2.1. The local Gromov–Hausdorff-vague topology

We introduce the local Gromov–Hausdorff-vague topology, which is used to discuss convergence of rooted-and-measured boundedly compact metric spaces. For details, see [Reference Noda33, Section 2] and [Reference Noda34].

Let $(S, d, \rho)$ be a rooted boundedly compact metric space.

We first recall the Hausdorff metric on $\mathcal{C}_{\mathrm{c}}(S)$ . For a subset $A \subseteq S$ , the (closed) $\varepsilon$ -neighborhood of A in (S, d) is given by

\begin{align*} A^{\varepsilon} \;:\!=\; \{ x \in S \mid \mbox{ there exists } y \in A\ \text{such that}\ d(x,y) \leq \varepsilon \}.\end{align*}

The Hausdorff metric $d_{\mathrm{H}}$ on $\mathcal{C}_{\mathrm{c}}(S)$ is defined by setting

\begin{align*} d_{\mathrm{H}}(A, B) \;:\!=\; \inf\{ \varepsilon \geq 0 \mid A \subseteq B^{\varepsilon},\, B \subseteq A^{\varepsilon} \},\end{align*}

where the infimum over the empty set is defined to be $\infty$ . It is known that $d_{\mathrm{H}}$ is indeed a metric (allowed to take the value $\infty$ because of the empty set) on $\mathcal{C}_{\mathrm{c}}(S)$ (see [Reference Burago, Burago and Ivanov12, Subsection 7.3.1]). We call the topology on $\mathcal{C}_{\mathrm{c}}(S)$ induced from $d_{\mathrm{H}}$ the Hausdorff topology.

A commonly used topology on $\mathcal{C}(S)$ is the Fell topology (see [Reference Molchanov31, Appendix C]). We define a metric inducing this topology using the Hausdorff metric as follows. Note that, for each subset $A \subseteq S$ and $r > 0$ , we write

(2.1) \begin{align} A^{(r)} \;:\!=\; \operatorname{cl}\!(A \cap B_{d}(\rho, r)),\end{align}

where $\operatorname{cl}\!(\!\cdot\!)$ denotes the closure of a subset.

Definition 2.2. For each $A, B \in \mathcal{C}(S)$ , define

(2.2) \begin{align} d_{\bar{\mathrm{H}}, \rho} (A, B) \;:\!=\; \int_{0}^{\infty} \mathrm{e}^{-r} \big( 1 \wedge d_{\mathrm{H}}(A^{(r)}, B^{(r)}) \big) \, \mathrm{d}r. \end{align}

The function $d_{\bar{\mathrm{H}}, \rho}$ is indeed a metric on $\mathcal{C}(S)$ and a natural extension of the Hausdorff metric for non-compact sets. The following is a basic property of $d_{\bar{\mathrm{H}}, \rho}$ .

For convergence of measures, we use the vague topology. So, we next introduce a metric inducing the vague topology. Recall that $(S, d, \rho)$ is a rooted boundedly compact metric space.

Recall that the Prohorov metric $d_{\mathrm{P}}$ between $\mu, \nu \in \mathcal{M}_{\mathrm{fin}}(S)$ is given by

\begin{align*} d_{\mathrm{P}}(\mu, \nu) \;:\!=\; \inf\{ \varepsilon \mid \mu(A) \leq \nu(A^{\varepsilon}) + \varepsilon,\, \nu(A) \leq \mu(A^{\varepsilon}) + \varepsilon \quad \mbox{for all } A \subseteq S \}.\end{align*}

By extending this Prohorov metric similarly to (2.2), we define a metric on $\mathcal{M}(S)$ as follows. Note that, for each $\mu \in \mathcal{M}(S)$ and $r > 0$ , we write $\mu^{(r)}$ for the restriction of $\mu$ to $B_{d}(\rho, r)$ .

Definition 2.4. For each $\mu, \nu \in \mathcal{M}(S)$ , we define

\begin{align*} d_{\mathrm{V}, \rho}(\mu, \nu) \;:\!=\; \int_{0}^{\infty} \mathrm{e}^{-r} ( 1 \wedge d_{\mathrm{P}}(\mu^{(r)}, \nu^{(r)}) )\, \mathrm{d}r. \end{align*}

We now introduce the local Gromov–Hausdorff-vague topology. We say that two rooted-and-measured boundedly compact metric spaces $G_{i} = (S_{i}, d_{i}, \rho_{i}, \mu_{i})$ , $i=1,2$ , are Gromov–Hausdorff-vague equivalent if and only if there exists a root-preserving isometry $f \colon S_{1} \to S_{2}$ such that $\mu_{2} = \mu_{1} \circ f^{-1}$ . Note that f being an isometry means that f is distance-preserving and surjective (and hence bijective) and f being root-preserving means that $f(\rho_{1})=\rho_{2}$ . We write $\mathbb{G}$ for the collection of Gromov–Hausdorff-vague-equivalence classes of rooted-and-measured boundedly compact metric spaces. We define $\mathbb{G}_{\mathrm c}$ to be the collection of $(S, d, \rho, \mu) \in \mathbb{G}$ such that (S, d) is compact.

Recall that the (pointed) Gromov–Hausdorff–Prohorov metric $d_{\mathrm{GHP}} $ on $\mathbb{G}_{\mathrm c}$ is given by setting, for $G_{i} = (S_{i}, d_{i}, \rho_{i}, \mu_{i}) \in \mathbb{G}_{\mathrm c},\, i=1,2$ ,

(2.3) \begin{align} d_{\mathrm{GHP}} (G_{1}, G_{2}) \;:\!=\; \inf_{f_{1}, f_{2}, M} \big\{ d(\,f_{1}(\rho_{1}), f_{2}(\rho_{2})) \vee d_{\mathrm{H}}(\,f_{1}(S_{1}), f_{2}(S_{2})) \vee d_{\mathrm{P}}\big(\mu_{1} \circ f_{1}^{-1}, \mu_{2} \circ f_{2}^{-1}\big) \big\},\end{align}

where the infimum is taken over all compact metric spaces (M, d) and all distance-preserving maps $f _i \colon S_{i} \to M,\, i=1,2$ . We equip $\mathbb{G}_{\mathrm c}$ with the topology induced by $d_{\mathrm{GHP}}$ , and call it the (pointed) Gromov–Hausdorff–Prohorov topology. It is known that this topology is Polish; further details about this metric can be found in [Reference Abraham, Delmas and Hoscheit1, Reference Khezeli25].

An extension of $d_{\mathrm{GHP}}$ to a metric on $\mathbb{G}$ was also studied in [Reference Abraham, Delmas and Hoscheit1, Reference Khezeli25]. It is defined in a manner similar to (2.2) as follows: for each $G_1, G_2 \in \mathbb{G}$ , set

\begin{align*}d'_{\mathrm{GHP}}(G_1, G_2)\;:\!=\;\int_0^\infty \mathrm{e}^{-r} \Bigl( 1 \wedge d_{\mathrm{GHP}}\big(G_1^{(r)}, G_2^{(r)}\big) \Bigr)\, \mathrm{d}r,\end{align*}

where we recall the restriction operator $\cdot^{(r)}$ from (1.1). This defines a metric on $\mathbb{G}$ that induces a Polish topology (see [Reference Khezeli25, Remark 3.20 and Theorem 3.27]). In what follows, however, we introduce another metric $d_{\mathbb{G}}$ on $\mathbb{G}$ in Definition 2.5. Although $d_{\mathbb{G}}$ induces the same topology as $d'_{\mathrm{GHP}}$ , we prefer $d_{\mathbb{G}}$ since its formulation is more flexible (see [Reference Noda34, Section 1] on this point). Indeed, the same philosophy also underlies the metrization of the space $\mathbb{M}_{\mathrm L}$ , which is discussed in the next subsection. The metric $d_{\mathbb{G}}$ is defined in a manner similar to (2.3), with a minor modification concerning the treatment of roots.

Definition 2.5. For $G_{i} = (S_{i}, d_{i}, \rho_{i}, \mu_{i}) \in \mathbb{G},\, i=1,2$ , we set

\begin{align*} d_{\mathbb{G}}(G_{1}, G_{2}) \;:\!=\; \inf_{f_{1}, f_{2}, M} \left\{ d_{\bar{\mathrm{H}}, \rho} (\,f_{1}(S_{1}), f_{2}(S_{2})) \vee d_{\mathrm{V}, \rho} (\mu_{1} \circ f_{1}^{-1}, \mu_{2} \circ f_{2}^{-1}) \right\}, \end{align*}

where the infimum is taken over all rooted boundedly compact metric spaces $(M, d, \rho)$ and all root-and-distance-preserving maps $f _i \colon S_{i} \to M,\, i=1,2$ .

Regarding convergence in $\mathbb{G}$ , we have the following result.

For later use, we introduce another space $\mathbb{D}$ by dropping measures from $\mathbb{G}$ . This is defined precisely as follows. We say that two rooted boundedly compact metric spaces are equivalent if and only if there exists a root-preserving isometry between them. We write $\mathbb{D}$ for the collection of equivalence classes of rooted boundedly compact metric spaces. If we define a metric on $\mathbb{D}$ similarly to $d_{\mathbb{G}}$ in Definition 2.5 but drop measure components, that is, if we define

\begin{align*} d_{\mathbb{D}}\bigl( (S_1, d_1, \rho_1), (S_2, d_2, \rho_2)\bigr) \;:\!=\; \inf_{f_{1}, f_{2}, M} d_{\bar{\mathrm{H}}, \rho} (\,f_{1}(S_{1}), f_{2}(S_{2})),\end{align*}

then we obtain a complete, separable metric $d_{\mathbb{D}}$ on $\mathbb{D}$ . The induced topology is called the local Gromov–Hausdorff topology; see [Reference Noda34, Section 4] for details.

2.2. The space $\mathbb{M}_{\mathrm L}$

In this subsection, following [Reference Noda33, Subsection 2.2], we define an extended version of the local Gromov–Hausdorff–vague topology on tuples consisting of a rooted-and-measured boundedly compact metric space together with a probability measure on the space of càdlàg paths and local-time-type functions.

To develop a Gromov–Hausdorff-type topology suitable for discussing the convergence of local times, we first introduce an extension of the compact-convergence topology for continuous functions to functions whose domains may differ. Although we focus here on local-time-type functions, the same discussion applies in a more general setting; see [Reference Noda34, Subsection 3.3].

Let $(S, d, \rho)$ be a rooted boundedly compact metric space. Recall from Definition 2.1 that $\mathcal{C}(S)$ denotes the collection of closed subsets of S.

