}\wedge\sigma}\left\lvert\varphi\right\rvert\circ X_{\tau_{n}\wedge\sigma}\right]<\infty$ for each $n$. Let $u_{n}$ be the probabilistic solution of the Dirichlet problem on $U-\Lambda\left(n\right)$, given by (1.2) for $\pi$-a.e. $x$, where $\tau_{n}=D_{\Lambda\left(n\right)}$. The following lemma gives conditions under which stable convergence of stopping times implies convergence for the corresponding solutions of the Dirichlet problem. Similar facts were proved in [4] for the Brownian motion case.

Lemma 6.1

Suppose that there exists a randomized stopping time $\tau$ such that $\tau$ has a rate measure and $\tau_{n}\wedge\zeta\rightarrow\tau\wedge\zeta$, $\mathbf{P}_{\pi}$-stably. Let $u$ be defined by

[TABLE]

Assume that $\mathbf{E}_{\pi}\left[\sup_{0\leq t<\sigma}e^{-\alpha t}\left\lvert\varphi\right\rvert\circ X_{t}\right]<\infty$, and $\mathbf{P}_{\pi}\left(t\mapsto\varphi\circ X_{t}\text{ is continuous on }[0,\sigma)\right)=1$. If $\varphi$ is nonzero and $\zeta$ is not identically equal to $\infty$, assume that the cemetery point $\partial$ is not a limit of points in $\bigcup_{n}\Lambda\left(n\right)$. Then $u_{n}$ converges weakly to $u$, in the sense that $\int u_{n}g\,d\pi\rightarrow\int ug\,d\pi$ for any $g\in L^{2}\left(\pi\right)$.

If the Markov process $X$ and $\pi$ are such that $\delta_{x}\,R_{\alpha}<<\pi$ for $\pi$-a.e. $x$ then $u_{n}\rightarrow u$ in $\pi$-measure.

Proof   Let $Y_{t}=g\left(X_{0}\right)\int_{0}^{t}e^{-\alpha r}f\left(X_{r}\right)\,dr$, $Z_{t}=g\left(X_{0}\right)e^{-\alpha t}\varphi\left(X_{t\wedge\sigma}\right)$, so that $\int Y_{\tau_{n}\wedge\sigma}\,d\mathbf{P}_{\mu}\,d\lambda^{\ast}+\int Z_{\tau_{n}}\,d\mathbf{P}_{\mu}\,d\lambda^{\ast}=\int u_{n}g\,d\pi$. By Lemma 3.1(i), we have immediately that $\int Y_{\tau_{n}\wedge\sigma}\,d\mathbf{P}_{\mu}\,d\lambda^{\ast}\rightarrow\int Y_{\tau\wedge\sigma}\,d\mathbf{P}_{\mu}\,d\lambda^{\ast}$. Suppose $\varphi\neq 0$ and $\zeta$ is not identically equal to $\infty$. Let $\tau^{\ast}$ be any $\mathbf{P}_{\pi}$-stable limit point. Let $F$ be the closure of $\bigcup_{n}\Lambda\left(n\right)$. By Lemma 4.5 with $B_{F}=\Omega$, $\mathbf{P}\times\lambda^{\ast}\left(\left\{X_{\tau^{\ast}}\notin F,\tau^{\ast}<\infty\right\}\right)=0$. Thus $\mathbf{P}\times\lambda^{\ast}\left(\zeta\leq\tau^{\ast}<\infty\right)=0$. Also, since $\tau$ has a rate measure, $\mathbf{P}_{\pi}\times\lambda^{\ast}\left(\zeta\leq\tau<\infty\right)=0$. Hence by Lemma 3.3, $\tau_{n}\rightarrow\tau$, $\mathbf{P}_{\pi}$-stably. It is easy to check that $t\mapsto Z_{t}$ has the continuity required by Lemma 3.1(i) at all times except for $t=\sigma<\infty$ when $\alpha>0$, and at all times except for $t=\sigma$ when $\alpha=0$. Since $\tau$ has a rate measure it follows $\mathbf{P}_{\pi}\left(\tau=\sigma<\infty\right)=0$, and if $\alpha=0$ it is assumed that $\mathbf{P}_{\pi}\left(\sigma=\infty\right)=0$. Thus Lemma 3.1(i) applies, so $\int Z_{\tau_{n}}\,d\mathbf{P}_{\mu}\,d\lambda^{\ast}\rightarrow\int Z_{\tau}\,d\mathbf{P}_{\mu}\,d\lambda^{\ast}$. The same conclusion holds more easily when $\zeta=\infty$. This proves the weak convergence.

If $\delta_{x}\,R_{\alpha}<<\pi$, then by Lemma 3.4 there exists a subsequence $n_{k}$ such that $\tau_{n_{k}}\wedge\zeta\rightarrow\tau\wedge\zeta$, $\mathbf{P}_{x}$-stably, for $\pi$-a.e. $x$. By what has already been proved, $u_{n_{k}}(x)\rightarrow u(x)$ for $\pi$-a.e. $x$. Since $n_{k}$ could be chosen as a subsequence from any other subsequence, it follows that $u_{n}\rightarrow u$ in $\pi$-measure.

The conclusion of Lemma 6.1 holds for any limit $\tau$ such that $S^{\tau}_{t}$ is a multiplicative functional, with the same proof. We note that the lemma requires no smoothness or continuity for $\varphi$ on $E-U$. Also, when $\varphi=0$ the proof of weak convergence is valid for any limit $\tau$.

Lemma 6.1 deals with the convergence of $u_{n}$ to $u$ when $u_{n}$ and $u$ are defined probabilistically on an open subset $U$ of $E$, for a general Markov process $X$. Suppose now that $X$ is properly associated with a Dirichlet form $\mathcal{E}$. Let $\varphi$ be an $\mathcal{E}$-quasi-continuous function in $D\left(\mathcal{E}\right)$. The next lemma gives analytical characterizations of $u_{n}$ and $u$ in terms of $\mathcal{E}$, for $\alpha>0$ and $f\in L^{2}\left(m\right)$. It follows that the probabilistic solutions for the Dirichlet problem agree with the analytical solutions which were studied in [8], [12], [9], [23]. The case of $\alpha=0$ can be dealt with similarly in situations where an inequality of Poincaré type holds.

Lemma 6.2

Let $\tau_{n}=D_{\Lambda\left(n\right)}$. Let $\alpha>0$, $f\in L^{2}\left(m\right)$, and let $\varphi$ be an $\mathcal{E}$-quasi-continuous function in $D\left(\mathcal{E}\right)$. If $\varphi$ is nonzero, assume for each $n$ that the cemetery point $\partial$ is not both a limit of points in $U-\Lambda\left(n\right)$ and a limit of points in $\left(E-U\right)\cup\Lambda\left(n\right)$. Let $u_{n}$ be defined by (1.2), for all $x\in E$ such that the expected values exist. Then $u_{n}$ is defined $\mathcal{E}$-q.e.

For $n=1,2,\ldots$ let $V_{n}$ denote the set of all $v\in D\left(\mathcal{E}\right)$ such that $v$ is $\mathcal{E}$-quasi-continuous and $v=0$ holds $\mathcal{E}$-q.e. on $U^{c}\cup\Lambda\left(n\right)$. Then $u_{n}$ is the $\mathcal{E}$-q.e. unique element in $D\left(\mathcal{E}\right)$ such that $u_{n}$ is $\mathcal{E}$-quasi-continuous, $u_{n}=\varphi$ holds $\mathcal{E}$-q.e. on $U^{c}\cup\Lambda\left(n\right)$ and

[TABLE]

for all $v\in V_{n}$. Furthermore $\left\lVert u_{n}\right\rVert_{\mathcal{E},\alpha}$ is bounded in $n$.

Let $\eta$ be a measure with finite energy, and let $\tau$ be the randomized stopping time with rate measure $\eta$. Let $V$ be the set of $\mathcal{E}$-quasi-continuous elements $v$ in $D\left(\mathcal{E}\right)\cap L^{2}\left(\eta\right)$ such that $v=0$ holds $\mathcal{E}$-q.e. on $U^{c}$. Assume that $\int_{U}\varphi^{2}\,d\eta<\infty$ and $\varphi\mathbf{1}_{U}\eta$ has finite energy. Let $u$ be defined by (6.1), for all $x\in E$ such that the expected values exist. Then $u$ is defined $\mathcal{E}$-q.e. and $u$ is the $\mathcal{E}$-q.e. unique element in $D\left(\mathcal{E}\right)$ with $\int_{U}u^{2}\,d\eta<\infty$ such that $u$ is $\mathcal{E}$-quasi-continuous, $u=\varphi$ holds $\mathcal{E}$-q.e. on $U^{c}$ and

[TABLE]

for all $v\in V$. The space $V$ can be replaced by the space $V^{\prime}$ consisting of all functions in $V$ which vanish $\mathcal{E}$-q.e. on the complement of a compact subset of $U$, or in the regular case by the space $V^{\prime\prime}$ of functions in $V\cap\mathcal{C}_{0}\left(E\right)$ with compact support in $U$.

Now assume that the absolute continuity condition (2.4) holds, and that for some probability measure $\pi$ with $m<<\pi$ on $U$, $\tau_{n}\wedge\zeta\rightarrow\tau\wedge\zeta$, $\mathbf{P}_{\pi}$-stably. If $\varphi$ is nonzero assume that $m(U)<\infty$. Then $\left\lVert u_{n}-u\right\rVert_{L^{2}\left(m\right)}\rightarrow 0$ and $u_{n}\rightarrow u$, $\mathcal{E}$-weakly.

Proof   We may assume $\varphi\geq 0$. By equation (9.4), $\mathbf{E}_{z}\left[e^{-\alpha\tau_{n}\wedge\sigma}\varphi\left(X_{\tau_{n}\wedge\sigma}\right)\right]={\varphi}^{\prime}_{U^{c}\cup\Lambda\left(n\right),\alpha}$ for $\mathcal{E}$-q.e. $z$. Also $R_{\alpha}\,\left\lvert f\right\rvert$ is $\mathcal{E}$-quasi-continuous, and hence finite $\mathcal{E}$-q.e.

Let $w_{n}=u_{n}-{\varphi}^{\prime}_{U^{c}\cup\Lambda\left(n\right),\alpha}$. By equation (1.2), $w_{n}=\mathbf{E}_{x}\left[\int_{0}^{\tau_{n}\wedge\sigma}e^{-\alpha t}f\left(X_{t}\right)\,dt\right]$. Also by equation (9.4) and the strong Markov property, $w_{n}=G_{\alpha}\,f-{\left(G_{\alpha}\,f\right)}^{\prime}_{U^{c}\cup\Lambda\left(n\right),\alpha}$. It follows that $u_{n}\in D\left(\mathcal{E}\right)$. By definition $u_{n}=\varphi$ on $U^{c}\cup\Lambda\left(n\right)$, and by equation (9.3), $\mathcal{E}_{\alpha}\left(u_{n},v\right)=\mathcal{E}_{\alpha}\left(G_{\alpha}\,f,v\right)=\int fv\,dm$ for any $v\in V$. This proves equation (6.2). If $u_{n},u_{n}^{\prime}$ are solution of equation (6.2) for the same $\varphi$ then $u_{n}-u_{n}^{\prime}\in V$, and hence $\left\lVert u_{n}-u_{n}^{\prime}\right\rVert_{\mathcal{E},\alpha}=0$ by equation (6.2), so the solution is unique.

