Hausdorff dimensions of inverse images and collision time sets for symmetric Markov processes

In this paper, we establish the Hausdorff dimensions of inverse images and collision time sets for a large class of symmetric Markov processes on metric measure spaces. We apply the approach in the works by Hawkes and Jain--Pruitt, and make full use of heat kernel estimates. In particular, the results efficiently apply to symmetric diffusion processes, symmetric stable-like processes, and symmetric diffusion processes with jumps in $d$-sets.

In particular, when D = {x} with some x ∈ M, this is reduced into the level set; on the other hand, the collision time set is defined by where X i := (X i t ) t≥0 , i = 1, 2, are two independent copies of X. Concrete examples of the Markov processes included in the framework of the present paper are symmetric diffusion processes, symmetric stable-like processes, symmetric diffusion processes with jumps in d-sets, and so on. Note that, as seen from the survey paper [37,Sections 6 and 7], dimension results and their proofs for level sets, inverse images and collision time sets are more complex than those for images. This work is inspired by the Hausdorff dimension results of the inverse images and collision time sets for stable processes on Euclidean space. Jain and Pruitt [15] established the Hausdorff dimensions on the collision sets of two independent stable processes X 1 := (X 1 t ) t≥0 and X 2 := (X 2 t ) t≥0 on R possibly with different indices. Their idea is to regard the collision of X 1 and X 2 as their direct product process X 1 ⊗ X 2 hitting the diagonal set in R 2 , and to compare the polarity of X 1 ⊗ X 2 with that of some stable process in R 2 . Jain and Pruitt [15,Introduction] also pointed out that, if X 1 and X 2 have the common index α ∈ (1, 2), then the collision time set of X 1 and X 2 has the same Hausdorff dimension as that of the level set of the onedimensional α-stable process. This property follows from the fact that the difference process (X 1 t − X 2 t ) t≥0 is also a α-stable process. However, if the indices of X 1 and X 2 are different, then it is unclear how to establish the Hausdorff dimension of the collision time set.
Motivated by [15], Hawkes [20,21] established the Hausdorff dimension of the inverse image for one-dimensional α-stable processes with α ∈ (1, 2). The idea of these works is to parametrize the stable indices by using the stable subordinators, and to utilize the regularity and polarity properties of the stable processes. Liu [26] applied this idea to the inverse images of compact sets for Lévy processes on Euclidean space. Recently, Knopova and Schilling [25] further applied this idea to the inverse image of Feller processes on Euclidean space, with application to the collision time sets of the two independent copies.
Our approach is based on heat kernel estimates for the associated Markov processes, together with the development of the ideas of [15,20,21,25] as mentioned before. More precisely, we will make full use of the subordinate processes and the associated potential theory. However, since the present paper is concerned with symmetric Markov processes on metric measure spaces, there are a few difficulties and differences compared with the papers cited above. For instance, (i) Concerning the inverse image, we follow the idea of Hawkes [20,21] to make use of the stable subordinator. However, since the subordinate process of a Markov process is not a stable process in general, we can not utilize the polarity property of stable processes as [20,21].
(ii) Inspired by [15] and [20,21], we determine the Hausdorff dimension of the collision time set by studying the regularity and polarity of the stable subordinate process of the direct product process. However, we need further consideration on the regularity property; it should be noted that, even for the direct product process of the two independent stable processes on R, its stable subordinate process is not a stable process on R 2 in general. Moreover, since the state space is a metric measure space, the approach with aid of the difference process is not applicable to the collision time set.
Due to these difficulties and differences above, we need some new ideas and some efforts in the present paper. To state the contribution of our paper, let us restrict on the following special setting. Theorem 1.1. Let (M, d, µ) be a connected d-set such that any closed ball in M is compact. For a subset F of M, let dim H (F ) denote its Hausdorff dimension. Let X := (X t ) t≥0 be the µ-symmetric diffusion process with walk dimension α or the symmetric α-stable-like process (that is, the associated scaling function of each process is φ(r) = r α ) on M. Then the following statements hold.