Definition 2.7. We define

\begin{align*} \,\widehat{\!C} (S \times \mathbb{R}_{\geq 0}, \mathbb{R}) \;:\!=\; \bigcup_{X \in \mathcal{C}(S)} C(X \times \mathbb{R}_{\geq 0}, \mathbb{R}). \end{align*}

Note that $\,\widehat{\!C}(S \times \mathbb{R}_{\geq 0}, \mathbb{R})$ contains the empty map $\emptyset_{\mathbb{R}} \colon \emptyset \times \mathbb{R}_{\geq 0} \to \mathbb{R}$ . For each $L \in \widehat{C}(S \times \mathbb{R}_{\geq 0}, \mathbb{R})$ , if $L \in C(X \times \mathbb{R}_{\geq 0}, \mathbb{R})$ then we write $\operatorname{dom}_{1}(L) \;:\!=\; X$ .

Definition 2.8. (The compact-convergence topology with variable domains.) Let $L, L_{1}, L_{2}, \ldots$ be elements in $\widehat{C}(S \times \mathbb{R}_{\geq 0}, \mathbb{R})$ . We say that $L_{n}$ converges to L in the compact-convergence topology with variable domains if and only if the sets $\operatorname{dom}_{1}(L_{n})$ converge to $\operatorname{dom}_{1}(L)$ in the Fell topology in S, and it holds that, for all $T>0$ and $r>0$ ,

\begin{align*} \lim_{\delta \to 0} \limsup_{n \to \infty} \sup\nolimits_{\substack{ x_{n} \in \operatorname{dom}_{1}(L_{n})^{(r)}, \\ x \in \operatorname{dom}_{1}(L)^{(r)}, \\ d(x_{n},x) < \delta}} \sup\nolimits_{0 \leq t \leq T} |L_{n}(x_{n}, t) - L(x, t)| =0, \end{align*}

where we recall the restriction $\cdot^{(r)}$ from (2.1).

By Theorem 2.18 in [Reference Noda33], the compact-convergence topology with variable domains is Polish. This topology is a natural extension of the compact-convergence topology on $C(S \times \mathbb{R}_{\geq 0}, \mathbb{R})$ in the following sense: the inclusion map

\begin{align*} C(S \times \mathbb{R}_{\geq 0}, \mathbb{R}) \ni L \mapsto L \in \widehat{C}(S \times \mathbb{R}_{\geq 0}, \mathbb{R})\end{align*}

is a topological embedding, i.e. a homeomorphism onto its image (see [Reference Noda33, Corollary 2.21]). For a precompactness criterion and a tightness criterion for $\widehat{C}(S \times \mathbb{R}_{\geq 0}, \mathbb{R})$ , see [Reference Noda33, Theorems 2.23 and 2.24].

Given two maps $f \colon A \to B$ and $f' \colon A' \to B'$ , we define $f \times f' \colon A \times A' \to B \times B'$ by setting

\begin{align*} (\,f \times f') (a, a') \;:\!=\; (\,f(a), f'(a')).\end{align*}

We write $\operatorname{id}_{A}$ for the identity map from A to itself.

Finally, it is possible to define the space $\mathbb{M}_{\mathrm L}$ . Let $\mathbb{M}_{\mathrm L}^{\circ}$ be the collection of $(S, d, \rho, \mu, \pi)$ such that $(S, d, \rho, \mu) \in \mathbb{G}$ and $\pi$ is a probability measure on $D(\mathbb{R}_{\geq 0}, S) \times \widehat{C}(S \times \mathbb{R}_{\geq 0}, \mathbb{R})$ , where we recall that $D(\mathbb{R}_{\geq 0}, S)$ denotes the space of càdlàg functions with values in S equipped with the usual $J_{1}$ -Skorohod topology. To introduce an equivalence relation on $\mathbb{M}_{\mathrm L}^{\circ}$ , we need some preparation. For a distance-preserving map $f \colon S_{1} \to S_{2}$ between boundedly compact metric spaces, we define

(2.4) \begin{align} &\qquad\qquad \tau_{f}^{J_{1}} \colon D(\mathbb{R}_{\geq 0}, S_{1}) \ni X \mapsto f \circ X \in D(\mathbb{R}_{\geq 0}, S_{2}), \\ &\tau_{f}^{\widehat{C}} \colon \widehat{C}(S_{1} \times \mathbb{R}_{\geq 0}, \mathbb{R}) \ni L \mapsto L \circ (\,f^{-1} \times \operatorname{id}_{\mathbb{R}_{\geq 0}}) \in \widehat{C}(S_{2} \times \mathbb{R}_{\geq 0}, \mathbb{R}),\nonumber\end{align}

where the inverse map $f^{-1}$ is restricted to $f(\!\operatorname{dom}_{1}(L))$ so that $L \circ (\,f^{-1} \times \operatorname{id}_{\mathbb{R}_{\geq 0}})$ is well defined. We then define $\tau_{f}^{J_{1} \times \widehat{C}} \;:\!=\; \tau_{f}^{J_{1}} \times \tau_{f}^{\widehat{C}}$ . For $\mathcal{X}_{i} = (S_{i}, d_{i}, \rho_{i}, \mu_{i}, \pi_{i}) \in \mathbb{M}_{\mathrm L}^{\circ},\, i=1,2$ , we say that $\mathcal{X}_{1}$ is $(\tau^{J_{1} \times \widehat{C}})^{-1}$ -equivalent to $\mathcal{X}_{2}$ if and only if there exists a root-preserving isometry $f \colon S_{1} \to S_{2}$ such that $\mu_{2} = \mu_{1} \circ f^{-1}$ and $\pi_{2} = \pi_{1} \circ (\tau_{f}^{J_{1} \times \widehat{C}})^{-1}$ .

We equip a Polish topology on $\mathbb{M}_{\mathrm L}$ defined in [Reference Noda33, Definition 2.27 and Theorem 2.28]. In particular, the topology is characterized in terms of convergence, as follows.

Let us prepare to describe precompactness and tightness criteria. For $\xi \in D(\mathbb{R}_{\geq 0}, S)$ , where (S, d) is a metric space, we define

\begin{align*} \tilde{w}_{S}(\xi, h, t) \;:\!=\; \inf_{(I_{k}) \in \Pi_{t}} \max_{k} \sup\nolimits_{r, s \in I_{k}} d(\xi(r), \xi(s)), \quad t, h > 0,\end{align*}

where $\Pi_{t}$ denotes the set of partitions of the interval [0, t) into subintervals $I_{k} = [u, v)$ with $v - u \geq h$ when $v < t$ .

Although it is not a space of our main interest, we introduce another space $\mathbb{M}$ for convenience. This space is used in the proofs of our main results. Roughly speaking, it is the space of measured metric spaces equipped with laws of stochastic processes, and is precisely defined as follows. Let $\mathbb{M}^{\circ}$ be the collection of $(S, d, \rho, \mu, \pi')$ such that $(S, d, \rho, \mu) \in \mathbb{G}$ and $\pi'$ is a probability measure on $D(\mathbb{R}_{\geq 0}, S)$ . Recall $\tau^{J_{1}}$ from (2.4). For $\mathcal{X}_{i} = (S_{i}, d_{i}, \rho_{i}, \mu_{i}, \pi'_{i}) \in \mathbb{M}^{\circ}, i=1,2$ , we say that $\mathcal{X}_{1}$ is $(\tau^{J_{1}})^{-1}$ -equivalent to $\mathcal{X}_{2}$ if and only if there exists a root-preserving isometry $f \colon S_{1} \to S_{2}$ such that $\mu_{2} = \mu_{1} \circ f^{-1}$ and $\pi'_{2} = \pi'_{1} \circ (\tau_{f}^{J_{1}})^{-1}$ .

We equip $\mathbb{M}$ with the Polish topology defined in [Reference Noda33, Definition 2.33 and Theorem 2.34]. In particular, the topology is characterized in terms of convergence, as follows.

The following theorems are analogues of Theorems 2.6 and 2.7, respectively.

3.1. Preliminaries

In this subsection, we recall some basic properties of resistance forms and resistance metrics, starting with their definitions. The reader is referred to [Reference Kigami27] for further background.

For later use, we recall two basic results from [Reference Kajino23].

Lemma 3.1. ([Reference Kajino23, Corollary 2.39(3)].) Let $(\mathcal{E}, \mathcal{F})$ be a resistance form on a non-empty set F. Fix $f, g \in \mathcal{F}$ . Then $f \wedge g \in \mathcal{F}$ and

\begin{align*} \mathcal{E}(\,f \wedge g, f \wedge g) \leq \mathcal{E}(\,f, f) + \mathcal{E}(g, g). \end{align*}

The same results hold when $f \wedge g$ is replaced with $f \vee g$ .

Lemma 3.2 ([Reference Kajino23, Corollary 2.39(3)].) Let $(\mathcal{E}, \mathcal{F})$ be a resistance form on a non-empty set F. Then, for any bounded functions $u, v \in \mathcal{F}$ , $u \cdot v \in \mathcal{F}$ and

\begin{align*} \mathcal{E}(u \cdot v, u \cdot v)^{\frac{1}{2}} \leq \|u\|_{\infty} \mathcal{E}(v, v)^{\frac{1}{2}} + \|v\|_{\infty} \mathcal{E}(u, u)^{\frac{1}{2}}. \end{align*}

For the following definition, recall the effective resistance on an electrical network with a finite vertex set from [Reference Levin and Peres29, Subsection 9.4] (see also [Reference Kigami26, Subsection 2.1]).

In Assumptions 1.1(ii) and 1.2(ii), we consider effective resistance between sets. This is precisely defined below.

Definition 3.3. (Effective resistance between sets.) Fix a resistance form $(\mathcal{E}, \mathcal{F})$ on F and write R for the corresponding resistance metric. For sets $A, B \subseteq F$ , we define

\begin{align*} R(A,B) \;:\!=\; \left( \inf\{ \mathcal{E}(\,f,f)\,:\, f \in \mathcal{F},\ f|_{A}=1,\ f|_{B}=0 \} \right)^{-1}, \end{align*}

which is defined to be zero if the infimum is taken over the empty set. Note that by (RF4) we clearly have $R( \{x\}, \{y\}) = R(x,y)$ .

Fix a resistance form $(\mathcal{E}, \mathcal{F})$ on a non-empty set F and write R for the corresponding resistance metric. We equip F with the topology induced from R. Since the condition (RF4) implies that

(3.1) \begin{align} | f(x) - f(y) |^{2} \leq \mathcal{E}(\,f,f) R(x,y) \quad \mbox{for all } f \in \mathcal{F},\end{align}

$\mathcal{F}$ is a subset of $C(F, \mathbb{R})$ . We henceforth assume that (F, R) is locally compact and separable, and the resistance form $(\mathcal{E}, \mathcal{F})$ is regular, as described by the following definition.