Since $\left\lVert{\left(G_{\alpha}\,f\right)}^{\prime}_{U^{c}\cup\Lambda\left(n\right),\alpha}\right\rVert_{\mathcal{E},\alpha}\leq K_{\alpha}\left\lVert G_{\alpha}\,f\right\rVert_{\mathcal{E},\alpha}$ and $\left\lVert{\varphi}^{\prime}_{U^{c}\cup\Lambda\left(n\right),\alpha}\right\rVert_{\mathcal{E},\alpha}\leq K_{\alpha}\left\lVert\varphi\right\rVert_{\mathcal{E},\alpha}$,

[TABLE]

for all $n$.

In considering (6.1), since replacing $\eta$ by $\mathbf{1}_{U}\eta$ leaves $\tau\wedge\sigma$ unchanged, we may assume without loss of generality that $\eta(U^{c})=0$, i.e. $\eta=\mathbf{1}_{U}\eta$. Let $\left(A_{t}\right)$ be the positive continuous additive functional with Revuz measure $\eta$ and let $\Phi^{\tau}$ be the random measure on $[0,\infty]$ such that $\Phi^{\tau}{\left((t,\infty]\right)}=e^{-A_{t}}$. Then for each $t$, $\int e^{-\alpha t}\mathbf{1}_{\left\{\tau>t\right\}}f\left(X_{t}\right)\,d\mathbf{P}_{x}\,d\lambda^{\ast}=\mathbf{E}_{x}\left[e^{-\alpha t}e^{-A_{t}}f\left(X_{t}\right)\right]$ by equation (3.1) with $Y_{t}=e^{-\alpha t}f\left(X_{t}\right)$. Also by taking $Y_{t}=e^{-\alpha(t\wedge\sigma)}\varphi\left(X_{t\wedge\sigma}\right)$ in equation (3.1) we have $\int e^{-\alpha\tau\wedge\sigma}\varphi\left(X_{\tau\wedge\sigma}\right)\,d\mathbf{P}_{x}\,d\lambda^{\ast}=\mathbf{E}_{x}\left[\int_{0}^{\sigma}e^{-\alpha t}e^{-A_{t}}\varphi\left(X_{t}\right)\,dA_{t}+e^{-\alpha\sigma}e^{-A_{\sigma}}\varphi\left(X_{\sigma}\right)\right]$. Then equation (6.1) implies

[TABLE]

Let $w(x)=\mathbf{E}_{x}\left[\int_{0}^{\sigma}e^{-\alpha t-A_{t}}f\left(X_{t}\right)\,dt\right]$, $g(x)=\mathbf{E}_{x}\left[\int_{0}^{\sigma}e^{-\alpha t-A_{t}}\varphi\left(X_{t}\right)\,dA_{t}\right]$, $h(x)=\mathbf{E}_{x}\left[e^{-\alpha\sigma-A_{\sigma}}\varphi\left(X_{\sigma}\right)\right]$, so that $u=w+g+h$. Clearly $u=\varphi$ holds on $\Lambda$.

We will show that each of the functions $w,g,h$ satisfies an equation similar to equation (6.3). First we will deal with $g$. Since $\varphi\eta$ has finite energy, $g\leq G_{\alpha}\,\left(\varphi\eta\right)$ by equation (9.4). It is easy to check that if $\xi$ is a bounded function then $\xi\eta$ has finite energy. Let $H_{j}=\left\{\varphi\leq j,\ G_{\alpha}\,\left(\varphi\eta\right)\leq j\right\}$, for $j=1,2,\ldots$. Let $\varphi_{j}=\mathbf{1}_{H_{j}}\varphi$. By the domination principle, $G_{\alpha}\,\left(\varphi_{j}\eta\right)\leq j$ holds $\mathcal{E}$-q.e. Let $g_{j}=\mathbf{E}_{x}\left[\int_{0}^{\sigma}e^{-\alpha t-A_{t}}\varphi_{j}\left(X_{t}\right)\,dA_{t}\right]$. By equation (9.1) we know that $g_{j}\leq G_{\alpha}\,\left(\varphi_{j}\eta\right)$, and hence $g_{j}(x)\leq j$ for $\mathcal{E}$-q.e. $x$. Also $g_{j}\nearrow g$, $\mathcal{E}$-q.e., by III.3.5 of [22].

Similarly to equation (4.1.7) in [10], one can show that $\mathbf{E}_{x}\left[\int_{0}^{\sigma}e^{-\alpha t}\varphi_{j}\left(X_{t}\right)\,dA_{t}\right]=g_{j}+\mathbf{E}_{x}\left[\int_{0}^{\sigma}e^{-\alpha t}g_{j}\left(X_{t}\right)\,dA_{t}\right]$ for $\mathcal{E}$-q.e. $x$.

Using equation (9.4), (9.1) and the strong Markov property, we then have $G_{\alpha}\,\left(\varphi_{j}\eta\right)-{\left(G_{\alpha}\,\left(\varphi_{j}\eta\right)\right)}^{\prime}_{\Lambda,\alpha}=g_{j}+G_{\alpha}\,\left(g_{j}\eta\right)-{\left(G_{\alpha}\,\left(g_{j}\eta\right)\right)}^{\prime}_{\Lambda,\alpha}$. Hence $g_{j}=G_{\alpha}\,\left(\varphi_{j}\eta\right)-{\left(G_{\alpha}\,\left(\varphi_{j}\eta\right)\right)}^{\prime}_{\Lambda,\alpha}-G_{\alpha}\,\left(g_{j}\eta\right)+{\left(G_{\alpha}\,\left(g_{j}\eta\right)\right)}^{\prime}_{\Lambda,\alpha}$.

Let $v\in V$. Then $\mathcal{E}_{\alpha}\left(G_{\alpha}\,\left(g_{j}\eta\right),v\right)=\int g_{j}v\,d\eta$, $\mathcal{E}_{\alpha}\left(G_{\alpha}\,(\varphi_{j}\eta),v\right)=\int\varphi_{j}v\,d\eta$, and $\mathcal{E}_{\alpha}\left({v_{1}}^{\prime}_{\Lambda,\alpha},v\right)=0$ for any $v_{1}\in D\left(\mathcal{E}\right)$. It follows that $\mathcal{E}_{\alpha}\left(g_{j},v\right)=\int\varphi_{j}v\,d\eta-\int g_{j}v\,d\eta$, so $\mathcal{E}_{\alpha}\left(g_{j},v\right)+\int g_{j}v\,d\eta=\int\varphi_{j}v\,d\eta$. Since $g_{j}\in V$, we have $\mathcal{E}_{\alpha}\left(g_{j},g_{j}\right)+\int g_{j}^{2}\,d\eta\leq\int\varphi_{j}g_{j}\,d\eta\leq\varphi g\,d\eta\leq\int\left(G_{\alpha}\,\left(\varphi\eta\right)\right)\varphi\,d\eta=\mathcal{E}_{\alpha}\left(G_{\alpha}\,\left(\varphi\eta\right),G_{\alpha}\,\left(\varphi\eta\right)\right)<\infty$ for all $j$. Letting $j\nearrow\infty$, using this equation and I.2.12 in [22], it follows that $g\in D\left(\mathcal{E}\right)\cap L^{2}\left(\eta\right)$ and $\mathcal{E}_{\alpha}\left(g,v\right)+\int gv\,d\eta=\int\varphi v\,d\eta$.

The same arguments show that $w\in D\left(\mathcal{E}\right)\cap L^{2}\left(\eta\right)$ and $\mathcal{E}_{\alpha}\left(w,v\right)+\int wv\,d\eta=\int fv\,dm$, for all $v\in V$.

Let $h_{j}(x)=\mathbf{E}_{x}\left[e^{-\alpha\sigma-A_{\sigma}}\varphi\wedge j\left(X_{\sigma}\right)\right]$. One can show that $\mathbf{E}_{x}\left[e^{-\alpha\sigma}\varphi\wedge j\left(X_{\sigma}\right)\right]=h_{j}+\mathbf{E}_{x}\left[\int_{0}^{\sigma}e^{-\alpha t}h_{j}\left(X_{t}\right)\,dA_{t}\right]$, which says that ${\left(\varphi\wedge j\right)}^{\prime}_{\Lambda,\alpha}=h_{j}+G_{\alpha}\,(h_{j}\eta)-{\left(G_{\alpha}\,{(h_{j}\eta)}\right)}^{\prime}_{\Lambda,\alpha}$. Thus $\mathcal{E}_{\alpha}\left(h_{j},\right){v}+\int h_{j}v\,d\eta=0$ for any $v\in V$. Clearly $\left\lVert\varphi\wedge j\right\rVert_{\mathcal{E},\alpha}\leq\left\lVert\varphi\right\rVert_{\mathcal{E},\alpha}$, and since $h_{j}\leq h$ we have $\left\lVert G_{\alpha}\,\left(h_{j}\eta\right)\right\rVert_{\mathcal{E},\alpha}\leq\left\lVert G_{\alpha}\,\left(h\eta\right)\right\rVert_{\mathcal{E},\alpha}$. Thus $\left\lVert h_{j}\right\rVert_{\mathcal{E},\alpha}$ is bounded in $j$. Hence by I.1.12 of [22], $h\in D\left(\mathcal{E}\right)$ and $\mathcal{E}_{\alpha}\left(h_{j},v\right)\rightarrow\mathcal{E}_{\alpha}\left(h,v\right)$. Since $\varphi\wedge j-h_{j}\in V$, $\mathcal{E}_{\alpha}\left(h_{j},\varphi\wedge j-h_{j}\right)+\int h_{j}\left(\varphi\wedge j-h_{j}\right)\,d\eta=0$, and hence $\int h_{j}^{2}\,d\eta\leq c_{j}+\left(\int h_{j}^{2}\,d\eta\right)^{1/2}\left(\int\varphi^{2}\,d\eta\right)^{1/2}$, where $c_{j}$ is bounded. It follows that $\int h_{j}^{2}\,d\eta$ is bounded, and hence that $\int h^{2}\,d\eta<\infty$. Hence $\mathcal{E}_{\alpha}\left(h,v\right)+\int hv\,d\eta=0$.

Adding the equations for $w,g,h$ gives equation (6.3). If $u,u^{\prime}$ are solutions of equation (6.3) for the same $\varphi$ then for any $v\in V$, $\mathcal{E}_{\alpha}\left(u-u^{\prime},v\right)+\int\left(u-u^{\prime}\right)v\,d\eta=0$ by equation (6.3). Since $u-u^{\prime}\in V$, $u=u^{\prime}$, and the solution is unique.