More generally, if F ⊂ M is a Borel set such that dim H (F ) > 0, then , P x -a.s. for any x ∈ M.
As mentioned above, the proof of Theorem 1.1 is partly based on heat kernel estimates for symmetric Markov processes, which are now developed greatly in recent years (see, e.g., [1,9,10,11,12,13,19]). Indeed, according to general results of our paper, we also can get by Remark 2.7 below and [11, Remark 1.12(iii) and Example 7.2] that -Let (M, d, µ) be a connected d-set such that any closed ball in M is compact. Let X := (X t ) t≥0 be the µ-symmetric diffusion process with jumps on M, where the scaling functions of diffusion part and jump part are given respectively by φ c (r) = r α and φ j (r) = r β for some 0 < β < α. Then, the conclusion (1) of Theorem 1.1 holds when d ≤ β, and the conclusion (2) of Theorem 1.1 still holds when d < β.
We make some comments on how to overcome the difficulties mentioned in (i) and (ii). For the inverse images, we derive the polarity of the subordinate processes by employing the Frostman lemma on the complete separable metric space in Subsection A.1. For the collision time sets, we first prove the zero-one law for the tail events (Proposition B.3), and then establish the Wiener tests for the recurrence and regularity of X 1 ⊗X 2 (Propositions B.6 and B.9). Under our general setting, we can see from Examples 3.12 and 4.11 that, the local properties of the volume growth and walk dimensions determine the Hausdorff dimensions of the inverse images and collision time sets. With regard to the collision time sets, our general results in Subsection 4.2 allow two independent symmetric Markov processes to be different. We also note that, as far as the authors know, the Wiener tests are unavailable for general symmetric Markov processes, even though those are well known for Brownian motion or other Lévy processes on Euclidean spaces (see, e.g., [28]).
We mention that Shieh [32,33] studied the possibility of collisions of two independent Hunt processes in terms of the heat kernels, with applications to Lévy processes on Euclidean space and Brownian motions on fractals. Our results in the present paper provide quantitative information on the collision times, and are applicable to symmetric jump processes of variable order on d-sets, fractals and ultra-metric spaces. We also characterize the Hausdorff dimension of the set of collision times on a given set by its Hausdorff dimension.
The rest of the paper is arranged as follows. In the next section, we present preliminaries and assumptions used in the paper. In Section 3, we obtain Hausdorff dimensions of level sets and inverse images, where we will first consider heat kernel and resolvent for the stablesubordinate process. In Section 4, we study Hausdorff dimensions of the collision time sets. For this, we establish estimates for the resolvent of stable-subordinate direct-product process. In the appendix, we collect some statements used in the proofs of our results, which include the Wiener tests for the recurrence and regularity of symmetric Markov processes on metric measure spaces.
We close this introduction with some words on notations. For nonnegative functions f and g on a set T , we write f (t) g(t) (resp. f (t) g(t)) for any t ∈ T if there exists a constant c > 0 such that f (t) ≤ cg(t) (resp. f (t) ≥ cg(t)) for any t ∈ T . We write f (t) ≃ g(t) for any t ∈ T if f (t) g(t) and f (t) g(t) for any t ∈ T .
We know by [17,Lemma 1.6.4 (iii)] that any irreducible Dirichlet form is either transient or recurrent.