We next introduce related Dirichlet forms and stochastic processes. First, suppose that we have a Radon measure $\mu$ of full support on F. Let $\mathcal{B}(F)$ be the Borel $\sigma$ -algebra on F and $\mathcal{B}^{\mu}(F)$ be the completion of $\mathcal{B}(F)$ with respect to $\mu$ .

We now define a bilinear form $\mathcal{E}_{1}$ on $\mathcal{F} \cap L^{2}(F, \mu)$ by setting

(3.2) \begin{align} \mathcal{E}_{1}(\,f,\, g) \;:\!=\; \mathcal{E}(\,f,\, g) + \int_{F}fg\, \mathrm{d}\mu.\end{align}

Then $(\mathcal{F} \cap L^{2}(F, \mu), \mathcal{E}_{1})$ is a Hilbert space (see [Reference Kigami26, Theorem 2.4.1]). The Dirichlet form $(\mathcal{E}, \mathcal{D})$ on $L^{2}(F, \mu)$ and the extended Dirichlet space $\mathcal{D}_{\mathrm{e}}$ associated with $(F, R, \mu)$ are given as follows.

Since we assume that the resistance form $(\mathcal{E},\mathcal{F})$ is regular, we have from [Reference Kigami27, Theorem 9.4] that the associated Dirichlet form $(\mathcal{E}, \mathcal{D})$ is regular (see [Reference Fukushima, Oshima, Takeda and Studies21] for the definition of a regular Dirichlet form). Moreover, standard theory gives us the existence of an associated Hunt process $((X_{t} )_{t \geq 0}, (\mathbb{P}_{x} )_{x \in F} )$ (see, e.g., [Reference Fukushima, Oshima, Takeda and Studies21, Theorem 7.2.1]), which we refer to as the Hunt process associated with $(F, R, \mu)$ . Note that such a process is, in general, specified uniquely only for starting points outside a set of zero capacity. However, in this setting, every point has strictly positive capacity (see [Reference Kigami27, Theorem 9.9]), and so the process is defined uniquely everywhere.

We now recall the definition of local times. Let $(\Omega, \mathscr{F})$ be the measurable space where the probability measures $(\mathbb{P}_{x})_{x \in F}$ are defined. We denote the minimum completed admissible filtration of the Hunt process X by $(\mathscr{F}_{t})_{t \geq 0}$ , the family of the translation (shift) operators for X by $(\theta_{t})_{t \geq 0}$ and the lifetime of X by $\zeta$ (see [Reference Fukushima, Oshima, Takeda and Studies21] for these definitions).

The metric-entropy condition (1.3) that is assumed for the spaces in $\check{\mathbb{F}}$ implies that the associated Hunt process admits a jointly continuous local time.

Proposition 3.1. ([Reference Noda33, p. 18 and Corollary 4.16].) In the above setting, the Hunt process X admits a jointly measurable local time $L=(L(x, t))_{x \in F,\, t \geq 0}$ satisfying the occupation density formula; that is, it holds that, for all $x \in F$ , $t \geq 0$ , and all non-negative measurable functions $f \colon F \to \mathbb{R}_{\geq 0}$ ,

(3.3) \begin{align} \int_{0}^{t} f(X_{s})\,\mathrm{d}s = \int_{F}f(y) L(y,t)\, \mu(\mathrm{d}y), \quad \mathbb{P}_{x} { \,almost\, surely.} \end{align}

Moreover, if the resistance metric space (F, R) is boundedly compact and satisfies (1.3) then the local time L can be chosen so that L is jointly continuous on $F \times \mathbb{R}_{\geq 0}$ , $\mathbb{P}_{x}$ almost surely for all $x \in F$ .

The recurrence and transience of the Dirichlet form $(\mathcal{E}, \mathcal{D})$ were studied in [Reference Croydon16], and we have the following characterization in terms of the resistance metric.

Lemma 3.3. ([Reference Croydon16, Lemma 2.3].) In the above setting, assume that (F, R) is boundedly compact. Then the associated regular Dirichlet form $(\mathcal{E}, \mathcal{D})$ is recurrent in the sense of [Reference Fukushima, Oshima, Takeda and Studies21, p. 55] if and only if

(3.4) \begin{align} \lim_{r \to \infty} R(\rho, B_{R}(\rho, r)^{c}) = \infty \end{align}

for some (or, equivalently, any) $\rho \in F$ .

By Lemma 3.3, the recurrence of the Dirichlet form $(\mathcal{E}, \mathcal{D})$ is independent of the measure $\mu$ and is characterized by the condition (3.4). Therefore, it is natural to introduce the notion of recurrent resistance forms and resistance metrics as follows.

Remark 3.2. Since we do not assume that (F, R) is boundedly compact in Definition 3.8, we use the sequence $(U_n)_{n \geq 1}$ instead of the balls $B_R(\rho, r)$ . It is straightforward to see that the definition does not depend on the particular choice of $(U_n)_{n \geq 1}$ or of $\rho$ . Namely, if $(\mathcal{E}, \mathcal{F})$ and R satisfy (RRF) then, for any increasing sequence $(\,\tilde{\!U}_n)_{n \geq 1}$ of relatively compact open subsets of F, we have

\begin{align*} \lim_{n \to \infty} R( \tilde{\rho}, \tilde{U}_n^c) = \infty \end{align*}

for any $\tilde{\rho} \in F$ .

Using the notion of recurrent resistance forms and resistance metrics introduced in Definition 3.8, we establish in Corollary 3.1 in the next subsection an extension of Lemma 3.3 to resistance metric spaces that are not necessarily boundedly compact.

3.2. Extended Dirichlet spaces of resistance forms

Recall from Subsection 3.1 that a resistance form equipped with a measure determines a Dirichlet form and an extended Dirichlet space. In this subsection, we prove that the extended Dirichlet space is independent of the measure and that the extended Dirichlet space coincides with the domain of the resistance form if and only if the resistance form is recurrent (see Theorems 3.2 and 3.3). This coincidence of domains will be useful for studying traces of resistance forms in Subsection 3.3.

Throughout this subsection, we fix a resistance form $(\mathcal{E}, \mathcal{F})$ on a non-empty set F, and write R for the associated resistance metric. Fix an element $x_{0} \in F$ and define an inner product $\mathcal{E}^{(1)}$ on $\mathcal{F}$ by setting

\begin{align*} \mathcal{E}^{(1)} (u,v) \;:\!=\; \mathcal{E}(u,v) + u(x_{0})v(x_{0}).\end{align*}

Note that the inner product $\mathcal{E}^{(1)}$ differs from $\mathcal{E}_1$ defined in (3.2). It is easy to check the following result with the use of (RF2).

Below, we verify that the topology on $\mathcal{F}^{(1)}$ induced by $\mathcal{E}^{(1)}$ is stronger than the compact-convergence topology.

Proof. Fix a non-empty compact subset $K \subseteq F$ . Using (3.1), we deduce that, for any $x \in K$ ,

\begin{align*} |u_{n}(x) - u(x)| \leq \sqrt{\mathcal{E}(u_{n}-u, u_{n}-u) R(x, x_{0})} + |u_{n}(x_{0}) - u(x_{0})|. \end{align*}

Since we have that $\sup_{x \in K} R(x, x_{0}) < \infty$ , we obtain the desired result.

In Theorem 3.2, we assume that the resistance form $(\mathcal{E}, \mathcal{F})$ is regular and that (F, R) is separable and locally compact. Let $\mu$ be a Radon measure on (F, R) of full support. We write $(\mathcal{E}, \mathcal{D})$ for the Dirichlet form associated with $(F, R, \mu)$ and $\mathcal{D}_{\mathrm{e}}$ for the extended Dirichlet space of $(\mathcal{E}, \mathcal{D})$ . From [Reference Fukushima, Oshima, Takeda and Studies21, Theorem 2.1.7] and [Reference Kigami27, Proposition 9.13], every function $u \in \mathcal{D}_{\mathrm{e}}$ has a continuous modification, and so we may regard $\mathcal{D}_{\mathrm{e}}$ as a subspace of $C(F, \mathbb{R})$ .

Proof. We first show $\mathcal{D}_{\mathrm{e}} \subseteq \mathcal{F}^{(1)}$ . Fix a continuous function $u \in \mathcal{D}_{\mathrm{e}}$ and let $(u_{n})_{n \geq 1}$ be an $\mathcal{E}$ -Cauchy sequence in $\mathcal{D}$ such that $u_{n}(x) \to u(x),$ $\mu$ -a.e. x. By the definition of $\mathcal{D}$ (see Definition 3.6), we may assume that $u_{n} \in C_{\mathrm c}(F, \mathbb{R}) \cap \mathcal{F}$ . Choose $x_{1} \in F$ such that $u_{n}(x_{1}) \to u(x_{1})$ . By (RF2), we can find $v \in \mathcal{F}$ such that

\begin{align*} \mathcal{E}(u_{n}-v, u_{n}-v) \to 0, \quad v(x_{1}) = u(x_{1}). \end{align*}

It is then the case that v is an element of $\mathcal{F}^{(1)}$ . Since we have from (3.1) that

\begin{align*} |u_{n}(x) - v(x)| \leq \sqrt{R(x,x_{1}) \mathcal{E}(u_{n} -v, u_{n} -v)} + |u_{n}(x_{1}) - v(x_{1})|, \end{align*}

it follows that $u_{n}(x) \to v(x)$ for all $x \in F$ . Therefore, $u = v$ , $\mu$ -a.e. The continuity of u and v and the fact that $\mu$ is of full support yield $u=v \in \mathcal{F}^{(1)}$ . The other inclusion $\mathcal{F}^{(1)} \subseteq \mathcal{D}_{\mathrm{e}}$ directly follows from Proposition 3.2.

By Theorem 3.2, the study of the extended Dirichlet space reduces to that of the resistance form. Henceforth, we return to the initial setting. In other words, $(\mathcal{E}, \mathcal{F})$ is simply a resistance form, which is not necessarily regular, and the corresponding resistance metric is not necessarily separable or locally compact.

Before proving the above theorem, we provide a corollary of this, which is an extension of Lemma 3.3 to the setting where the resistance metric space is not necessarily boundedly compact.