Using I.2.12 in [22], it is straightforward to show that $V^{\prime}$ is dense in $V$ with respect to $\left\lVert\cdot\right\rVert_{\mathcal{E},\alpha}$-norm, and that when $\mathcal{E}$ is regular, $V^{\prime\prime}$ is dense.

Since $\varphi$ is $\mathcal{E}$-quasi-continuous, $t\mapsto\varphi\circ X_{t}$ is continuous with $\mathbf{P}_{\pi}$-probability one on $[0,\sigma)$ for every smooth $\pi$. Assume $\pi<<m$ and that $\tau_{n}\wedge\zeta\rightarrow\tau\wedge\zeta$, $\mathbf{P}_{\pi}$-stably. If $\varphi$ is nonzero assume that $m(U)<\infty$. In proving the final statement of the lemma we may assume using (6.4) and IV.4.17 of [22] that $\varphi$ is bounded. By Lemma 6.1, $u_{n}\rightarrow u$ in $m$-measure. By (1.2), $u_{n}(x)\leq G_{\alpha}\,\left\lvert f\right\rvert+\text{ a constant}$, so $\left\lVert u_{n}-u\right\rVert_{L^{2}\left(m\right)}\rightarrow 0$ by dominated convergence. Hence also $u_{n}\rightarrow u$, $\mathcal{E}$-weakly, by I.2.12 in [22].

Comparing (6.2) with (6.3), we see that the Dirichlet boundary condition on $\Lambda\left(n\right)$ has been replaced by a penalty term associated with the measure $\eta$, together with an additional source term if $\varphi$ does not vanish on $\Lambda\left(n\right)$. For any measure $\eta$, not necessarily finite, which does not charge $\mathcal{E}$-exceptional sets, equation (6.3) with $\varphi=0$ is said to describe the relaxed Dirichlet problem for $u$ with zero boundary conditions, having penalty measure $\eta$. The function $u_{n}$ defined by (6.2) with $\varphi=0$ can also be regarded as the solution of a relaxed Dirichlet problem, namely

[TABLE]

where the measure $\eta_{n}$ is infinite on all subsets of $\Lambda\left(n\right)$ which have positive capacity, and is zero otherwise. This is the setting in which the convergence of $u_{n}$ to $u$ has been studied as a special case of convergence for solutions of relaxed Dirichlet problems, in the analytical papers cited in Section 2.4. General relaxed Dirichlet problems are not considered in the present paper, although a probabilistic representation of the solution of a general relaxed Dirichlet problem has been given in [17].

7 Transformations

7.1 Using Girsanov’s Theorem

In order to prove stable convergence of stopping times, it may be possible to use a Girsanov transformation (cf. 7.6.4 in [21]) to reduce the problem to the case of a simpler process. Let $X,Y$ be processes which can be defined on the same sample space $\Omega$. Let $\mathbf{P}$ and $\mathbf{Q}$ be probabilities on $\Omega$ for $X$ and $Y$ respectively.

Let $\mathcal{G}_{t}$ be $\sigma\left(X_{s},\,s\leq t\right)$, $\mathcal{G}=\sigma\left(X_{s},\ s\in[0,\infty)\right)$. Let $\tau_{n}$ be a sequence of $\left(\mathcal{G}_{t}\right)$-stopping times, and let $\tau$ be a randomized stopping time. Assume that it has been shown, by any method, that $\tau_{n}\rightarrow\tau$, $\mathbf{Q},\mathcal{G}$-stably. If $\mathbf{P}$ can be obtained by a Girsanov transformation from $\mathbf{Q}$, then $\mathbf{P}<<\mathbf{Q}$ on $\mathcal{G}_{t}$ for each $t$. In this case, since $\bigcup_{t}\mathcal{G}_{t}$ is $\mathbf{P}$-dense in $\mathcal{G}$, it follows that $\tau_{n}\rightarrow\tau$, $\mathbf{P},\mathcal{G}$-stably.

7.2 Time changes

A time change can be used to change the measure $m$ for the $L^{2}$-space containing $D\left(\mathcal{E}\right)$.

Lemma 7.1

Assume that $\mathcal{E}$ is regular and symmetric. Let $D_{e}\left(\mathcal{E}\right)$ denote the extended Dirichlet space associated with $\mathcal{E}$, and let $\mathcal{E}^{e}$ be the extension of $\mathcal{E}$ to $D_{e}\left(\mathcal{E}\right)$, as defined after Definition 1.1.4 and Theorem 1.1.5 of [10]. Let $b$ be a locally bounded function in $\mathcal{B}^{+}$ with $b>0$ everywhere, and define the positive continuous additive functional $B$ by $B_{t}=\int_{0}^{t}b\left(X_{s}\right)\,ds$. Define the partial inverse $\kappa_{t}$ on $[0,\infty]$ by $\kappa_{t}=\inf\left\{s:B_{s}>t\right\}$ for $t<B_{\zeta-}$ and $\kappa_{t}=\infty$ for $t\geq B_{\zeta-}$. Let $\check{X}_{t}=X_{\kappa_{t}}$. Then $\check{X}$ is a Markov process on $\Omega$ with lifetime $\check{\zeta}=B_{\zeta-}$, which is properly associated with the Dirichlet form $\left(\check{\mathcal{E}},D\left(\check{\mathcal{E}}\right)\right)$, where $D\left(\check{\mathcal{E}}\right)=D_{e}\left(\mathcal{E}\right)\cap L^{2}\left(E,bm\right)$ and $\check{\mathcal{E}}\left(f,g\right)=\mathcal{E}^{e}\left(f,g\right)$ for all $f,g$ in $D\left(\check{\mathcal{E}}\right)$. A subset of $E$ is $\check{\mathcal{E}}$-exceptional if and only if it is $\mathcal{E}$-exceptional.

Let $\Lambda\left(n\right)$ be a sequence of Borel sets. Let $\tau_{n}$ be the entrance or hitting time of $\Lambda\left(n\right)$ by $X$, and correspondingly let $\check{\tau}_{n}$ be the entrance or hitting time of $\Lambda\left(n\right)$ by $\check{X}$. Then there is a $\mathcal{E}$-exceptional set $N$ such that if $x\notin N$, then with $\mathbf{P}_{x}$-probability one we have $\check{\tau}_{n}=B_{\tau_{n}}$ if $\tau_{n}<\zeta$, $\check{\tau}_{n}=\infty$ otherwise. Let $\nu$ be a probability measure such that $\nu(N)=0$. If $\tau_{n}\wedge\zeta$ converges $\mathbf{P}_{\nu}$-stably to $\tau\wedge\zeta$, for some randomized stopping time $\tau$, let $\check{\tau}=B_{\tau}$ for $\tau<\zeta$, $\check{\tau}=\infty$ otherwise. Then $\check{\tau}_{n}\wedge\check{\zeta}$ converges $\mathbf{P}_{\nu}$-stably to $\check{\tau}\wedge\check{\zeta}$. If $S^{\tau}_{t}=e^{-A_{t}}$ for some positive continuous additive functional $A$, then $S^{\check{\tau}}_{t}=e^{-\check{A}_{t}}$, where $\check{A}$ is the positive continuous additive functional defined by $\check{A}_{t}=A_{\kappa_{t}-}$. The Revuz measure $\mu_{\check{A}}$ for $\check{A}$ with respect to $\check{X}$ is equal to the Revuz measure $\mu_{A}$ for $A$ with respect to $X$.

Proof   Theorem 5.2.2 of [10] shows that $\check{X}$ is the Markov process associated with $\check{\mathcal{E}}$. By Theorem 5.2.8 of [10], a subset of $E$ is $\check{\mathcal{E}}$-exceptional if and only if it is $\mathcal{E}$-exceptional. It follows from the definitions that $\check{\tau}_{n}=B_{\tau_{n}}$ if $\tau_{n}<\zeta$, $\check{\tau}_{n}=\infty$ otherwise. Let $C_{t}=B_{t-}$ for $t\in[0,\infty]$. On a defining set for $B$, $C_{t}=B_{t}$ for $t<\infty$. $\check{\tau_{n}}\wedge\check{\zeta}=C_{\tau_{n}\wedge\zeta}$, $\check{\tau}\wedge\check{\zeta}=C_{\tau\wedge\zeta}$, and $t\mapsto C_{t}(\omega)$ is continuous on $[0,\infty]$ for $\omega$ in a defining set for $B$. Lemma 3.1 (i) implies that $\check{\tau}_{n}\wedge\check{\zeta}$ converges $\mathbf{P}_{\nu}$-stably to $\check{\tau}\wedge\check{\zeta}$.

It is easy to check that $S^{\sigma}_{t}(\omega)=1-\sup\left(\left\{r:\ r\in(0,1),\ \sigma(\omega,r)\leq t\right\}\cup\left\{0\right\}\right)$, for any randomized stopping time $\sigma$. Applying this formula to $\sigma=\check{\tau}$ shows that $S^{\check{\tau}}_{t}=e^{-\check{A}_{t}}$. By Lemma 6.2.8 of [16], $\mu_{\check{A}}=\mu_{A}$.

Lemma 7.1 is applied in the proofs of Lemmas 8.2 and 8.3, to reduce the convergence argument to the case in which $m$ is Lebesgue measure $\lambda_{d}$.

7.3 Localization

Lemma 7.2

Let $X$ be a Hunt process. Let $U$ be an open subset of $E$ such that the cemetery point $\partial$ is not in the closure of $U$. Let $m^{U}$ be the restriction of $m$ to the measurable subsets of $U$. Let $\mathcal{V}$ be the set of $f\in D\left(\mathcal{E}\right)$ such that $f=0$ holds $\mathcal{E}$-q.e. on $U^{c}$. Let $\mathcal{V}^{U}$ be the set of functions $h^{U}$ on $U$, where $h^{U}$ is the restriction to $U$ of a function $h$ in $\mathcal{V}$. Let $\mathcal{E}^{U}$ be the form with domain $\mathcal{V}^{U}$ such that $\mathcal{E}^{U}\left(h^{U},g^{U}\right)=\mathcal{E}\left(h,g\right)$ for all $h,g\in\mathcal{V}$. Let $\zeta^{U}$ be the first exit time for $U$, i.e. $\zeta^{U}=D_{E_{\partial}-U}$. Let $X^{U}$ be the process defined by $X^{U}_{t}=X_{t}$ for $t<\zeta^{U}$, $X_{t}^{U}=\partial$ otherwise, with filtration $\left(\mathcal{F}_{t}^{U}\right)$ equal to the closure of the natural filtration associated with $X^{U}$. Then: $\left(\mathcal{E}^{U},D\left(\mathcal{E}^{U}\right)\right)$ is a quasi-regular Dirichlet form on $L^{2}\left(U,m^{U}\right)$, and $X^{U}$ is a special standard process with lifetime $\zeta^{U}$, which is properly associated with $\left(\mathcal{E}^{U},D\left(\mathcal{E}^{U}\right)\right)$. With an appropriate topology on $U\cup\left\{\partial\right\}$, $X^{U}$ is a Hunt process. Let $C$ be the set of functions $g^{U}$, where $g\in D\left(\mathcal{E}\right)$ is such that $g=0$ holds $\mathcal{E}$-q.e. on the complement of a compact subset of $U$. Then $C$ is dense in $D\left(\mathcal{E}^{U}\right)$. Any $\mathcal{E}$-exceptional subset of $U$ is $\mathcal{E}^{U}$-exceptional. If $G_{\alpha}^{U}$ denotes the potential operator for $\mathcal{E}^{U}$, then for any $f\in L^{2}\left(m\right)$ such that $f=0$ on $U^{c}$,

[TABLE]

and for any measure $\nu$ on $E$ with finite energy, such that $\nu=0$ on all subsets of $U^{c}$, if $\nu^{U}$ denotes the restriction of $\nu$ to subsets of $U$ then $\nu^{U}$ has finite $\mathcal{E}^{U}$-energy and

[TABLE]

Furthermore, if $\left(A_{t}\right)_{t\geq 0}$ is a positive continuous additive functional for $X$ with Revuz measure $\mu$, then $\left(A_{t\wedge\zeta^{U}}\right)$ is the positive continuous additive functional for $X^{U}$ with Revuz measure $\mu^{U}$, where $\mu^{U}$ is the restriction of $\mu$ to subsets of $U$.