Let F e denote the totality of µ-measurable functions u on M such that |u| < ∞ µ-a.e. on M and there exists a sequence {u n } ⊂ F such that lim n→∞ u n = u µ-a.e. on M and lim m,n→∞ E(u n − u m , u n − u m ) = 0. The sequence {u n } is called an approximating sequence of u. For any u ∈ F e and its approximating sequence {u n }, the limit E(u, u) = lim n→∞ E(u n , u n ) exists, and does not depend on the choice of the approximating sequence for u ([17, Theorem 1.5.2]). We call (F e , E) the extended Dirichlet space of (E, F ) ( [17, p.41]). We also know by [17,Lemma 1.5.5] that, if (E, F ) is transient, then F e is complete with respect to √ E. We next recall from [17] the notion of the capacity relative to (E, F ). Let C 0 (M) denote the totality of continuous functions on M with compact support. In what follows, we suppose that We then define the (1-)capacity of any subset A of M by We say that a statement S(x) depending on x ∈ M holds quasi everywhere (q.e. in short) if there exists a set N ⊂ M with Cap(N ) = 0 such that S(x) holds for any x ∈ M \ N . For f ∈ F , letf be its quasi-continuous µ-version; that is, f =f , µ-a.e. on M, and for any ε > 0, there exists a closed subset F of M such that Cap(M \ F ) < ε andf is finite continuous on F ([17, Section 2.1]).
Let ν be a positive Radon measure on M. According to [17, p.77, (2.2.1)], we say that ν is of finite energy integral, if there exists C > 0 such that Let S 0 denote the totality of measures of finite energy integral on M. Then, there exists a unique function U 1 ν ∈ F such that The function U 1 ν is called the 1-potential of ν. We note that any measure in S 0 charges no set of zero capacity ( [17,Theorem 2.2.3]). Moreover, if K is a compact subset of M, then there exist a unique element e K ∈ F and a unique measure ν K ∈ S 0 such that e K = U 1 ν K and Cap(K) = E 1 (e K , e K ) = ν K (K) (see [17, (2.2.13)]). The element e K and the measure ν K are called the 1-equilibrium potential and the 1-equilibrium measure of K, respectively. Let We then see by [17, p. [17, p.85], we can also introduce the notions of a class of measures of finite (0-order) finite energy integral (S (0) 0 in notation), and of (0-order) potential of the measure ν ∈ S (0) 0 (Uν in notation). In particular, if K is a compact subset of M, then we have the corresponding 0order equilibrium potential e (0) K ∈ F e and the 0-order equilibrium measure ν K ∈ S (0) 0 such that e K = Uν K and Cap (0) (K) = E(e K , e K ) = ν K (K).

Hunt process and measurability
In this subsection, we first recall from [5] classes of measurable subsets of M associated with Hunt processes. As in Subsection 2.1, (M, d) is a locally compact separable metric space, and µ is a positive Radon measure on M with full support. Let Here θ t : Ω → Ω is the shift operator of the paths defined by X s • θ t = X s+t for every s > 0, and ζ = inf{t > 0 : X t = ∆} is the lifetime.
A subset A of M is called nearly Borel measurable (relative to the process X), if for any probability measure ν on M, there exist Borel subsets B 1 and B 2 of M such that B 1 ⊂ A ⊂ B 2 and P ν (X t ∈ B 2 \ B 1 for some t ≥ 0) = 0 ([5, Definition 10.2 in Chapter I]). Let B n (M) denote the totality of nearly Borel measurable subsets of M. For A ∈ B n (M), let σ A be the hitting time of X to A; that is, σ A = inf{t > 0 : X t ∈ A}. We say that a point x ∈ M is regular for A, if P x (σ A = 0) = 1. Let A r denote the totality of regular points for A, i.e., Then, A r is nearly Borel measurable ([5, Corollary 2.13 in Chapter II]). If A is a subset of M, then A r is defined as the totality of points regular for all nearly Borel subsets containing A. We call A r the regular set for A (relative to the process X). Recall that µ is a positive Radon measure on M with full support. Since the state space M is locally compact and separable, there exists a strictly positive Borel measurable function g on M such that µ g = g · µ is a Borel probability measure on M and thus B µ g (M) = B µ (M). Using this relation, we can uniquely extend the measure µ to B * (M). We use the same notation µ for such an extension.
We next recall from [17] the relation between symmetric Hunt processes and Dirichlet forms. Let {p t } t>0 be the transition function of a Hunt process X on M defined by with the convention that f (∆) = 0. The left hand side above is written as p t f . We now assume that the process X is µ-symmetric, i.e., (p t u, v) = (u, p t v) for any t > 0 and nonnegative functions u, v ∈ B(M). According to [17, p.30 and p.160], we can extend {p t } t>0 uniquely to a strongly continuous Markovian semigroup {T t } t>0 on L 2 (M; µ). Then, by (2.1), we can associate a Dirichlet form (E, F ) on L 2 (M; µ).