To prove Theorem 3.3, we introduce the notion of a Cesàro mean sequence. Given a sequence $(u_{n})_{n \geq 1}$ in $C(F, \mathbb{R})$ , the Cesàro mean sequence $(v_{n})_{n \geq 1}$ in $C(F, \mathbb{R})$ is defined by setting

\begin{align*} v_{n}(x) \;:\!=\; \frac{1}{n} \sum_{l =1}^{n} u_{l}(x).\end{align*}

As consequences of Proposition 3.3, we obtain two corollaries that are useful to approximate functions in $\mathcal{F}^{(1)}$ by more tractable functions.

Proof. Choose $u_{n} \in C_{\mathrm c}(F, \mathbb{R}) \cap \mathcal{F}$ such that $u_{n} \to u$ with respect to $\mathcal{E}^{(1)}$ . Define $\tilde{u}_{n} \in C_{\mathrm c}(F, \mathbb{R}) \cap \mathcal{F}$ by setting

\begin{align*} \tilde{u}_{n} \;:\!=\; \bigl( (\!- |u|) \vee u_n \bigr) \wedge |u|. \end{align*}

Clearly, $|\tilde{u}_n(x)| \leq |u(x)|$ for all $x \in F$ and $n \geq 1$ . We then have that $\tilde{u}_{n} \to u$ pointwise, and Lemma 3.1 yields that

\begin{align*} \sup\nolimits_{n} \mathcal{E}(\tilde{u}_{n}, \tilde{u}_{n}) \leq \sup\nolimits_{n} \mathcal{E}(u_{n}, u_{n}) + 2\, \mathcal{E}(|u|, |u|) < \infty. \end{align*}

Therefore, the desired result follows from Proposition 3.3.

We are ready to prove Theorem 3.3.

Proof of Theorem 3.3. Assume (i). Then, by definition, we can find an increasing sequence $(U_n)_{n \geq 1}$ of relatively compact open subsets and functions $\varphi_n \in \mathcal{F}$ such that $\bigcup_{n \geq 1} U_n = F$ , $\varphi_n(x_{0}) = 1$ , $\varphi_n|_{U_n^c} = 0$ , and $\mathcal{E}(\varphi_n, \varphi_n) \to 0$ as $n \to \infty$ . In particular, $\varphi_n \in C_{\mathrm c}(F, \mathbb{R})$ for all n, and $\mathcal{E}(1_{F} - \varphi_n, 1_{F} - \varphi_n) \to 0$ as $n \to \infty$ , which implies (ii).

Next, assume (ii). Using Corollary 3.2, we can find functions $\varphi_{n} \in C_{\mathrm c}(F, \mathbb{R}) \cap \mathcal{F}$ such that $\varphi_{n} \to 1_{F}$ with respect to $\mathcal{E}^{(1)}$ and $\| \varphi_{n} \|_{\infty} \leq 1$ . Fix $u \in \mathcal{F}$ . If u is bounded then Lemma 3.2 yields

\begin{align*} \mathcal{E}(u \cdot \varphi_{n}, u \cdot \varphi_{n}) \leq 2\|u\|_{\infty} \mathcal{E}(\varphi_{n}, \varphi_{n}) + 2\|\varphi_{n}\|_{\infty} \mathcal{E}(u, u). \end{align*}

The right-hand side of the above inequality is uniformly bounded; hence, we deduce that $u \in \mathcal{F}^{(1)}$ by Proposition 3.3. When u is not bounded, we use Corollary 3.3 and choose functions $u_{n} \in \mathcal{F}$ such that $u_{n} \to u$ with respect to $\mathcal{E}^{(1)}$ and $\|u_{n}\|_{\infty} < \infty$ . Since we have that $u_{n} \in \mathcal{F}^{(1)}$ and $\mathcal{F}^{(1)}$ is closed with respect to $\mathcal{E}^{(1)}$ (by its definition), we obtain $u \in \mathcal{F}^{(1)}$ . Therefore, we deduce that $\mathcal{F} \subseteq \mathcal{F}^{(1)}$ , which implies (iii).

Finally, assume (iii). It is then the case that $1_{F} \in \mathcal{F}^{(1)}$ , and hence it follows from the definition of $\mathcal{F}^{(1)}$ that there exists $\varphi_{n} \in C_{\mathrm c}(F, \mathbb{R}) \cap \mathcal{F}$ such that $\varphi_{n} \to 1_{F}$ with respect to $\mathcal{E}^{(1)}$ . If necessary, by considering sufficiently large n with $\varphi_n(x_0) > 0$ and replacing $\varphi_{n}$ by $\varphi_{n}/\varphi_{n}(x_{0})$ , we may assume that $\varphi_{n}(x_{0}) = 1$ for all n. By Proposition 3.3, we can find a subsequence $(\varphi_{n(k)})_{k \geq 1}$ whose Cesàro mean sequence $(\psi_k)_{k \geq 1}$ converges to $1_F$ with respect to $\mathcal{E}^{(1)}$ . Note that $\psi_k(x_0) = 1$ for all k. Write $U_k$ for the interior of the support of $\psi_k$ . Taking the Cesàro mean sequence ensures that the sequence $(U_k)_{k \geq 1}$ is increasing. We have from Proposition 3.2 that $\psi_k \to 1_F$ in the compact-convergence topology. This implies that $\bigcup_{k \geq 1} U_k = F$ . Moreover, by Definition3.3, we have that

\begin{align*} R(x_{0}, U_k^{c}) \geq \mathcal{E}(\psi_k, \psi_k)^{-1} \quad \mbox{for all } k \geq 1. \end{align*}

Since $\psi_k \to 1_F$ with respect to $\mathcal{E}$ and $\mathcal{E}(1_F, 1_F) = 0$ , we deduce that

\begin{align*} \lim_{k \to \infty} R(x_0, U_k^c) = \infty. \end{align*}

Therefore, we obtain (i).

As another corollary of Theorem 3.3 other than Corollary 3.1, we obtain the following.

Proof. Fix $u \in C_{\mathrm c}(F, \mathbb{R})$ and $\varepsilon > 0$ . It suffices to find a function $w \in C_{\mathrm c}(F, \mathbb{R}) \cap \mathcal{F}$ such that $\|u - w\|_\infty < \varepsilon$ . By Theorem 3.3, there exists a sequence $(\varphi_{n})_{n \geq 1}$ in $C_{\mathrm c}(F, \mathbb{R}) \cap \mathcal{F}$ such that $\varphi_{n} \to 1_{F}$ with respect to $\mathcal{E}^{(1)}$ . It follows from Proposition 3.2 that

\begin{align*} c_{n} \;:\!=\; \inf_{x \in \operatorname{supp}(u)} \varphi_{n}(x) > 0 \end{align*}

for all sufficiently large n, where $\operatorname{supp}(\!\cdot\!)$ denotes the support of functions. Fix such an n and write $K \;:\!=\; \operatorname{supp}(\varphi_{n})$ . Note that $\operatorname{supp}(u) \subseteq K$ . Define $\psi \in C_{\mathrm c}(F, \mathbb{R}) \cap \mathcal{F}$ by setting $\psi(x) \;:\!=\; 0 \vee (c_{n}^{-1} \varphi_{n}(x) \wedge 1)$ . Note that $\psi|_{\operatorname{supp}(u)} \equiv 1$ and $\| \psi \|_{\infty} \leq 1$ . A general version of the Stone–Weierstrass theorem (see [Reference Dudley19, Theorem 2.4.11]) yields that $\{v|_{K} \mid v \in C_{\mathrm c}(F, \mathbb{R}) \cap \mathcal{F} \}$ is dense in $C_{\mathrm c}(K, \mathbb{R})$ (see the proof of Theorem 6.3 in [Reference Kigami27]). Thus, we can find $v \in C_{\mathrm c}(F, \mathbb{R}) \cap \mathcal{F}$ satisfying $\sup_{x \in K} |u(x) - v(x)| < \varepsilon$ . Define $w \;:\!=\; v \cdot \psi$ , which belongs to $C_{\mathrm c}(F, \mathbb{R}) \cap \mathcal{F}$ by Lemma 6.5 in [Reference Kigami27]. For $x \in \operatorname{supp}(u)$ , we have that $|u(x) - w(x)| = |u(x) - v(x)| < \varepsilon$ . For $x \in K \setminus \operatorname{supp}(u)$ , since we have that $u(x) = 0$ , it follows that

\begin{align*} |u(x) - w(x)| = |u(x) \cdot \psi(x) - v(x) \cdot \psi(x)| \leq |u(x) - v(x)| < \varepsilon. \end{align*}

For $x \notin K$ , we have that $|u(x) - w(x)| = 0$ . Therefore, we deduce that $\| u-w \|_{\infty} < \varepsilon$ , which completes the proof.

3.3. Traces of resistance forms

In this subsection, we study extended Dirichlet spaces associated with traces of resistance forms introduced in [Reference Kigami27, Chapter 8]. In particular, in Theorem 3.5 we establish that processes associated with traces of resistance forms coincide with trace processes. Throughout this subsection, we fix a resistance form $(\mathcal{E}, \mathcal{F})$ on a non-empty set F, and write R for the corresponding resistance metric. Moreover, we fix a non-empty subset $B \subseteq F$ .

Definition 3.10. (The space $\mathcal{F}^{(1)}|_{B}$ .) We define a subspace of $C(B, \mathbb{R})$ by

\begin{align*} \mathcal{F}^{(1)}|_{B} \;:\!=\; \big\{ u|_{B} \mid u \in \mathcal{F}^{(1)} \big\}. \end{align*}

The following two assertions are proved in the same way as in [Reference Kajino23, Theorem 2.29] and hence we omit the proofs. (Indeed, replacing $\mathcal{F}$ by $\mathcal{F}^{(1)}$ in the proof works. See also [Reference Kigami27, Lemmas 8.2 and 8.5], where the separability of (F, R) is additionally assumed.)