The proof is omitted. It follows from the definitions and the properties given in [22] and Section 9. We will refer to the Dirichlet form $\left(\mathcal{E}^{U},D\left(\mathcal{E}^{U}\right)\right)$ described in this lemma as the restriction of $\left(\mathcal{E},D\left(\mathcal{E}\right)\right)$ to $U$. The process $X^{U}$ has the same sample space as $X$, with an appropriate change in the shift operator.

Combining localizations.

When stable convergence can be proved for a large enough class of localized versions of a process, global convergence can be obtained by combining restrictions, as in the following lemma.

Lemma 7.3

Let $X$ be a Hunt process on a separable metric space such that for $\mathcal{E}$-q.e. $x$, $t\mapsto X_{t}$ is continuous on $[0,\zeta)$ with $\mathbf{P}_{x}$-probability one. Let $U_{\ell}$, $\ell=1,2,\ldots$ be a locally finite open cover for $E$. Let $\zeta^{\ell}$ be the first exit time for $U_{\ell}$, i.e. $\zeta^{\ell}=D_{E_{\partial}-U_{\ell}}$. Let $X^{U_{\ell}}_{t}=X_{t}$ for $t<\zeta^{\ell}$, $X_{t}^{U_{\ell}}=\partial$ otherwise. Suppose that the absolute continuity condition (2.4) holds.

Let $\tau_{n}$ be a terminal time for each $n$. Let $\eta$ be a smooth measure on $E$. For each $\ell$, let $\eta^{\ell}$ be the restriction of $\eta$ to $U_{\ell}$. Suppose that for each $\ell$ and any smooth probability measure $\pi^{\ell}$ on $U^{\ell}$, $\tau_{n}\wedge\zeta^{\ell}$ converges $\mathbf{P}_{\pi^{\ell}}$-stably to $\tau^{\ell}\wedge\zeta^{\ell}$, where $\tau^{\ell}$ is the randomized stopping time which has rate measure $\eta^{\ell}$ for $X^{U_{\ell}}$.

Let $\tau$ be the randomized stopping time with rate measure $\eta$. Then for any smooth probability measure $\pi$ on $E$, $\tau_{n}\wedge\zeta\rightarrow\tau\wedge\zeta$, $\mathbf{P}_{\pi}$-stably.

The proof is omitted. The idea of the proof is the following. By piecing together the stopping times $\zeta^{\ell}$ one obtains a sequence of stopping times $\sigma_{k}$ such that $\sigma_{k}\nearrow\zeta$. Using Lemmas 3.1 and 3.4 one can show by induction that $\tau_{n}\wedge\sigma_{k}\rightarrow\tau\wedge\sigma_{k}$, $\mathbf{P}_{\pi}$-stably as $n\rightarrow\infty$, and Lemma 3.2 then gives the result.

8 Examples

Details for proofs of most of the statements in this section, and other examples, are given in [7].

Let $\mathcal{E}$ be regular. Let $\left(\tilde{\Omega}_{1},\tilde{\mathbf{P}}_{1}\right)$ be a probability space. For each $x\in E$ and $n=1,2,\ldots$, let $\Gamma^{x}_{n}$ be a map from $\tilde{\Omega}_{1}$ to the collection of closed subsets of $E$. Let $F(E)$ be the space of compact subsets of $E$ with Hausdorff metric. It is assumed that for any $K\in F(E)$, the map $\left(x,\tilde{\omega}_{1}\right)\mapsto\Gamma^{x}_{n}\left(\tilde{\omega}_{1}\right)\cap K$ is jointly measurable from $E\times\tilde{\Omega}_{1}$ into $F(E)$. The case that $\tilde{\Omega}_{1}$ is a one-point space, so that $\Gamma^{x}_{n}$ is nonrandom, is an important special case.

Let $\mu$ be a probability measure on $E\cup\left\{\partial\right\}$, where $\partial$ is the cemetery point. For each $n$, let $\xi_{1}(n,\mu),\ldots,\xi_{\kappa_{n}}(n,\mu)$ be iid random variables with distribution $\mu$, defined on some probability space $\left(\tilde{\Omega}_{2},\tilde{\mathbf{P}}_{\mu}\right)$, where $\lim_{n\rightarrow\infty}\kappa_{n}=\infty$. Let $\left(\tilde{\Omega},\tilde{\mathbf{P}}\right)=\left(\Omega_{1}\times\Omega_{2},\tilde{\mathbf{P}}_{1}\times\tilde{\mathbf{P}}_{\mu}\right)$, and let $\Lambda^{\mu}_{i}\left(n\right)$ be the random set $\Gamma^{\xi_{i}(n,\mu)}_{n}$, where we define $\Gamma^{\partial}_{n}=\emptyset$. The family $\Lambda^{\mu}_{i}\left(n\right)$ will be said to be the random center model associated with $\left(\Gamma^{x}_{n}\right),\mu$. The terminology is intended to suggest that $\Lambda^{\mu}_{i}\left(n\right)$ could be randomly chosen by first selecting the random “center” $x=\xi_{i}(n,\mu)$ and then choosing a possibly random set $\Gamma^{x}_{n}$ near $x$. Let $\Lambda^{\mu}\left(n\right)=\bigcup_{i}\Lambda^{\mu}_{i}\left(n\right)$. We are interested in random center models such that $D_{\Lambda^{\mu}\left(n\right)}$ and $T_{\Lambda^{\mu}\left(n\right)}$ converge $\mathbf{P}_{\pi}$-stably in $\tilde{\mathbf{P}}$-probability. Random center models were studied in [19], [28], [27], [1], [2].

Let $B_{r}(x)$ denote the open metric ball in $E$ with center $x$ and radius $r$. It is assumed that the sets $\Gamma^{x}_{n}$ become small, meaning that there exists a nonrandom sequence $\varrho_{n}\in(0,\infty)$ such that $\varrho_{n}\rightarrow 0$ and such that for $\mu$-a.e. $x\in E$, $\tilde{\mathbf{P}}_{1}\left(\Gamma^{x}_{n}\subset B_{\varrho_{n}}(x)\right)=1$.

Lemma 8.1

Let $\mu$ be a probability measure on $E\cup\left\{\partial\right\}$. Suppose that for each $n$ there is a constant $\chi_{n}\in[0,\infty)$ such that

[TABLE]

Let $B_{\ell}\in\mathcal{B}$ be nondecreasing and such that $\mu\left(E-B_{\ell}\right)\rightarrow 0$. Let $\mu_{\ell}$ be the probability measure on $E\cup\left\{\partial\right\}$ such that $\mu_{\ell}\left(K\right)=\mu\left(K\cap B_{\ell}\right)$ for every $K\in\mathcal{B}$. Let $\tau_{n}=D_{\Lambda^{\mu}\left(n\right)}$ or $\tau_{n}=T_{\Lambda^{\mu}\left(n\right)}$, and let $\tau_{n}^{\ell}=D_{\Lambda^{\mu_{\ell}}\left(n\right)}$ if $\tau_{n}=D_{\Lambda^{\mu}\left(n\right)}$, $\tau_{n}^{\ell}=T_{\Lambda^{\mu_{\ell}}\left(n\right)}$ if $\tau_{n}=T_{\Lambda^{\mu}\left(n\right)}$. Let $\pi$ be a smooth probability measure on $E$, such that for each $\ell$, $\tau_{n}^{\ell}$ converges $\mathbf{P}_{\pi}$-stably in $\tilde{\mathbf{P}}$-probability to a randomized stopping time $\tau^{\ell}$. Then $\tau^{\ell}$ decreases to a limit $\tau$, $\mathbf{P}_{\pi}$-a.e., and $\tau_{n}$ converges $\mathbf{P}_{\pi}$-stably in $\tilde{\mathbf{P}}$-probability to $\tau$ as $n\rightarrow\infty$. If $\eta^{\ell}$ is a rate measure for $\tau^{\ell}$, and $\eta$ is a smooth measure such that $\eta^{\ell}\nearrow\eta$, then $\eta$ is a rate measure for $\tau$.

The proof is straightforward, and is given in [7]. Condition (8.1) gives a uniform bound on the total capacity of the sets $\Lambda_{j}\left(n\right)$. When $\mu$ is a smooth measure such that (8.1) holds, and we wish to prove convergence for $D_{\Lambda^{\mu}\left(n\right)}$ or $T_{\Lambda^{\mu}\left(n\right)}$, Lemma 8.1 allows us to assume that the measure $\mu$ has finite energy and compact support in $E$.

We now consider particular random center models for which $E$ is an open subset $U$ of ${\mathbb{R}}^{d}$, $d\geq 2$. Let $b_{ij}$, $i,j=1,\ldots,d$, be bounded measurable functions on $U$, $d\geq 2$, with $b_{ij}=b_{ji}$, such that for some constant $e_{0}>0$, $\sum_{ij}w_{i}b_{ij}(x)w^{j}\geq e_{0}\sum_{i}w_{i}^{2}$ for all $x\in U$, $w\in{\mathbb{R}}^{d}$. We will denote the matrix function $\left(b_{ij}\right)$ by $b$. Let $\sigma$ be a positive function on $U$ which is measurable, bounded and bounded away from zero. Let $\mathcal{E}^{b,\sigma,U}$ be the Dirichlet form on $L^{2}\left(U,\sigma\lambda_{d}\right)$ such that $\mathcal{E}^{b,\sigma,U}\left(f,g\right)=\int_{U}\sum_{ij}b_{ij}\left(\partial_{i}f\right)\left(\partial_{j}g\right)\,d\lambda_{d}$ for any smooth functions $f,g$ on $U$ which have compact support in $U$, where $\lambda_{d}$ denotes Lebesgue measure on ${\mathbb{R}}^{d}$ and $D\left(\mathcal{E}^{b,\sigma,U}\right)$ is the closure of the space of such functions. $\mathcal{E}^{b,\sigma,U}$ exists by II.2 in [22]. Let $X=X^{b,\sigma,U}$ be the Markov process with lifetime $\zeta$ and cemetery point $\partial$ which is properly associated with $\mathcal{E}^{b,\sigma,U}$.