Conversely, if (E, F ) is a regular Dirichlet form on L 2 (M; µ) associated with a strongly continuous Markovian semigroup {T t } t>0 on L 2 (M; µ), then there exists a µ-symmetric Hunt process X on M such that Let X be a µ-symmetric Hunt process on M generated by a regular Dirichlet form (E, F ). A set N ⊂ M is called exceptional, if there exists a nearly Borel setÑ ⊃ N such that P x (σÑ < ∞) = 0 for µ-a.e. x ∈ M. A set N ⊂ M is called properly exceptional, if it is nearly Borel measurable such that µ(N ) = 0 and M \ N is X-invariant; that is,

Heat kernel
, ζ) be a µ-symmetric Hunt process on M associated with the regular Dirichlet form (E, F ) on L 2 (M; µ). In what follows, we always impose the following Assumption (H) on the process X.  • For any t > 0, x ∈ M \ N and A ∈ B(M), • For any t > 0 and x, y ∈ M \ N , p(t, x, y) = p(t, y, x).
• For any s, t > 0 and x, y ∈ M \ N , The function p(t, x, y) in Assumption (H) is called the heat kernel in the literature. While (2.5) determines p(t, x, y) for µ-a.e. y ∈ M, we can regularize p(t, x, y) under the so-called ultracontractivity condition so that the condition (ii) in Assumption (H) is fulfilled (see, e.g., [ and recurrent if x, y) > 0 for any t > 0 and x, y ∈ M 0 .
We note that [34,Remark 2.2] refers to the condition (2.7) with x ∈ M 0 and sup y∈M 0 p(t, x, y) replaced by x ∈ M and sup y∈M p(t, x, y), respectively; however, the argument there shows that the condition (2.7) suffices for transience.
(ii) Suppose that the heat kernel p(t, x, y) satisfies (WUHK) and (HR). If u is a bounded continuous function on M, then so is p t u for any t > 0. In particular, there exists a version of the process X such that all the conditions in Assumption (H) (ii) are valid by replacing M \ N with M. If (WUHK) and (HR) are imposed on the heat kernel, then we take the process X as the version above.
(iii) We see by the proof of [12,Proposition 5.4] that, if the heat kernel p(t, x, y) satisfies (WUHK) and (HR), then it satisfies (NDLHK) as well.
Remark 2.7. The form in the right hand side of (2.12) for the definition (WUHK) comes from two-sided heat kernel estimates for the mixture of symmetric stable-like (jump) processes in metric measure spaces; see [10,12]. We should emphasize that this kind of heat kernel upper bounds are satisfied for a large class of symmetric Markov processes, including symmetric diffusion processes generated by strongly local Dirichlet forms (see [1,19]), symmetric diffusion processes with jumps in metric measure spaces (see [11]), and symmetric jump processes that allowed to have light tails of polynomial decay at infinity or to have (sub-or super-) exponential decay jumps (see [13,9]). To verify the assertion above, below we take the µ-symmetric diffusion process X on an Ahlfors d-regular set (M, d, µ) with walk dimension α ≥ 2 for example. Similar arguments work for all the processes mentioned above. In this example, V (x, r) ≃ r d , and the heat kernel p(t, x, y) of the process X enjoys the following two-sided estimates: .
Here, we write f (s, x) ≍ g(s, x), if there exist constants c k > 0, k = 1, 2, 3, 4, such that for the specified range of (s, x). Then, by some calculations, one can see that there are constants c 5 > 0 such that for all x, y ∈ M and t > 0, This implies that for all x, y ∈ M and t > 0, In particular, (WUHK) holds with φ(r) = r α .