Proposition 3.4. For each $\varphi \in \mathcal{F}^{(1)}|_{B}$ , there exists a unique function $h_{B}^{(1)}(\varphi) \in \mathcal{F}^{(1)}$ such that $h_{B}^{(1)}(\varphi)|_{B} = \varphi$ and

\begin{align*} \mathcal{E}\big(h_{B}^{(1)}(\varphi), h_{B}^{(1)}(\varphi)\big) = \inf\big\{ \mathcal{E}(u,u) \mid u \in \mathcal{F}^{(1)},\ u|_{B}=\varphi \big\}. \end{align*}

Henceforth, we assume that (F, R) is separable and locally compact, and $(\mathcal{E}, \mathcal{F})$ is regular. Fix a fully supported Radon measure $\mu$ on F, and write $(\mathcal{E}, \mathcal{D})$ and $\mathcal{D}_{\mathrm{e}}$ for the associated Dirichlet form and the extended Dirichlet space, respectively. Moreover, we suppose that we have another Radon measure $\nu$ on F such that the (topological) support of $\nu$ is B. Note that this implies that B is closed. Set

\begin{align*} &\check{\mathcal{D}}_{\mathrm{e}}^{\nu} \;:\!=\; \{ \varphi \in \mathcal{L}(B, \nu) \mid \mbox{ there exists } u \in \mathcal{D}_{\mathrm{e}} \ \text{such that}\ \varphi=u|_{B}, \ \nu \text{-a.e.} \}, \\ &\qquad\qquad\qquad\quad\qquad \check{\mathcal{D}}^{\nu} \;:\!=\; \check{\mathcal{D}}_{\mathrm{e}}^{\nu} \cap L^{2}(B, \nu),\end{align*}

where we recall the space $\mathcal{L}(B, \nu)$ from Definition 3.5.

Write $X = ((X_{t})_{t \geq 0}, (\mathbb{P}_{x})_{x \in F})$ for the Hunt process associated with $(\mathcal{E}, \mathcal{D})$ . Let $\sigma_{B}$ denote the hitting time of B, i.e.

(3.5) \begin{align} \sigma_{B} \;:\!=\; \inf\{ t > 0 \mid X_{t} \in B \}.\end{align}

For $\varphi \in \check{\mathcal{D}}_{\mathrm{e}}^{\nu}$ , its harmonic extension via the process X is defined by

\begin{align*} \check{h}_{B}(\varphi)(x) \;:\!=\; \mathbb{E}_{x} \big[u(X_{\sigma_{B}}) \cdot 1_{\{\sigma_{B} < \infty\}}\big],\end{align*}

where we choose $u \in \mathcal{D}_{\mathrm{e}}$ such that $u|_{B}=\varphi,\, \nu$ -a.e. Note that $\check{h}_{B}(\varphi)$ is independent of the choice of u (see [Reference Fukushima, Oshima, Takeda and Studies21, Lemma 6.2.1]). The following result says that the harmonic extension $\check{h}_{B}(\varphi)$ via the associated process X coincides with the harmonic extension $h_{B}^{(1)}(\varphi)$ as the energy minimizer over $\mathcal{F}^{(1)}$ defined in Proposition 3.4.

For $\varphi, \psi \in \check{\mathcal{D}}_{\mathrm{e}}^{\nu}$ , set

\begin{align*} \check{\mathcal{E}}^{\nu}(\varphi, \psi) \;:\!=\; \mathcal{E}(\check{h}_{B}(\varphi), \check{h}_{B}(\psi)).\end{align*}

Note that $(\check{\mathcal{E}}^{\nu}, \check{\mathcal{D}}^{\nu})$ is the trace of $(\mathcal{E}, \mathcal{D})$ on B with respect to $\nu$ (see [Reference Fukushima, Oshima, Takeda and Studies21, Subsection 6.2]). Define a PCAF $A=(A_{t})_{t \geq 0}$ and its right-continuous inverse $\tau=(\tau(t))_{t \geq 0}$ by setting

\begin{align*} A_{t} \;:\!=\; \int_B L(x,t)\, \nu(\mathrm{d}x), \qquad \tau(t) \;:\!=\; \inf\{ s > 0 \,:\, A_{s} > t \},\end{align*}

where $(L(x,t))_{x \in F, t \geq 0}$ is the jointly measurable local time of X satisfying the occupation density formula (3.3). The trace $\check{X}^{\nu}=(\check{X}^{\nu}_{t})_{t \geq 0}$ of X on B (with respect to $\nu$ ) is defined by setting $\check{X}^{\nu}_{t} \;:\!=\; X_{\tau(t)}$ . Note that $(\check{X}^{\nu}, (\mathbb{P}_{x})_{x \in B})$ is a strong Markov process (see [Reference Fukushima, Oshima, Takeda and Studies21, Theorem A.2.12]).

The following is the main result of this section. The assertion is almost the same as in Theorem 2.5 and Lemma 2.6 in [Reference Croydon, Hambly and Kumagai18], but it is new that we do not assume that B is compact.

Theorem 3.5. Assume that the resistance form $(\mathcal{E}, \mathcal{F})$ is recurrent. Set

\begin{align*} \mathcal{F}|_{B} &\;:\!=\; \{ u|_{B} \mid u \in \mathcal{F} \}, \\ \mathcal{E}|_{B}(\varphi, \psi) &\;:\!=\; \mathcal{E}\big(h_{B}^{(1)}(\varphi), h_{B}^{(1)}(\psi)\big), \qquad \varphi, \psi \in \mathcal{F}|_{B}. \end{align*}

Then $(\mathcal{E}|_{B}, \mathcal{F}|_{B})$ is a recurrent resistance form, and the corresponding resistance metric is $R|_{B \times B}$ . The process associated with $(B, R|_{B \times B}, \nu)$ is $\check{X}^{\nu}$ .

Proof. By Remark 3.5 and Theorem 8.4 in [Reference Kigami27], we have that $(\mathcal{F}|_{B}, \mathcal{E}|_{B})$ is a regular resistance form and the corresponding resistance metric is $R|_{B \times B}$ . Moreover, by use of the recurrence of $(\mathcal{E}, \mathcal{F})$ , it is straightforward to verify the recurrence of $(\mathcal{E}|_B, \mathcal{F}|_B)$ . This proves the first assertion.

For the second assertion, note that Theorem 3.3 yields that $\mathcal{F}^{(1)} = \mathcal{F}$ . Combining this with Lemmas 3.5 and 3.6, we deduce that the extended Dirichlet space of $\check{X}^{\nu}$ is $\mathcal{F}|_{B}$ . Moreover, by Theorem 3.4, we have that $\mathcal{E}|_{B}=\check{\mathcal{E}}^{\nu}$ on $\mathcal{F}|_{B}$ . On the other hand, from Theorem 3.2, we have that $(\mathcal{F}|_{B})^{(1)}$ is the extended Dirichlet space for the process associated with the tuple $(B, R|_{B \times B}, \nu)$ . Therefore, the last assertion follows by proving that $(\mathcal{F}|_{B})^{(1)} = \mathcal{F}|_{B}$ , as this implies the coincidence of the Dirichlet forms of the processes.

By definition, we have $(\mathcal{F}|_{B})^{(1)} \subseteq \mathcal{F}|_{B}$ . Fix $\varphi \in \mathcal{F}|_{B}$ and choose $u \in \mathcal{F}$ such that $u|_{B} = \varphi$ . By Theorem 3.3, there exists a sequence $(u_{n})_{n \geq 1}$ in $C_{\mathrm c}(F, \mathbb{R}) \cap \mathcal{F}$ such that $u_{n} \to u$ with respect to $\mathcal{E}^{(1)}$ . Set $\varphi_{n} \;:\!=\; u_{n}|_{B}$ . It is then the case that $\varphi_{n} \in C_{\mathrm c}(B, \mathbb{R}) \cap \mathcal{F}|_{B}$ . Obviously, $\varphi_{n} \to \varphi$ pointwise (on B). Moreover, we deduce that

\begin{align*} \mathcal{E}|_{B}(\varphi_{n} - \varphi, \varphi_{n} - \varphi) &= \mathcal{E}\big(h_{B}^{(1)}(\varphi_{n} - \varphi), h_{B}^{(1)}(\varphi_{n} - \varphi)\big)\\ &= \inf\{ \mathcal{E}(w,w) \mid w \in \mathcal{F}\ \text{such that}\ w|_{B}=\varphi_{n} - \varphi \}\\ &\leq \mathcal{E}(u_{n}- u, u_{n}-u), \end{align*}

which implies that $\varphi_{n} \to \varphi$ with respect to $\mathcal{E}|_{B}$ . Therefore, it follows that $\varphi \in (\mathcal{F}|_{B})^{(1)}$ and hence $(\mathcal{F}|_{B})^{(1)} = \mathcal{F}|_{B}$ , which completes the proof.

The resistance form $(\mathcal{E}|_B, \mathcal{F}|_B)$ defined above is called the trace of $(\mathcal{E}, \mathcal{F})$ onto B.

4.1. Resistance forms of electrical networks

In this subsection, we study some basic properties of resistance forms associated with electrical networks. Recall the definition of electrical networks G and related bilinear forms $(\mathcal{E}_G, \mathcal{F}_G)$ from Definitions 1.4 and 1.5.

Proof. The conditions (RF1), (RF2), and (RF5) can be checked similarly to the proofs of Proposition 1.21 and Lemma 1.27 in [Reference Barlow8]. Since we have that

(4.1) \begin{align} \mathcal{E}_{G}\big(1_{\{x\}}, 1_{\{x\}}\big) = \frac{1}{2} \sum_{z, w \in V_{G}} c_{G}(z, w) \big(1_{\{x\}}(z) - 1_{\{x\}}(w)\big)^{2} = \sum_{z \in V_{G}} c_{G}(x, z) = c_{G}(x) < \infty, \end{align}

we deduce that $1_{\{x\}} \in \mathcal{F}_{G}$ , which implies (RF3). Using Rayleigh’s monotonicity principle (see [Reference Levin and Peres29, Theorem 9.12]) and the fact that the electrical network is connected, we can verify that $R_G(x,y) \leq d(x,y)$ , where d(x, y) is the shortest path distance on a weighted graph $(V_{G}, E_{G}, (c_{G}(z,w)^{-1})_{\{z,w\} \in E_{G}})$ , which implies (RF4). By (4.1) and (RF4), we have that, for any $x \neq y$ ,

\begin{align*} R_{G}(x, y) \geq \mathcal{E}_G\big(1_{\{x\}}, 1_{\{x\}}\big)^{-1} = c_{G}(x)^{-1} > 0. \end{align*}

Therefore, the topology on $V_G$ induced from $R_{G}$ is the discrete topology. In particular, a subset K of $V_{G}$ is compact if and only if $|K| < \infty$ . Since $1_{\{x\}} \in \mathcal{F}_{G}$ for each $x \in V_{G}$ , it follows that $1_{K} \in \mathcal{F}_{G}$ for any compact subset K of $V_{G}$ . Therefore, using Theorem 6.3 in [Reference Kigami27], we deduce that $(\mathcal{E}_{G}, \mathcal{F}_{G})$ is regular.

Below, we verify that processes associated with resistance forms on electrical networks coincide with naturally associated reversible Markov chains on them.