When studying $X$, it will be convenient to consider to extend $b_{ij}$ and $\sigma$ to all of ${\mathbb{R}}^{d}$, in such a way that $b_{ij}$ is bounded on ${\mathbb{R}}^{d}$ and $\sum_{ij}w_{i}b_{ij}(x)w^{j}\geq e_{0}\sum_{i}w_{i}^{2}$ for all $x,w\in{\mathbb{R}}^{d}$ and $\sigma$ is bounded and bounded away from zero on ${\mathbb{R}}^{d}$. We can define the Dirichlet form $\mathcal{E}^{b,\sigma,V}$ for any open subset $V$ of ${\mathbb{R}}^{d}$ analogously to $\mathcal{E}^{b,\sigma,U}$. Let $X^{b,\sigma,V}$ be the process associated with $\mathcal{E}^{b,\sigma,V}$. By Lemma 7.2, we may assume that the process $X=X^{b,\sigma,U}$ is the restriction of $X^{b,\sigma,{\mathbb{R}}^{d}}$ to $U$, so that $\zeta$ is the exit time of $U$ by $X^{b,\sigma,{\mathbb{R}}^{d}}$, and $X_{t}=X^{b,\sigma,{\mathbb{R}}^{d}}_{t}$ when $t<\zeta$, $X_{t}=\partial$ if $t\geq\zeta$. We will use this version of $X$ in what follows.

It will be assumed from now on that $\Gamma^{x}_{n}$ is compact with $\tilde{\mathbf{P}}_{1}$-probability one for $\mu$-a.e. $x$. If $\Gamma^{x}_{n}$ is not compact, redefine $\Gamma^{x}_{n}=\emptyset$, and also extend $\Gamma^{x}_{n}$ to all $x\in{\mathbb{R}}^{d}$ by setting $\Gamma^{x}_{n}=\emptyset$ for $x\in U^{c}$. For any open subset $V$ of ${\mathbb{R}}^{d}$ and any $x\in V$, let $\psi_{n,\alpha}^{x,b,\sigma,V}$ denote the $\alpha$-equilibrium measure for $\Gamma^{x}_{n}$ using $\mathcal{E}^{b,\sigma,V}$. The map $\left(x,\tilde{\omega}_{1}\right)\mapsto\int f\,d\psi_{n,\alpha}^{x,b,\sigma,V}$ is jointly measurable on $V\times\tilde{\Omega}_{1}$ for any $f\in b\mathcal{B}$, by regularity. Similarly the map $\left(x,\tilde{\omega}_{1}\right)\mapsto\int\left(G_{\alpha}\,{\psi_{n,\alpha}^{x,b,\sigma,V}}\right)\,d\psi_{n,\alpha}^{x,b,\sigma,V}$ is jointly measurable. Define the average measure $\bar{\psi}^{x,b,\sigma,V}_{n,\alpha}$ by $\bar{\psi}_{n,\alpha}^{x,b,\sigma,V}(W)=\tilde{\mathbf{E}}_{1}\left[\psi^{x,b,\sigma,V}_{n,\alpha}(W)\right]$. Using the notation of Theorem 2.1, for $X=X^{b,\sigma,U}$, we have $\gamma^{n}_{j}=\psi_{n,\alpha}^{\xi_{j}(n,\mu)}$ and

[TABLE]

We first consider the translation-invariant case. By translation-invariance we mean here that $b$ is constant on $U$, and the distribution of the sets $\Gamma^{x}_{n}-x$ is the same for $\mu$-a.e. $x$. Translation-invariant models in the Brownian motion setting were considered in [19], [28], [27], [1], with the sets $\Gamma^{x}_{n}$ equal to nonrandom scaled translates of a fixed compact set. The next lemma differs from earlier results in some technical aspects, since $\mu$ is only required to be smooth, the sets $\Gamma^{x}_{n}$ are allowed to be random with different shapes for each $n$, and the rate at which the sets shrink is only constrained by (8.1). It seems of interest as an example for Corollary 2.2 because of the simplicity of the proof.

Lemma 8.2

Let $\mu$ be a smooth probability measure on $U$. Suppose that the random center model for $U$ associated with $\left(\Gamma^{x}_{n}\right),\mu$ is translation-invariant. Then $\left\lVert\bar{\gamma}^{n}\right\rVert_{\text{tv}}=\kappa_{n}\left\lVert\bar{\psi}_{n,\alpha}^{x,b,1,{\mathbb{R}}^{d}}\right\rVert_{\text{tv}}$ for $\mu$-a.e. $x$. Suppose that (8.1) holds and $\lim_{n\rightarrow\infty}\left\lVert\bar{\gamma}^{n}\right\rVert_{\text{tv}}=c\in[0,\infty)$. Let $\tau_{n}=D_{\Lambda\left(n\right)}$ or $\tau_{n}=T_{\Lambda\left(n\right)}$, using the process $X=X^{b,\sigma,U}$. For any smooth probability measure $\pi$ on $U$, $\tau_{n}\rightarrow\tau$, $\mathbf{P}_{\pi}$-stably in $\tilde{\mathbf{P}}$-probability, where $\tau$ is the randomized stopping time with rate measure $c\mu$.

Proof   By the definition of translation-invariance, for each $n$ there is a measure $\pi_{n}$ such that $\bar{\psi}_{n,\alpha}^{b,1,{\mathbb{R}}^{d}}(W)=\pi_{n}(W-x)$ for $\mu$-a.e. $x$. By (8.2), $\kappa_{n}\left\lVert\pi_{n}\right\rVert_{\text{tv}}=\left\lVert\bar{\gamma}^{n}\right\rVert_{\text{tv}}\rightarrow c$.

By Lemma 7.3 we may assume without loss of generality that $U={\mathbb{R}}^{d}$. By Lemma 7.1 we may then assume that $\sigma=1$. Then $X=X^{b,1,{\mathbb{R}}^{d}}$ is essentially Brownian motion, and the absolute continuity condition (2.4) holds.

By Lemma 8.1, we can assume that $\mu$ has finite energy and has compact support, so that with $\tilde{\mathbf{P}}$-probability one all the sets $\Lambda\left(n\right)$ are contained in a compact set. Since $\varrho_{n}\rightarrow 0$, $\kappa_{n}\pi_{n}\rightarrow c\delta_{0}$ weakly as a sequence of measures.

Since $\bar{\gamma}^{n}=\kappa_{n}\pi_{n}\ast\mu$, $\bar{\gamma}^{n}\rightarrow c\mu$ weakly as a sequence of measures. Let $\mu^{x}$ be the translated measure defined by $\mu^{x}(B)=\mu(B-x)$. We have

[TABLE]

Hence $\limsup_{n}\kappa_{n}^{2}\mathcal{E}_{\alpha}\left(G_{\alpha}\,\left(\pi_{n}\ast\mu\right),G_{\alpha}\,\left(\pi_{n}\ast\mu\right)\right)\leq c^{2}\left\lVert G_{\alpha}\,\mu\right\rVert_{\mathcal{E},\alpha}^{2}=\mathcal{E}_{\alpha}\left(G_{\alpha}\,\left(c\mu\right),G_{\alpha}\,\left(c\mu\right)\right)$. Since $\left\lVert\kappa_{n}G_{\alpha}\,{\left(\pi_{n}\ast\mu\right)}\right\rVert_{\mathcal{E},\alpha}$ is bounded, $\kappa_{n}G_{\alpha}\,\left(\pi_{n}\ast\mu\right)\rightarrow G_{\alpha}\,\left(c\mu\right)$, $\mathcal{E}$-weakly, by Lemma 9.6. Hence also $\liminf_{n}\kappa_{n}^{2}\mathcal{E}_{\alpha}\left(G_{\alpha}\,\left(\pi_{n}\ast\mu\right),G_{\alpha}\,\left(\pi_{n}\ast\mu\right)\right)\geq\mathcal{E}_{\alpha}\left(G_{\alpha}\,\left(c\mu\right),G_{\alpha}\,\left(c\mu\right)\right)$, so $\kappa_{n}G_{\alpha}\,\left(\pi_{n}\ast\mu\right)\rightarrow G_{\alpha}\,\left(c\mu\right)$ in energy norm. Thus Corollary 2.2 applies and gives convergence.

From a physical standpoint, one can think of the holes $\Lambda_{i}\left(n\right)$ is representing fixed defects in some material, but one might also consider the case of moving obstacles in a fluid medium (“dust particles” in [19]). These moving holes would presumably travel slowly in comparison to Brownian motion, but if their movement is considered it would at least affect the formula for the limit of the stopping times $\tau_{n}$. It seems to be an interesting problem to actually prove convergence of $\tau_{n}$ in the Brownian motion case when the holes are moving. When all holes move with identical velocity $v(t)$, Girsanov’s theorem can be applied to show that if convergence holds without the movement of the holes, then convergence holds for the moving case also. It is more reasonable physically to consider the case in which the holes $\Lambda_{i}\left(n\right)$, $i=1,\ldots,\kappa_{n}$, move independently. In this case it is plausible that the $\tau_{n}$ would still converge in probability under suitable conditions. We have no result of this sort, however.

One can measure the asymptotic capacity of the sets $\Gamma^{x}_{n}$ in various ways. For any sequences $t_{n},u_{n}$ of nonzero numbers, let $t_{n}\sim u_{n}$ mean that $\lim_{n\rightarrow\infty}t_{n}/u_{n}=1$. It is not hard to show using the estimates in [30] that for any open subset $V$ of ${\mathbb{R}}^{d}$ with $x\in V$ and any $\alpha,\beta\in(0,\infty)$,

[TABLE]

with a corresponding asymptotic equivalence for the average measures $\left\lVert\bar{\psi}_{n,\alpha}^{x,b,\sigma,V}\right\rVert_{\text{tv}}$ and $\left\lVert\bar{\psi}_{n,\beta}^{x,b,1,{\mathbb{R}}^{d}}\right\rVert_{\text{tv}}$. Equation (8.3) holds whether or not the model is translation-invariant. In particular it shows that the constant $c$ in Lemma 8.2 can be expressed in terms of $\alpha$-equilibrium measures with respect to $\mathcal{E}^{b,\sigma,U}$ if desired, although these measures may not be as easy to compute.

The same arguments used to show (8.3) also show that for general coefficients $b^{ij}$ satisfying the stated assumptions, $\alpha$-capacity with respect to $\mathcal{E}^{b,\sigma,V}$ is locally comparable to classical capacity. That is, given any compact subset $K$ of $V$ there exists a constant $c^{\prime}$ such that when $\varrho_{n}<1$, for all $x$ such that $B_{\varrho_{n}}(x)\subset K$ and all $n$,

[TABLE]

When $V={\mathbb{R}}^{d}$, (8.4) holds for all $x\in{\mathbb{R}}^{d}$. If the coefficients $b^{ij}$ happen to be continuous on $U$, using (8.3) and equation (9.5) one also finds easily that for $x\in V$,

[TABLE]

where $b^{x}$ is the constant matrix function equal to $b(x)$ everywhere. Following an idea in [1] and [2] one can then relate $\left\lVert\psi_{n,\alpha}^{x,b,\sigma,U}\right\rVert_{\text{tv}}$ to the classical capacity of the sets $\Gamma^{x}_{n}$. For a compact subset $K$ of ${\mathbb{R}}^{d}$, let $\mathsf{Q}^{\mathsf{cl}}_{d}\left(K\right)$ be the classical capacity of $K$, where the classical capacity is calculated using the potential kernel $\wp_{d}\left(y,z\right)=1/\left(\left\lvert z-y\right\rvert^{d-2}(d-2)\omega_{d}\right)$ if $d>2$, $\wp_{d}\left(y,z\right)=-\log\left\lvert z-y\right\rvert/\omega_{2}$ if $d=2$, and $\omega_{d}$ here denotes the surface area of the unit hypersphere in ${\mathbb{R}}^{d}$, so that for example $\omega_{3}=4\pi$. Then

[TABLE]

where $b(x)^{-1/2}$ is the inverse of the positive square root of the matrix $b(x)$ and $b(x)^{-1/2}\Gamma^{x}_{n}$ denotes the set of points $b(x)^{-1/2}z$, $z\in\Gamma^{x}_{n}$. Also, when $U$ is bounded, one can show that

[TABLE]

where we define $\mathsf{Q}^{U}\left(K\right)=\inf\left\{\mathcal{E}^{I,1,U}\left(f,f\right):\ f\in D\left(\mathcal{E}^{I,1,U}\right),\ f\geq 1\text{ q.e. on }K\right\}$, for any compact subset $K$ of $U$, and $I$ is the $d\times d$ identity matrix.

Given Lemma 8.2, one would naturally hope that convergence holds for a more general case of the random center model for subsets of ${\mathbb{R}}^{d}$. However, the easy proof of Lemma 8.2 used the translation-invariance of $\bar{\psi}_{n,\alpha}^{x}$ heavily. A similar proof, using the analog of translation-invariance, is applicable when $\mathcal{E}$ is associated with the Laplace-Beltrami operator on a homogeneous Riemannian manifold. In the general case a proof can be given by strengthening the bound in (8.1), as in (8.8) below.

Lemma 8.3

Let $\mu$ be a smooth probability measure on $U\cup\left\{\partial\right\}$. In the random center model for $U$ associated with $\left(\Gamma^{x}_{n}\right)$ and $\mu$, let $q_{n}(x)=\left\lVert\bar{\psi}^{x,b,1,{\mathbb{R}}^{d}}_{n,\alpha}\right\rVert_{\text{tv}}$, for $\mu$-a.e. $x$, and let $\nu_{n}=q_{n}\mu$. Assume that there exists $\varrho_{n}\in[0,\infty)$ with $\varrho_{n}\searrow 0$, such that $\tilde{\mathbf{P}}_{1}\left(\Gamma^{x}_{n}\subset B_{\varrho_{n}}(x)\right)=1$ for $\mu$-a.e. $x$, and

[TABLE]

Let $\tau_{n}=D_{\Lambda\left(n\right)}$ or $\tau_{n}=T_{\Lambda\left(n\right)}$, using the process $X=X^{b,\sigma,U}$. Assume that $\nu_{n}$ converges weakly as a sequence of measures to a finite measure $\eta$. Then for any smooth probability measure $\pi$ on $U$, $\tau_{n}\rightarrow\tau$, $\mathbf{P}_{\pi}$-stably in $\tilde{\mathbf{P}}$-probability, where $\tau$ is the randomized stopping time with rate measure $\eta$. The same conclusion holds using $q_{n}(x)=\left\lVert\bar{\psi}^{x,b,\sigma,U}_{n,\alpha}\right\rVert_{\text{tv}}$.

Girsanov’s theorem can be used to extend Lemma 8.3 to examples with drift.

Assumption (8.8) is a uniform smallness condition on the sets $\Gamma^{x}_{n}$, and is satisfied by the iid models in [19], [28], [27], [1], [2]. This condition is equivalent to the statement that $\sup_{n}\kappa_{n}\left\lVert\bar{\psi}^{x,b,\sigma,U}_{n,\alpha}\right\rVert_{\text{tv}}<\infty$ for some point $x\in U$. By (8.4), (8.8) implies that $\chi_{n}$ exists such that (8.1) holds. Equations (8.3) and (8.4) and Lemma 8.1 show that the conclusion of Lemma 8.3 also holds if $q_{n}$ is defined by $q_{n}(x)=\left\lVert\bar{\psi}^{x,b,\sigma,U}_{n,\alpha}\right\rVert_{\text{tv}}$.

The proof of Lemma 8.3 uses that fact that when $U={\mathbb{R}}^{d}$ and $\sigma=1$, a nice potential kernel exists ([30]). Lemmas 7.2 and 7.1 are again used to reduce the proof to that setting.

When the state space is a manifold rather than a subset of ${\mathbb{R}}^{d}$, convergence should still be determined by local behavior. Thus Lemma 7.3 allows one to extend Lemma 8.3 to the case of a diffusion on a $d$-dimensional Riemannian manifold, $d\geq 2$, whose topology has a countable base. This gives a more general form of Theorem 4.2 of [2], which deals with the Laplace-Beltrami operator on a compact Riemannian manifold with boundary, when the sets $\Lambda\left(n\right)$ are unions of iid random geodesic balls.

9 Dirichlet form properties

Here we summarize facts which are used, with references or proofs. $X$ is assumed to be as in Section 2.3. Smooth measures were defined in that section. The proof of Theorem 2.3.15 in [10] gives:

Lemma 9.1

For any finite smooth measure $\mu$ and any $\varepsilon>0$ there exists $F\in\mathcal{B}$ such that $\mu\left(E-F\right)<\varepsilon$ and $\mathbf{1}_{F}\mu$ has finite energy.

Let $\eta$ be a smooth measure which is the Revuz measure for the positive continuous additive functional $\left(A_{t}\right)$, i.e. for any $f\in\mathcal{B}^{+}$, $\lim_{t\downarrow 0}\tfrac{1}{t}\mathbf{E}_{m}\left[\int_{0}^{t}f\left(X_{s}\right)\,dA_{s}\right]=\int fd\eta$. The proof of Theorem 4.1.1 in [10] or Theorem 4.1.13 in [26] shows that this equation holds if and only if for all $\alpha\in(0,\infty)$ and all $f,h\in\mathcal{B}^{+}$, $\mathbf{E}_{hm}\left[\int_{0}^{\infty}e^{-\alpha t}f\left(X_{t}\right)\,dA_{t}\right]=\int f\left(\hat{R}_{\alpha}\,h\right)\,d\eta$. It follows that when $f\eta$ has finite energy, for $\mathcal{E}$-q.e. $x$ we have

[TABLE]

$\alpha$-excessive functions are defined in III.1.1 of [22]. By III.1.2(iii), $G_{\alpha}\,\mu$ is $\alpha$-excessive for any measure $\mu$ with finite energy. Let $u$ be a function with $u\geq 0$ and $e^{-\alpha t}p_{t}\,u\leq u$, $m$-a.e., for all $t>0$. Suppose also that $u$ has an $\mathcal{E}$-quasi-continuous version $\tilde{u}$. Then $e^{-\alpha t}p_{t}\,u\leq\tilde{u}$ holds $\mathcal{E}$-q.e. on $E$ by IV.3.3 (iii) of [22], since $p_{t}\,u$ is $\mathcal{E}$-quasi-continuous by IV.2.9 of [22]. By the right continuity of $t\mapsto\tilde{u}\circ X_{t}$ we have for $\mathcal{E}$-q.e. $x$ that $\liminf_{t\searrow 0}e^{-\alpha t}p_{t}\,u(x)\geq\tilde{u}(x)$. Similar facts hold for $\beta R_{\beta+\alpha}\,{u}$. Thus for $\mathcal{E}$-q.e. $x$,

[TABLE]

It is also easy to prove the following.

Lemma 9.2

Let $v\geq 0$ be $\mathcal{E}$-quasi-continuous and such that $e^{-\alpha t}p_{t}\,v\leq v$ holds $\mathcal{E}$-q.e. for each $t$ (for example, let $\nu$ be $\alpha$-excessive and $\mathcal{E}$-quasi-continuous). Then $t\mapsto e^{-\alpha t}v\left(X_{t}\right)$ is a supermartingale with respect to $\mathbf{P}_{x}$ for $\mathcal{E}$-q.e. $x$.

For $A\subset E$, by solving III.3.10 of [22] we can define reduced functions on $A$, as follows. For any function $f$ on $E$ which has an $\mathcal{E}$-quasi-continuous version $\tilde{f}$, let $\mathcal{L}_{f,A}$ denote the set of all $w\in D\left(\mathcal{E}\right)$ such that $\tilde{w}\geq\tilde{f}$ holds $\mathcal{E}$-quasi-everywhere on $A$. Assuming that $\mathcal{L}_{f,A}\neq\emptyset$, let $g$ be the unique element in $\mathcal{L}_{f,A}$ such that $\mathcal{E}_{\alpha}\left(g,w\right)\geq\mathcal{E}_{\alpha}\left(g,g\right)$ for all $w\in\mathcal{L}_{f,A}$. The function $g$ is an $\alpha$-excessive member of $D\left(\mathcal{E}\right)$, and

[TABLE]

We denote any $\mathcal{E}$-quasi-continuous version of $g$ by ${f}_{A,\alpha}$, and refer to ${f}_{A,\alpha}$ as the $\mathcal{E},\alpha$-reduced function for $f$ on $A$. If $h\wedge{f}_{A,\alpha}$ is an $\alpha$-excessive member of $D\left(\mathcal{E}\right)$ (in particular if $h$ itself is an $\alpha$-excessive member of $D\left(\mathcal{E}\right)$), and $h\geq f$ holds $\mathcal{E}$-q.e. on $A$, then $h\geq{f}_{A,\alpha}$, $\mathcal{E}$-q.e. on $E$. If $f\wedge{f}_{A,\alpha}$ is an $\alpha$-excessive member of $D\left(\mathcal{E}\right)$ then taking $h=f\wedge{f}_{A,\alpha}$ shows ${f}_{A,\alpha}=f$ holds $\mathcal{E}$-q.e. on $A$.