Proof. Note that it is assumed in the definition of Hunt processes that $(X_{t})_{t \geq 0}$ is minimal. Let $Q=(q_{xy})_{x,y \in V_{G}}$ be the Q-matrix of the Hunt process. By direct calculations, we deduce that

\begin{align*} \mathcal{E}_{G}(1_{\{x\}}, 1_{\{y\}}) = \bigg\{\begin{array}{l@{\quad}l} -c_{G}(x,y), &x \neq y, \\[3pt] c_{G}(x), &x = y. \end{array} \end{align*}

Let $(\mathcal{E}_{G}, \mathcal{D}_{G})$ be the Dirichlet form determined by the resistance form $(\mathcal{E}_{G}, \mathcal{F}_{G})$ and the measure $\nu$ . We write $(T_t)_{t > 0}$ for the associated semigroup, i.e. $T_{t}f(x) \;:\!=\; \mathbb{E}_{x}\, f(X_{t})$ . Using the approximating forms of $(\mathcal{E}_{G}, \mathcal{D}_{G})$ (see [Reference Fukushima, Oshima, Takeda and Studies21, Lemma 1.3.4]), we obtain

\begin{align*} \mathcal{E}_{G}(1_{\{x\}}, 1_{\{y\}}) &= \lim_{t \downarrow 0} \frac{1}{t} \sum_{z \in V_{G}} (1_{\{x\}}(z) - T_{t}1_{\{x\}}(z)) \cdot 1_{\{y\}}(z) \nu(\{z\})\\ &= \lim_{t \downarrow 0} \frac{1_{\{x\}}(y)-T_{t}1_{\{x\}}(y)}{t} \nu(\{y\})\\ &= \lim_{t \downarrow 0} \frac{T_{0}1_{\{x\}}(y)-T_{t}1_{\{x\}}(y)}{t} \nu(\{y\})\\ &= -q_{yx} \nu(\{y\}), \end{align*}

where we use the backward equation for X in the last equality (see [Reference Kallenberg24, Theorem 13.9]). The desired result is straightforward.

Let G be an electrical network. Define $X_{G} = (X_{G}(t))_{t \geq 0}$ to be the Hunt process associated with $(V_{G}, R_{G}, \mu_{G})$ , where we recall that $\mu_G$ denotes the associated conductance measure. By Theorem 4.2, $X_{G}$ is the constant-speed random walk on G. In this setting, we have an explicit formula for the local time $L_{G} = (L_{G}(x,t))_{t \geq 0, x \in V_{G}}$ of $X_{G}$ as follows:

\begin{align*} L_{G}(x, t) = \frac{1}{c_{G}(x)} \int_{0}^{t} 1_{ \{x\} } (X_G(s))\, \mathrm{d}s.\end{align*}

Recall from Section 1 that $Y_{G}=(Y_{G}(k))_{k \geq 0}$ denotes the discrete-time Markov chain on G. For convenience, in the following discussions, we always suppose that $Y_{G}$ is defined on the same probability measure space as $X_{G}$ via the following relation:

(4.2) \begin{align} Y_{G}(k) = X_{G}\big(J_{G}^{(k)}\big),\end{align}

where $(J_{G}^{(k)})_{k \geq 0}$ is the sequence of jump times of $X_{G}$ with $J_{G}^{(0)} \;:\!=\; 0$ .

4.2. Traces of electrical networks

In this subsection, we introduce the notion of traces of electrical networks, which will be used in the proof of the main results of this paper to approximate Markov chains on infinite electrical networks by Markov chains on finite electrical networks.

Throughout this subsection, we fix an electrical network G such that the associated resistance form $(\mathcal{E}_{G}, \mathcal{F}_{G})$ is recurrent. We define the associated Laplacian $\mathcal{L}_{G}$ by setting

\begin{align*} \mathcal{L}_{G}\,f(x) \;:\!=\; \sum_{y \in V_{G}} P_{G}(x,y)f(y) - f(x) = \mathbb{E}_{x}^{G}\,f(Y_{G}(1)) - f(x)\end{align*}

for each function $f \colon V_G \to \mathbb{R}$ satisfying $\mathbb{E}_{x}^{G}|f(Y_{G}(1))| < \infty$ for all $x \in V_{G}$ . In particular, for any bounded function f, $\mathcal{L}_{G}f$ is defined.

If $\int_{V_{G}} |\mathcal{L}_{G}u| |v|\, \mathrm{d}\mu_{G} < \infty$ then it holds that

\begin{align*} \mathcal{E}_{G}(u, v) = - \int_{V_{G}} \mathcal{L}_{G}u \cdot v\, \mathrm{d}\mu_{G}. \end{align*}

Proof. Note that if v is compactly supported then the assertion follows from Theorem 1.24 in [Reference Barlow8]. Assume that $\| v \|_{\infty} < \infty$ . Since the resistance form $(\mathcal{E}_G, \mathcal{F}_G)$ is assumed to be recurrent, we have from Theorem 3.3 that $\mathcal{F}_G^{(1)} = \mathcal{F}_G$ . Thus, we can use Corollary 3.2 to find a sequence $v_{n}$ in $C_{\mathrm c}(V_{G}, \mathbb{R}) \cap \mathcal{F}_{G}$ such that $v_{n} \to v$ with respect to $\mathcal{E}_{G}^{(1)}$ and $|v_n(x) |\leq |v(x)|$ for all $x \in V_G$ and $n \geq 1$ . Since each $v_n$ is compactly supported, we have that

\begin{align*} \mathcal{E}_{G}(u, v_{n}) = - \int_{V_{G}} \mathcal{L}_{G}u \cdot v_{n} \, \mathrm{d}\mu_{G}. \end{align*}

Letting $n \to \infty$ in the above equality and using the dominated convergence theorem, we deduce the desired result. In the case that $\| v\|_{\infty}=\infty$ , by Corollary 3.3, the same result is verified.

Fix a non-empty subset $B \subseteq V_{G}$ . We now introduce the notion of trace of electrical networks. Recall the definition of the resistance form $(\mathcal{E}_{G}|_{B}, \mathcal{F}_{G}|_{B})$ from Theorem 3.5. Set $V(G|_{B}) \;:\!=\; B$ and, for $x,y \in V(G|_{B})$ ,

\begin{align*} c_{G|_{B}}(x,y) \;:\!=\; \begin{cases} -\mathcal{E}_{G}|_{B}(1_{\{x\}}, 1_{\{y\}}), &x \neq y,\\ 0, &x = y. \end{cases}\end{align*}

Following the argument in the proof of Theorem A.33 in [Reference Barlow8], we can readily verify that $c_{G|_B}(x,y)$ is non-negative. We define the edge set $E(G|_{B})$ on $V(G|_{B})$ by declaring $\{x,y\}$ is an edge if and only if $c_{G|_{B}}(x,y) > 0$ .

To prove Theorem 4.3, we first establish an explicit formula for the conductances on the trace. We define $T_B$ and $T_B^{+}$ to be the first hitting time and return time of $Y_{G}$ to B, i.e.

(4.3) \begin{align} T_{B} \;:\!=\; \{ n \geq 0 \mid Y_{G}(n) \in B \}, \quad T_{B}^{+} \;:\!=\; \{ n > 0 \mid Y_{G}(n) \in B \}.\end{align}

Note that both are finite almost surely. Indeed, since we assume that the associated resistance form $(\mathcal{E}_{G}, \mathcal{F}_{G})$ is recurrent, the Dirichlet form of $X_G$ is recurrent by Lemma 3.3. Moreover, since the graph G is connected, the Dirichlet form is irreducible in the sense of [Reference Fukushima, Oshima, Takeda and Studies21, p. 55]. Consequently, by Theorem 4.7.1(iii) in [Reference Fukushima, Oshima, Takeda and Studies21], we deduce that the jump chain $Y_G$ visits B infinitely often almost surely.

Theorem 4.4. For any $x,y \in B$ ,

\begin{align*} -\mathcal{E}_G|_B(1_{\{x\}}, 1_{\{y\}}) = \begin{cases} c_{G}(x) \mathbb{P}_{x}^{G} (Y_{G}(T_{B}^{+})=y), & x \neq y,\\ -c_G(x) \mathbb{P}_x^G(Y_G(T_B^+) \in B \setminus \{x\}), & x = y. \end{cases} \end{align*}

In particular, if $x \neq y$ then

\begin{align*} c_{G|_B}(x,y) = c_{G}(x) \mathbb{P}_{x}^{G} (Y_{G}(T_{B}^{+})=y). \end{align*}

Proof. Fix $x, y \in B$ . Recall that $X_G$ denotes the constant-speed random walk on G and $\sigma_B$ denotes the first hitting time of B by $X_G$ . Since we have that $X_{G}(\sigma_{B}) = Y_{G}(T_{B})$ , by using Theorem 3.4, we deduce that, for each $z \in B$ ,

\begin{align*} h_{B}^{(1)}(1_{\{x\}})(z) = \mathbb{P}_{z}^{G}( Y_{G}(T_{B}) = x ) = \begin{cases} 0, & z \in B\setminus \{x\}, \\ 1, & z =x. \end{cases} \end{align*}

The Markov property yields that, for any $z \in V_{G}$ ,

\begin{align*} \sum_{w \in V_{G}} P_{G}(z,w) h_{B}^{(1)}(1_{\{y\}})(w) &= \sum_{w \in V_{G}} P_{G}(z,w) \mathbb{P}_{w}^{G}(Y_{G}(T_{B}) = y) \\ &= \mathbb{P}_{z}^{G}(Y_{G}(T_{B}^{+}) = y), \end{align*}

which implies that

\begin{align*} \mathcal{L}_{G}(h_{B}^{(1)}(1_{\{y\}})) (z) = \begin{cases} 0, & z \notin B, \\ \mathbb{P}_{z}^{G}(Y_{G}(T_{B}^{+}) = y), & z \in B \setminus \{y\},\\ -\mathbb{P}_y^G(Y_G(T_B^+) \in B \setminus \{y\}), & z = y. \end{cases} \end{align*}

From Proposition 4.1, it follows that

\begin{align*} - \mathcal{E}_{G}\big(h_{B}^{(1)}(1_{\{y\}}), h_{B}^{(1)}(1_{\{x\}})\big) = \mathcal{L}_{G}\big(h_{B}^{(1)}(1_{\{y\}})\big)(x)\, c_{G}(x). \end{align*}

Now, the result is immediate.

Corollary 4.1. For any $x \in B$ , it holds that

\begin{align*} 0 \leq c_{G}(x) - c_{G|_B}(x) = c_{G}(x) \mathbb{P}_{x}^{G} (Y_{G}(T_{B}^{+})=x) \leq \sum_{y \notin B} c_{G}(x,y). \end{align*}

Proof. By Theorem 4.4, we obtain

\begin{align*} 0 \leq c_{G}(x) - c_{G|_B}(x) &= c_{G}(x) \Bigg( 1 - \sum_{y \in B \setminus \{x\}} \mathbb{P}_{x}^{G} (Y_{G}(T_{B}^{+})=y) \Bigg) \\ &= c_{G}(x) \mathbb{P}_{x}^{G} (Y_{G}(T_{B}^{+})=x) \\ &\leq c_{G}(x) \sum_{y \notin B} P_{G}(x,y) \mathbb{P}_{y}^{G} (Y_{G}(T_{B})=x) \\ &\leq \sum_{y \notin B} c_{G}(x,y), \end{align*}

where we use the Markov property in the third line.

Proof of Theorem 4.3. By Theorem 4.4 and Corollary 4.1, we deduce that the graph $(V(G|_B), E(G|_B))$ is connected and $c_{G|_B}(x) < \infty$ for all $x \in B$ . Thus, $(V(G|_B), E(G|_B), c_{G|_B})$ is an electrical network. Write $(\mathcal{E}_{G|_B}, \mathcal{F}_{G|_B})$ for the associated resistance form; that is, for any functions $f, g \colon B \to \mathbb{R}$ ,

\begin{align*} \mathcal{E}_{G|_B}(\,f, g) \;:\!=\; \frac{1}{2} \sum_{x, y \in B} c_{G|_B}(x,y) (\,f(x) - f(y)) (g(x) - g(y)) \end{align*}

(if the right-hand side exists), and $\mathcal{F}_{G|_B} \;:\!=\; \{f \mid \mathcal{E}_{G|_B}(\,f,f) < \infty\}$ .

By Theorem 4.4, we have that

\begin{align*} \mathcal{E}_G|_B(1_{\{x\}}, 1_{\{x\}}) = \sum_{y \in B} c_{G|_B}(x,y) \quad \mbox{for all } x \in B. \end{align*}

We deduce from the above identity that, for any function $f \colon B \to \mathbb{R}$ supported on a finite subset $B_0 \subseteq B$ ,

\begin{align*} \mathcal{E}_G|_B(\,f, f) &= \sum_{x,y \in B_0} f(x)\, f(y)\, \mathcal{E}_G|_B(1_{\{x\}}, 1_{\{y\}})\\ &= \sum_{x,y \in B} f(x)\, f(y)\, \mathcal{E}_G|_B(1_{\{x\}}, 1_{\{y\}})\\ &= \sum_{x \in B} f(x)^2 \sum_{y \in B} c_{G|_B}(x,y) - \sum_{x,y \in B} f(x)\, f(y)\, c_{G|_B}(x, y) \\ &= \frac{1}{2} \sum_{x, y \in B} c_{G|_B}(x,y) (\,f(x) - f(y))^2\\ &= \mathcal{E}_{G|_B}(\,f, f). \end{align*}

Thus,

\begin{align*} \mathcal{E}_G|_B(\,f, f) = \mathcal{E}_{G|_B}(\,f, f) \end{align*}

for all functions $f \in C_{\mathrm c}(B, \mathbb{R})$ . (Note that the space B is equipped with the discrete topology, and so every $f \in C_{\mathrm c}(B, \mathbb{R})$ is supported on a finite subset.) Combining this with the recurrence of $(\mathcal{E}_G|_B, \mathcal{F}_G|_B)$ , we find that $(\mathcal{E}_{G|_B}, \mathcal{F}_{G|_B})$ is also recurrent. Thus, by Theorem 3.3, the domains $\mathcal{F}_G|_B$ and $\mathcal{F}_{G|_B}$ are the closures of $C_{\mathrm c}(B, \mathbb{R})$ with respect to $\mathcal{E}_G|_B^{(1)}$ and $\mathcal{E}_{G|_B}^{(1)}$ , respectively. However, the coincidence of the forms on $C_{\mathrm c}(B, \mathbb{R})$ implies that $(\mathcal{E}_G|_B, \mathcal{F}_G|_B) = (\mathcal{E}_{G|_B}, \mathcal{F}_G|_B)$ . This completes the proof.

We now provide a coupling of the discrete-time Markov chains $Y_{G|_B}$ on $G|_B$ and $Y_{G}$ on G. Set $\tilde{T}_{B}^{(0)} \;:\!=\; 0$ and inductively for each $k \geq 0$

\begin{align*} \tilde{T}_{B}^{(k+1)} \;:\!=\; \inf \left\{ n \geq \tilde{T}_{B}^{(k)} + 1\, \middle |\, Y_{G}(n) \in B \setminus \big\{ Y_{G}\big(\tilde{T}_{B}^{(k)}\big)\big\} \right\}.\end{align*}

If the infimum is taken over the empty set then we set $\tilde{T}_{B}^{(k+1)} \;:\!=\; \tilde{T}_{B}^{(k)}$ .

Theorem 4.5. It holds that, for any $\rho \in B$ ,

\begin{align*} \mathbb{P}^{G}_{\rho} \left( \operatorname{tr}_{B} Y_{G} \in \cdot \right) = \mathbb{P}^{G|_B}_{\rho} \left( Y_{G|_B} \in \cdot \right) \end{align*}

as probability measures on $D(\mathbb{R}_{\geq 0}, B)$ .

Proof. If $|B| =1$ then the assertion is obvious. We assume that there are at least two vertices in B. The strong Markov property of $Y_{G}$ yields that $\left( \operatorname{tr}_{B} Y_{G}(k) \right)_{k \geq 0}$ is a Markov chain on B. Thus, it suffices to show that

\begin{align*} \mathbb{P}_{\rho}^{G} \left( Y_{G}(\tilde{T}_{B}^{(1)}) = x \right) = \frac{c_{G|_B}(\rho, x)}{c_{G|_B}(\rho)} \quad \mbox{for all } x \in B \setminus \{\rho\}. \end{align*}

Define $T_{B}^{(0)} \;:\!=\; 0$ and inductively for each $k \geq 0$

\begin{align*} T_{B}^{(k+1)} \;:\!=\; \inf \left\{ n \geq T_{B}^{(k)} + 1 \mid Y_{G}(n) \in B \right\}\!. \end{align*}

Fix $x \in B \setminus \{\rho\}$ . By the strong Markov property, we deduce that

\begin{align*} \mathbb{P}_{\rho}^{G} \left( Y_{G}(\tilde{T}_{B}^{(1)}) = x \right) &= \sum_{k=1}^{\infty} \mathbb{P}_{\rho}^{G} \left( Y_{G}(T_{B}^{(1)}) = \rho, \ldots, Y_{G}(T_{B}^{(k-1)}) = \rho, Y_{G}(T_{B}^{(k)}) = x \right)\\ &= \sum_{k=1}^{\infty} \mathbb{P}_{\rho}^{G} \left( Y_{G}(T_{B}^{+}) = \rho \right)^{k-1} \mathbb{P}_{\rho}^{G} \left( Y_{G}(T_{B}^{+}) = x \right)\\ &= \frac{\mathbb{P}_{\rho}^{G} \left( Y_{G}(T_{B}^{+}) = x \right)} { 1 - \mathbb{P}_{\rho}^{G} \left( Y_{G}(T_{B}^{+}) = \rho \right)} \\ &= \frac{c_{G|_B}(\rho, x)}{c_{G|_B}(\rho)}, \end{align*}

where we use Theorem 4.4 to obtain the last equality.

4.3. Measurability

This subsection is devoted to a measurability problem that we face when dealing with random electrical networks. Recall the space $\mathbb{F}$ from Definition 1.1. Define a subspace of $\mathbb{F}$ consisting of rooted-and-measured resistance metric spaces associated with electrical networks by setting

\begin{align*} \mathbb{F}^{E} \;:\!=\; \left\{ (V_{G}, R_{G}, \rho_{G}, \mu_{G}) \in \mathbb{F} \mid G\ \text{is a rooted electrical network} \right\}.\end{align*}

Given $(V_{G}, R_{G}, \rho_{G}, \mu_{G}) \in \mathbb{F}^{E}$ , we write

\begin{align*} \mathbb{P}_{G} (\!\cdot\!) \;:\!=\; \mathbb{P}_{\rho_{G}}^{G} \left( ( Y_{G}, \ell_{G} ) \in \cdot \right), \quad \mathcal{Y}_{G} \;:\!=\; (V_{G}, R_{G}, \rho_{G}, \mu_{G}, \mathbb{P}_{G}),\end{align*}

where we recall from (1.6) that $\ell_G$ denotes the local time of $Y_G$ . Note that $\mathbb{P}_{G}$ is a probability measure on $D(\mathbb{R}_{\geq 0}, V_{G}) \times C(V_{G} \times \mathbb{R}_{\geq 0}, \mathbb{R})$ and $\mathcal{Y}_{G}$ is an element of $\mathbb{M}_{\mathrm L}$ (recall the space $\mathbb{M}_{\mathrm L}$ from Subsection 2.2). The aim of this subsection is to prove that $\mathcal{Y}_G$ is measurable with respect to G. This is verified in Corollary 4.2, through approximation of $\mathcal{Y}_G$ by traces.

We first recall that, on finite electrical networks, convergence of resistance metrics and conductances are equivalent.

Lemma 4.1. ([Reference Cao13, Lemma 2.8].) Fix a non-empty finite set V. Define

\begin{align*} \mathscr{R}\,(V) &\;:\!=\; \left\{ (R_{G}(x,y))_{x,y \in V} \in \mathbb{R}^{V \times V} \mid G\ \textit{is an electrical network with}\ V_{G} = V \right\}, \\ \mathscr{C}\,(V) &\;:\!=\; \left\{ (c_{G}(x, y))_{x,y \in V} \in \mathbb{R}^{V \times V} \mid G\ \textit{is an electrical network with}\ V_{G} = V \right\}. \end{align*}

We equip both $\mathscr{R}\,(V)$ and $\mathscr{C}\,(V)$ with the Euclidean topology induced from $\mathbb{R}^{V \times V}$ . Then the map $\mathscr{R}(V) \ni (R_{G}(x,y))_{x,y \in V} \mapsto (c_{G}(x,y))_{x, y \in V} \in \mathscr{C}(V)$ is a homeomorphism (note that the map is well defined by Theorem 3.1).

Using Lemma 4.1, we derive a simplified version of Theorem 1.1; that is, we show that convergence of finite electrical networks implies the convergence of the associated discrete-time Markov chains and their local times, under uniform finiteness of the conductances.