For any $f\in D\left(\mathcal{E}\right)$, $\alpha\in(0,\infty)$ and any set $A$, let ${f}^{\prime}_{A,\alpha}$ be the unique $g\in D\left(\mathcal{E}\right)$ such that $\tilde{f}=\tilde{g}$ holds $\mathcal{E}$-q.e. on $A$ and equation (9.3) holds. An $\mathcal{E}$-quasi-continuous version of ${f}^{\prime}_{A,\alpha}$ is used whenever pointwise values are needed. It is easy to check from the definitions that $\left\lVert{f}^{\prime}_{A,\alpha}\right\rVert_{\mathcal{E},\alpha}\leq K_{\alpha}\left\lVert f\right\rVert_{\mathcal{E},\alpha}$. Also, if $f\wedge{f}_{A,\alpha}$ is an $\alpha$-excessive member of $D\left(\mathcal{E}\right)$ then ${f}^{\prime}_{A,\alpha}={f}_{A,\alpha}$, and so ${f}^{\prime}_{A,\alpha}$ is an $\alpha$-excessive element of $D\left(\mathcal{E}\right)$.

Let $A$ be a closed subset of $E$ such that the cemetery point $\partial$ is not in the closure of both $A$ and $E-A$. Let $u\in D\left(\mathcal{E}\right)$. Then

[TABLE]

holds for $\mathcal{E}$-q.e. $z$. To prove equation (9.4), we note that since $X$ is a Hunt process, there exist open sets $U_{k}$ with $U_{k}\searrow A$ and $D_{U_{k}}\nearrow D_{A}$, $\mathbf{P}_{z}$-a.e. It is enough to prove equation (9.4) when $u$ is bounded. The first equality can be obtained by applying V.1.6 of [22] to the open sets $U_{k}$, and then using a convergence argument. The second equality can be derived from the first since $\tilde{u}$ and ${u}^{\prime}_{A,\alpha}$ are $\mathcal{E}$-quasi-continuous and $\varepsilon+D_{A}\circ\theta_{\varepsilon}\rightarrow T_{A}$ as $\varepsilon\searrow 0$.

Lemma 9.3

Let $\mu$ a measure with finite energy. Then $e^{-\alpha t}G_{\alpha}\,\mu\left(X_{t}\right)$ is a supermartingale with respect to $\mathbf{P}_{x}$ for $\mathcal{E}$-q.e. $x$. Let $A$ be a closed set with $\mu\left(A^{c}\right)=0$. Then ${\left(G_{\alpha}\,\mu\right)}_{A,\alpha}=G_{\alpha}\,\mu$. If $A$ is also such that $\partial$ is not in the closure of both $A$ and $E-A$, then $G_{\alpha}\,\mu(x)=\mathbf{E}_{x}\left[e^{-\alpha D_{A}}G_{\alpha}\,\mu\left(X_{D_{A}}\right)\right]$ for $\mathcal{E}$-q.e. $x$, and $e^{-\alpha t\wedge D_{A}}G_{\alpha}\,\mu\left(X_{t\wedge D_{A}}\right)$ is a martingale with respect to $\mathbf{P}_{x}$ for $\mathcal{E}$-q.e. $x$.

Proof   By Lemma 9.2, $e^{-\alpha t}G_{\alpha}\,\mu\left(X_{t}\right)$ is a supermartingale with respect to $\mathbf{P}_{x}$ for $\mathcal{E}$-q.e. $x$.

Let $A$ be a closed set with $\mu\left(A^{c}\right)=0$. Using equation (2.3) and equation (9.3) one has $\mathcal{E}_{\alpha}\left(G_{\alpha}\,\mu-{\left(G_{\alpha}\,\mu\right)}_{A,\alpha},G_{\alpha}\,\mu-{\left(G_{\alpha}\,\mu\right)}_{A,\alpha}\right)=0$, and hence ${\left(G_{\alpha}\,\mu\right)}_{A,\alpha}=G_{\alpha}\,\mu$. Assume $A$ is also such that $\partial$ is not in the closure of both $A$ and $E-A$. By equation (9.4) with ${u}^{\prime}_{A,\alpha}=u=G_{\alpha}\,\mu$, $G_{\alpha}\,\mu(x)=\mathbf{E}_{x}\left[e^{-\alpha D_{A}}G_{\alpha}\,\mu\left(X_{D_{A}}\right)\right]$ for $\mathcal{E}$-q.e. $x$. Since $e^{-\alpha t}G_{\alpha}\,\mu\left(X_{t}\right)$ is a supermartingale with respect to $\mathbf{P}_{x}$ for $\mathcal{E}$-q.e. $x$, it follows that that $e^{-\alpha t\wedge D_{A}}G_{\alpha}\,\mu\left(X_{t\wedge D_{A}}\right)$ is a martingale with respect to $\mathbf{P}_{x}$ for $\mathcal{E}$-q.e. $x$.

For $\alpha\in(0,\infty)$ and any $\alpha$-excessive $u\in D\left(\mathcal{E}\right)$, by VI.2.1 of [22] there exists a measure $\mu$ with finite energy such that $u=G_{\alpha}\,\mu$. If $u\leq G_{\alpha}\,\nu$, then $\mu\left(E\right)\leq\nu\left(E\right)$ by Lemma 9.5 below.

Let $\mathsf{C}$ be the collection of all closed sets $A$ with $\mathcal{L}_{1,A}\neq\emptyset$. Let $A\in\mathsf{C}$. Since ${1}_{A,\alpha}$ is an $\alpha$-excessive member of $D\left(\mathcal{E}\right)$, ${1}_{A,\alpha}\wedge 1$ is also an $\alpha$-excessive member of $\mathcal{L}_{1,A}$, and so ${1}_{A,\alpha}=1$ holds $\mathcal{E}$-q.e. on $A$. The unique measure $\gamma$ with finite energy such that ${1}_{A,\alpha}=G_{\alpha}\,\gamma$ will be referred to as the $\alpha$-equilibrium measure for the set $A$, and ${1}_{A,\alpha}$ will be called the $\alpha$-equilibrium potential for $A$. Then $\mathcal{E}_{\alpha}\left({1}_{A,\alpha},{1}_{A,\alpha}\right)=\int\left(G_{\alpha}\,\gamma\right)\,d\gamma=\int 1\,d\gamma=\gamma\left(E\right)$, $\gamma$, so $\gamma$ is a finite measure. Because $A$ is closed, $\gamma\left(A^{c}\right)=0$, and $\gamma$ is the unique measure such that $G_{\alpha}\,\gamma=1$ holds $\mathcal{E}$-q.e. on $A$ and $\gamma\left(A^{c}\right)=0$. Define the $\alpha$-capacity of $A$, denoted by $\mathsf{Cap}_{\alpha}\left(A\right)$, to be $\gamma(E)$.

For symmetric $\mathcal{E}$, $w\in\mathcal{L}_{1,A}$ implies $\mathcal{E}_{\alpha}\left(w,w\right)=\mathcal{E}_{\alpha}\left(w-{1}_{A,\alpha},w-{1}_{A,\alpha}\right)+2\mathcal{E}_{\alpha}\left({1}_{A,\alpha},w-{1}_{A,\alpha}\right)+\mathcal{E}_{\alpha}\left({1}_{A,\alpha},{1}_{A,\alpha}\right)\geq\mathcal{E}_{\alpha}\left({1}_{A,\alpha},{1}_{A,\alpha}\right)$, using the definition of reduction. Hence in the symmetric case,

[TABLE]

We can prove that $\alpha$-capacity is monotone, in the sense that if $A_{1},A_{2}\in\mathsf{C}$ with $A_{1}\subset A_{2}$, then $\mathsf{Cap}_{\alpha}\left(A_{1}\right)\leq\mathsf{Cap}_{\alpha}\left(A_{2}\right)$. Hence it is convenient to to extend the definition of capacity. If $W$ is a closed set which is a countable union of sets in $\mathsf{C}$, define $\mathsf{Cap}_{\alpha}\left(W\right)=\sup\left\{\mathsf{Cap}_{\alpha}\left(B\right):\ B\in\mathsf{C},\ B\subset W\right\}$. Capacities for the nonsymmetric case are defined differently in III.2.8 of [22], but have similar properties. The proof that $\alpha$-capacity is monotone follows easily from (9.5) in the symmetric case, and in general by the next lemma, which is known as the domination principle, together with Lemma 9.5.

Lemma 9.4

Let $v\geq 0$ be $\mathcal{E}$-quasi-continuous and such that $e^{-\alpha t}p_{t}\,v\leq v$ holds $\mathcal{E}$-q.e. for each $t$. Let $\mu$ be a finite measure with finite energy such that $G_{\alpha}\,\mu\leq v$ holds $\mu$-a.e. on $E$. Then $G_{\alpha}\,\mu\leq v$ holds $\mathcal{E}$-q.e. on $E$.

Proof   Let $f\in L^{2}\left(m\right)$, $f\geq 0$. There exists a nondecreasing sequence of compact subsets $A_{n}$ of $E$, such that $G_{\alpha}\,\mu\leq v$ everywhere on $A_{n}$, and such that $\int_{A_{n}}\left(\hat{G}_{\alpha}\,f\right)\,d\mu\nearrow\int\left(\hat{G}_{\alpha}\,f\right)\,d\mu$. Let $\mu_{n}=\mathbf{1}_{A_{n}}\,\mu$, so that $\mu_{n}\left(A_{n}^{c}\right)=0$. Let $W_{t}=e^{-\alpha t\wedge D_{A_{n}}}v\left(X_{t\wedge D_{A_{n}}}\right)$. By Lemma 9.2, $W$ is a supermartingale with respect to $\mathbf{P}_{x}$ for $\mathcal{E}$-q.e. $x$, so $\mathbf{E}_{x}\left[W_{0}\right]\geq\mathbf{E}_{x}\left[e^{-\alpha D_{A_{n}}}W_{D_{A_{n}}}\right]\geq\mathbf{E}_{x}\left[e^{-\alpha D_{A_{n}}}G_{\alpha}\,\mu_{n}\left(X_{D_{A_{n}}}\right)\right]$. By Lemma 9.3, $G_{\alpha}\,\mu_{n}(x)=\mathbf{E}_{x}\left[e^{-\alpha D_{A_{n}}}G_{\alpha}\,\mu_{n}\left(X_{D_{A_{n}}}\right)\right]$ for $\mathcal{E}$-q.e. $x$. Hence $v(x)\geq G_{\alpha}\,\mu_{n}(x)$ for $\mathcal{E}$-q.e. $x$. Also $\int f\left(G_{\alpha}\,\mu_{n}\right)\,dm=\int\left(\hat{G}_{\alpha}\,f\right)\,d\mu_{n}=\int_{A_{n}}\left(\hat{G}_{\alpha}\,f\right)\,d\mu\nearrow\int\left(\hat{G}_{\alpha}\,f\right)\,d\mu=\int f\left(G_{\alpha}\,\mu\right)\,dm$. Thus $\int f\left(G_{\alpha}\,\mu\right)\,dm\leq\int fv\,dm$. Since this is true for every nonnegative $f\in L^{2}\left(m\right)$, $G_{\alpha}\,\mu\leq v$ holds $m$-a.e., and so by IV.3.3 of [22], $G_{\alpha}\,\mu\leq v$ holds $\mathcal{E}$-q.e.

Lemma 9.5

Let $\mu,\nu$ be finite measures with finite energy, and $\alpha\in(0,\infty)$ such that $G_{\alpha}\,\mu\leq G_{\alpha}\,\nu$ holds $m$-a.e. Then $\mu\left(E\right)\leq\nu\left(E\right)$.

Proof   By V.1.7 there exists an $\mathcal{E}$-quasi-continuous $f\in D\left(\mathcal{E}\right)$ with $f>0$, $\mathcal{E}$-q.e. on $E$. Then $\hat{G}_{\alpha}\,f=\hat{R}_{\alpha}\,f>0$, $\mathcal{E}$-q.e. on $E$. Let $u_{n}=\left(n\hat{G}_{\alpha}\,f\right)\wedge 1$. Since $u_{n}$ is $\alpha$-coexcessive, $\mathcal{E}_{\alpha}\left(G_{\alpha}\,\mu,u_{n}\right)\leq\mathcal{E}_{\alpha}\left(G_{\alpha}\,\nu,u_{n}\right)$ by III.1.2(iii) of [22]. Hence $\int u_{n}\,d\mu\leq\int u_{n}\,d\nu$ and $u_{n}\nearrow 1$ $\mathcal{E}$-q.e.

Lemma 9.6

Let $\mathcal{E}$ be regular. Let $\mu_{n}$, $n=1,2,\ldots$ be finite measures with finite energy, and let $\mu$ be a finite smooth measure. Suppose $\sup_{n}\left\lVert G_{\alpha}\,\mu_{n}\right\rVert_{\mathcal{E},\alpha}<\infty$ and $\mu_{n}\rightarrow\mu$ vaguely as a sequence of measures. Then $\mu$ has finite energy and $G_{\alpha}\,\mu_{n}\rightarrow G_{\alpha}\,\mu$, $\mathcal{E}$-weakly.

Proof   Let $v$ be $\mathcal{E}$-quasi-continuous and let $w\in D\left(\mathcal{E}\right)$ such that $0\leq v\leq\tilde{w}$ holds $\mathcal{E}$-q.e. Let $\left(F_{k}\right)$ be an $\mathcal{E}$-nest such that $F_{k}$ is compact and the restriction of $v$ to $F_{k}$ is nonnegative and continuous for each $k$. Let $v_{k}(x)=v(x)$ for $x\in F_{k}$, $v_{k}(x)=0$ otherwise. Then $v_{k}$ is the limit of a decreasing sequence of functions in $\mathcal{C}_{0}\left(E\right)$. Hence $\limsup_{n\rightarrow\infty}\int v_{k}\,d\mu_{n}\leq\int v_{k}\,d\mu\leq\int v\,d\mu$. Let $z_{k}\in D\left(\mathcal{E}\right)$ be such that $z_{k}=0$ holds $\mathcal{E}$-q.e. on $F_{k}^{c}$ and $\left\lVert w-z_{k}\right\rVert_{\mathcal{E},\alpha}\rightarrow 0$. Then $\left\lVert\left\lvert w-z_{k}\right\rvert\right\rVert_{\mathcal{E},\alpha}\rightarrow 0$. Let $c=\sup_{n}\left\lVert G_{\alpha}\,\mu_{n}\right\rVert_{\mathcal{E},\alpha}$. $\int\left\lvert\tilde{w}-\tilde{z}_{k}\right\rvert\,d\mu_{n}=\mathcal{E}_{\alpha}\left(G_{\alpha}\,\mu_{n},\left\lvert w-z_{k}\right\rvert\right)\leq K_{\alpha}c\left\lVert\left\lvert w-z_{k}\right\rvert\right\rVert_{\mathcal{E},\alpha}\leq K_{\alpha}c\left\lVert w-z_{k}\right\rVert_{\mathcal{E},\alpha}$. Hence $\left\lvert\int v\,d\mu_{n}-\int v_{k}\,d\mu_{n}\right\rvert=\int_{F_{k}^{c}}v\,d\mu_{n}\leq\int_{F_{k}^{c}}\tilde{w}\,d\mu_{n}=\int_{F_{k}^{c}}\left(\tilde{w}-\tilde{z}_{k}\right)\,d\mu_{n}\leq K_{\alpha}c\left\lVert w-z_{k}\right\rVert_{\mathcal{E},\alpha}$, and so $\limsup_{n\rightarrow\infty}\int v\,d\mu_{n}\leq\int v\,d\mu$. Suppose that $\lim_{n\rightarrow\infty}\int\tilde{w}\,d\mu_{n}=\int\tilde{w}\,d\mu$. Applying what has been shown for $v$ to $\tilde{w}-v$, $\limsup_{n\rightarrow\infty}\int(\tilde{w}-v)\,d\mu_{n}\leq\int(\tilde{w}-v)\,d\mu$, so $\lim_{n\rightarrow\infty}\int v\,d\mu_{n}=\int v\,d\mu$.

Now let $v$ be any $\mathcal{E}$-quasi-continuous function with $v\geq 0$. Let $f\in\mathcal{C}_{0}\left(E\right)$ with $f\geq 0$. Then $f\wedge v$ is nonnegative and $\mathcal{E}$-quasi-continuous. Since $\mathcal{E}$ is regular, there exists $w\in D\left(\mathcal{E}\right)\cap\mathcal{C}_{0}\left(E\right)$ with $f\leq w$. Since $\lim_{n\rightarrow\infty}\int w\,d\mu_{n}=\int w\,d\mu$, $\lim_{n\rightarrow\infty}\int(f\wedge v)\,d\mu_{n}=\int(f\wedge v)\,d\mu$. Assume that $v\in D\left(\mathcal{E}\right)$. We have $\int(f\wedge v)\,d\mu\leq\limsup_{n\rightarrow\infty}\int v\,d\mu_{n}\leq K_{\alpha}c\left\lVert v\right\rVert_{\mathcal{E},\alpha}$. It follows that $\int v\,d\mu\leq K_{\alpha}c\left\lVert v\right\rVert_{\mathcal{E},\alpha}$ for every $v\in D\left(\mathcal{E}\right)$ with $v\geq 0$. Hence for every $v\in D\left(\mathcal{E}\right)$, $\left\lvert\int\tilde{v}\,d\mu\right\rvert\leq\int\left\lvert\tilde{v}\right\rvert\,d\mu\leq c\left\lVert\,\left\lvert v\right\rvert\,\right\rVert_{\mathcal{E},\alpha}\leq c\left\lVert v\right\rVert_{\mathcal{E},\alpha}$, so $\mu$ has finite energy.

Since $\mathcal{E}$ is regular, $D\left(\mathcal{E}\right)\cap\mathcal{C}_{0}\left(E\right)$ is dense in $D\left(\mathcal{E}\right)$ with respect to $\left\lVert\cdot\right\rVert_{\mathcal{E},\alpha}$-norm. By vague convergence, for $v\in D\left(\mathcal{E}\right)\cap\mathcal{C}_{0}\left(E\right)$ we have $\lim_{n\rightarrow\infty}\int v\,d\mu_{n}=\int v\,d\mu$, i.e. $\mathcal{E}_{\alpha}\left(G_{\alpha}\,\mu_{n},v\right)\rightarrow\mathcal{E}_{\alpha}\left(G_{\alpha}\,\mu,v\right)$. Since $\sup_{n}\left\lVert G_{\alpha}\,\mu_{n}\right\rVert_{\mathcal{E},\alpha}<\infty$, this convergence holds for all $v\in D\left(\mathcal{E}\right)$ by a $3\varepsilon$ argument.

Lemma 9.7

Let $\alpha>0$, and let $B=\left\{x:\ \delta_{x}\,R_{\alpha}<<m\right\}$. Let $u_{n},u$ be nonnegative $\mathcal{E}$-quasi-continuous functions in $D\left(\mathcal{E}\right)$ such that $u_{n}\rightarrow u$, $\mathcal{E}$-weakly. Assume for each $n$ that $\beta R_{\alpha+\beta}\,{u_{n}}\leq u_{n}$ holds $\mathcal{E}$-q.e. on $B$ for all $\beta>0$. Then $\liminf_{n\rightarrow\infty}u_{n}\geq u$ holds $\mathcal{E}$-q.e. on $B$.

If (2.4) holds then $\lim_{n\rightarrow\infty}\int\left\lvert u_{n}-u\right\rvert h\,dm=0$ for all $h\in L^{2}\left(m\right)$.

Proof   $\lim_{n\rightarrow\infty}\int u_{n}h\,dm=\mathcal{E}_{\alpha}\left(u_{n},\hat{G}_{\alpha}\,h\right)\rightarrow\mathcal{E}_{\alpha}\left(u,\hat{G}_{\alpha}\,h\right)=\int uh\,dm$ for every $h\in L^{2}\left(m\right)$. If $x\in B$ and $\beta>0$, let $f_{x,\beta}$ be a density for $\delta_{x}\,R_{\alpha+\beta}$ with respect to $m$. For each $x\in B$ and all $c,\beta>0$, let $h_{x,\beta}^{c}=\beta f_{x,\alpha+\beta}\wedge c$. Then for $x\in B$, $\int u_{n}h_{x,\beta}^{c}\,dm\leq\int u_{n}\beta f_{x,\alpha+\beta}\,dm=\int u_{n}\beta\,d\left(\delta_{x}\,R_{\alpha+\beta}\right)=\beta R_{\alpha+\beta}\,u_{n}(x)$. Also $\beta R_{\alpha+\beta}\,u_{n}(x)\leq u_{n}(x)$, $\mathcal{E}$-q.e. Thus for $\mathcal{E}$-q.e. $x\in B$, $\liminf_{n\rightarrow\infty}u_{n}(x)\geq\lim_{n\rightarrow\infty}\int u_{n}h_{x,\beta}^{c}\,dm=\int uh_{x,\beta}^{c}\,dm$ for all $c>0$, $\beta>0$. Letting $c\rightarrow\infty$, $\liminf_{n\rightarrow\infty}u_{n}(x)\geq\beta R_{\alpha+\beta}\,u(x)=\mathbf{E}_{x}\left[\int_{0}^{\infty}\beta e^{-(\alpha+\beta)t}u\left(X_{t}\right)\right]$. Letting $\beta\uparrow\infty$, by Fatou we have $\liminf_{n\rightarrow\infty}u_{n}\geq u$, $\mathcal{E}$-q.e. on $B$. Now suppose that (2.4) holds. Then $\liminf_{n\rightarrow\infty}u_{n}\geq u$, $m$-a.e. Let $h\in L^{2}\left(m\right)$. Then $\lim_{n\rightarrow\infty}\int h\left(u_{n}-u\right)^{-}\,dm=0$ by dominated convergence. Since $\lim_{n\rightarrow\infty}\int h\left(u_{n}-u\right)\,dm=0$, $\lim_{n\rightarrow\infty}\int h\left(u_{n}-u\right)^{+}\,dm=0$ as well.