Proposition 4.2. For each $n \in \mathbb{N}$ , let $G_{n}$ be a rooted electrical network with a finite vertex set. Assume that $(V_{G_{n}}, R_{G_{n}}, \rho_{G_{n}})$ converges to $(V_{G}, R_{G}, \rho_{G})$ in the pointed Gromov–Hausdorff topology for some rooted electrical network G with $|V_{G}| < \infty$ . Furthermore, assume that

\begin{align*} \sup\nolimits_{n} \sup\nolimits_{x \in V_{G_{n}}} c_{G_{n}}(x) < \infty. \end{align*}

Then it holds that $\mathcal{Y}_{G_{n}} \to \mathcal{Y}_{G}$ in $\mathbb{M}_{\mathrm L}$ .

Proof. We may assume that $(V_{G_{n}}, R_{G_{n}}, \rho_{G_{n}})$ and $(V_{G}, R_{G}, \rho_{G})$ are embedded into a common compact metric space $(M, d^{M})$ in such a way that $V_{G_{n}} \to V_{G}$ in the Hausdorff topology and $\rho_{G_{n}} \to \rho_{G}$ (as objects embedded into M). Recall the metric entropy from Definition 1.2. By Theorem 4.15 in [Reference Noda34], for all but countably many $\delta$ , we have that

(4.4) \begin{align} \lim_{n \to \infty} N_{R_{G_{n}}}(V_{G_{n}}, \delta) = N_{R_{G}}(V_{G}, \delta). \end{align}

We choose $\delta$ sufficiently small so that $N_{R_{G}}(V_{G}, \delta) = |V_{G}|$ , which is possible by the finiteness of $V_G$ . From (4.4), it follows that $N_{R_{G_{n}}}(V_{G_{n}}, \delta) = |V_{G}|$ for all sufficiently large n, which implies that $\liminf_{n \to \infty} | V_{G_{n}} | \geq |V_{G}|$ . Assume that

\begin{align*} \limsup_{n \to \infty} | V_{G_{n}} | > |V_{G}|. \end{align*}

Then, by the Hausdorff convergence $V_{G_{n}} \to V_{G}$ , it is possible to find a subsequence $(n_{k})_{k \geq 1}$ and $x^{(n_{k})}, y^{(n_{k})} \in V_{G_{n_{k}}}$ such that $x^{(n_k)} \neq y^{(n_k)}$ and both $x^{(n_{k})}$ and $y^{(n_{k})}$ converge to a common point $x \in V_{G}$ . This implies that $R_{G_{n_{k}}}(x^{(n_{k})}, y^{(n_{k})}) \to 0$ . However, we have that

\begin{align*} R_{G_{n_k}}(x^{(n_{k})}, y^{(n_{k})})^{-1} &= \inf \{ \mathcal{E}_{G_{n_{k}}}(\,f, f) \mid f(x^{(n_{k})}) = 1,\, f(y^{(n_{k})})=0 \} \\ &\leq \mathcal{E}_{G_{n_{k}}}\big(1_{\{x^{(n_{k})} \}}, 1_{\{ x^{(n_{k})} \}}\big) \\ &= c_{G_{n_{k}}}(x^{(n_{k})}) \\ &\leq \sup_{x \in V_{G_{n_{k}}}} c_{G_{n_{k}}}(x), \end{align*}

from which we deduce that $\sup_{x \in V_{G_{n_{k}}}} c_{G_{n_{k}}}(x) \to \infty$ . This contradicts the assumption. Therefore, it holds that $|V_{G_{n}}| = |V_{G}|$ for all sufficiently large n.

When $|V_G| = 1$ , the result is trivial. We thus assume that $|V_G| \geq 2$ . By the above result and the Hausdorff convergence $V_{G_{n}} \to V_{G}$ , we can write $V_{G_{n}} = (x_{i}^{(n)})_{i=1}^{N}$ and $V_{G} = (x_{i})_{i=1}^{N}$ in such a way that $x_{i}^{(n)} \to x_{i}$ in M for each i. This yields that $R_{G_{n}}(x_{i}^{(n)}, x_{j}^{(n)}) \to R_{G}(x_{i}, x_{j})$ for all i, j. From Lemma 4.1, it follows that $c_{G_{n}}(x_{i}^{(n)}, x_{j}^{(n)}) \to c_{G}(x_{i}, x_{j})$ , which implies the convergence of the associated measures $\mu_{G_{n}} \to \mu_{G}$ in the weak topology, and the convergence of transition probabilities:

\begin{align*} P_{G_{n}} \big( x_{i}^{(n)}, x_{j}^{(n)} \big) = \frac{c_{G_{n}}\big(x_{i}^{(n)}, x_{j}^{(n)}\big)}{c_{G_{n}}\big(x_{i}^{(n)}\big)} \xrightarrow{n \to \infty} \frac{c_{G}{(x_{i}, x_{j})}}{c_{G}(x_{i})} = P_{G} ( x_{i}, x_{j} ) \quad \mbox{for all } i,j. \end{align*}

Thus, we deduce that the convergence of finite-dimensional distributions of $Y_{G_{n}}$ to those of $Y_{G}$ as processes in M. Since we consider the discrete-time processes, the tightness of $(Y_{G_{n}})_{n \geq 1}$ is obvious (see [Reference Kallenberg24, Theorem 23.11]). Therefore, $Y_{G_{n}}$ started at $\rho_{G_{n}}$ converges weakly to $Y_{G}$ started at $\rho_{G}$ in the usual $J_{1}$ -Skorohod topology as processes in M. By the Skorohod representation theorem, we may assume that $Y_{G_{n}}$ started at $\rho_{G_{n}}$ converges to $Y_{G}$ started at $\rho_{G}$ almost surely on some probability space. Fix $T > 0$ arbitrarily. We can check that, with probability 1, it holds that

\begin{align*} \mbox{for all } t \in [0,T], \quad Y_{G_{n}}(t) = x_{i}^{(n)} \Leftrightarrow Y_{G}(t) = x_{i} \end{align*}

for all sufficiently large n. This yields that

\begin{align*} \int_{0}^{t} 1_{\{ x_{i}^{(n)} \}} (Y_{G_{n}}(s))\, \mathrm{d}s = \int_{0}^{t} 1_{\{ x_{i} \}} (Y_{G}(s))\, \mathrm{d}s \quad \mbox{for all } t \in [0,T], \end{align*}

which, combined with the convergence $c_{G_{n}}(x_{i}^{(n)}) \to c_{G}(x_{i})$ , implies that

\begin{align*} \sup\nolimits_{i} \sup\nolimits_{0 \leq t \leq T} \left| \ell_{G_{n}}(x_{i}^{(n)}, t) - \ell_{G}(x_{i}, t) \right| \to 0. \end{align*}

Therefore, we find that $\ell_{G_{n}} \to \ell_{G}$ in $\widehat{C}(M \times \mathbb{R}_{\geq 0}, \mathbb{R})$ almost surely, which completes the proof.

Given a rooted electrical network G and $r > 0$ , we define $\tilde{G}^{(r)}$ to be the trace of G onto $B_{R_{G}}(\rho_{G}, r)$ equipped with the root $\rho_{\tilde{G}^{(r)}} \;:\!=\; \rho_{G}$ . In the following discussions, note that $\mu_{\tilde{G}^{(r)}}$ , which is the conductance measure associated with $\tilde{G}^{(r)}$ , is in general different from $\mu_{G}^{(r)}$ defined in (1.1), which is the measure $\mu_{G}$ restricted to $B_{R_{G}}(\rho_{G}, r)$ . Using Proposition 4.2, we verify the measurability of $\mathcal{Y}_{\tilde{G}^{(r)}}$ in Proposition 4.3.

Proof. Fix a measurable function $f \colon \mathbb{M}_{\mathrm L} \to \mathbb{R}$ assumed to be bounded and continuous. Given $\varepsilon > 0$ , define a map $F_{\varepsilon} \colon (0, \infty) \times \mathbb{F}^E \to \mathbb{R}$ by setting

\begin{align*} F_{\varepsilon} (r, (V_{G}, R_{G}, \rho_{G}, \mu_{G})) \;:\!=\; \frac{1}{\varepsilon \wedge (r / 2)} \int_{0}^{\varepsilon \wedge (r / 2)} f ( \mathcal{Y}_{\tilde{G}^{(r-s)}} )\, \mathrm{d}s. \end{align*}

Note that the integral on the right-hand side of the above equation is well defined by Lemma4.2. Suppose that $r_{n} \to r$ and $(V_{G_{n}}, R_{G_{n}}, \rho_{G_{n}}, \mu_{G_{n}}) \to (V_{G}, R_{G}, \rho_{G}, \mu_{G})$ . Then, for Lebesgue almost-every $s \in (0, r/2)$ , it holds that

\begin{align*} \big(V_{\tilde{G}_{n}^{(r_{n} - s)}}, R_{\tilde{G}_{n}^{(r_{n} - s)}}, \rho_{\tilde{G}_{n}^{(r_{n} - s)}}, \mu_{G_{n}}^{(r_{n} - s)}\big) \to \big(V_{\tilde{G}^{(r-s)}}, R_{\tilde{G}^{(r-s)}}, \rho_{\tilde{G}^{(r-s)}}, \mu_{G}^{(r-s)}\big) \end{align*}

in the Gromov–Hausdorff–Prohorov topology (see [Reference Noda33, Lemma 2.15]). We then have that

(4.5) \begin{align} \limsup_{n \to \infty} \mu_{\tilde{G}_{n}^{(r_{n} - s)}} \big(V_{\tilde{G}_{n}^{(r_{n} - s)}}\big) \leq \limsup_{n \to \infty} \mu_{G_{n}}^{(r_{n} - s)} \big(V_{\tilde{G}_{n}^{(r_{n} - s)}}\big) = \mu_{G}^{(r-s)}\big(V_{\tilde{G}^{(r - s)}}\big) < \infty, \end{align}

where we use Corollary 4.1 to obtain the first inequality. From Proposition 4.2, it follows that $\mathcal{Y}_{\tilde{G}_{n}^{(r_{n} - s)}} \to \mathcal{Y}_{\tilde{G}^{(r - s)}}$ for Lebesgue almost-every $s \in (0, r/2)$ . This yields that the map $F_{\varepsilon}$ is continuous. Moreover, by Lemma 4.2, $F_{\varepsilon}$ converges to the map $(r, (V_{G}, R_{G}, \rho_{G}, \mu_{G})) \mapsto f(\mathcal{Y}_{\tilde{G}^{(r)}})$ as $\varepsilon \to 0$ pointwise, which completes the proof.