Furthermore, according to results in all the cited papers, we know that, for all the processes mentioned above, (ODHK), (NDLHK), (WUHK) and (HR) are satisfied.
3 Hausdorff dimensions of level sets and inverse images 3.1 Heat kernel and resolvent for stable-subordinate processes For γ ∈ (0, 1), let S γ := ({τ t } t≥0 , P γ ) be the γ-stable subordinator which is independent of the process X. Let π t (s) denote the density function of τ t . According to [8,Theorem 4.4] (or the proof of [7, Theorem 3.1]), there exist positive constants c 1 and c 2 such that and Let X γ t = X τt for any t ≥ 0, and let X γ := (X γ t ) t≥0 be the γ-stable subordinate process of X. Then, the process X γ is a µ-symmetric Hunt process. Let (E γ , F γ ) be a Dirichlet form on L 2 (M; µ) associated with X γ . Then, by [30, Theorem 2.1 (ii) and Theorem 3.1 (i)-(ii)], (E γ , F γ ) is also regular, irreducible and conservative. We note that M 0 = M \ N is X γ -invariant by definition, and N is of zero capacity relative to (E γ , F γ ) by [30, Theorem 2.2 (i)]; hence N is also properly exceptional with respect to X γ . Moreover, the subordinate process X γ possesses the density function q(t, x, y) with respect to the measure µ so that q(t, x, y) = ∞ 0 p(s, x, y) π t (s) ds, t > 0, x, y ∈ M 0 . Therefore, the process X γ satisfies Assumption (H) as well.
(2) Under (NDLHK), Moreover, and Moreover, Remark 3.2. According to Lemma 3.1 above, if the original process X fulfills one of the conditions in Definition 2.4, then the subordinate process X γ also satisfies the corresponding one, with φ in (2.10) replaced by φ γ .
Proof of Lemma 3.1.
(1) Suppose that (2.10) holds. Then, by (3.1) and the change of variables formula with u = t/s γ , By (2.15), there exist positive constants c 2 and η 1 such that Similarly, there exist positive constants c 3 and η 2 such that Combining this with (3.6), we get the desired upper bound of q(t, x, x).
On the other hand, it follows by (3.2) that Fix a constant θ > 1 and let θ n = t 1/γ θ n . Then, by (2.15) again, there exist positive constants c 6 and η 3 such that .
We thus arrive at the desired lower bound of q(t, x, x).
Therefore, we arrive at the desired lower bound of q(t, x, y).
Using the lower bound of q(t, x, y) above, we obtain dt. Since p(s, x, y)π t (s) ds = I 1 + I 2 .
Then, by (WUHK) and (3.1), The last inequality above follows by the same calculation as (3.8). Hence .
Following the calculation in the proof of (1), we also have so that (3.5) follows. The upper bounds of u γ 1 (x, y) follow by the same calculations as in (2). Suppose that the process X satisfies one of the conditions in Definition 2.4. For γ ∈ (0, 1], let Then, by Remark 2.3(ii) and Lemma 3.1, the process X γ is recurrent if the process X satisfies (NDLHK) and I γ (x) = ∞ for any x ∈ M; X γ is transient if X satisfies (WUHK) and I γ (x) < ∞ for any x ∈ M. The next lemma provides the Green function (or 0-order resolvent) estimates of the process X γ .
Suppose that the process X satisfies Assumption (H). Then for any γ ∈ (0, 1], the following estimates hold. (1) Under (NDLHK), We omit the proof of Lemma 3.3 because it is similar to that of Lemma 3.1.

Hausdorff dimensions of level sets
In this subsection, we will determine the Hausdorff dimensions of the level sets for the process X. First, we recall the definition of the Hausdorff dimension. Let ϕ be a continuous Hausdorff function of finite order such that ϕ(0) = 0 (see Definition A.1). Let H ϕ denote the associated Hausdorff measure on the metric measure space M. If ϕ(t) = t p for some p > 0, then we write For any fixed a ∈ M, let γ a (s) = inf γ > 0 : Then, the main result of this part can be stated as follows.
Theorem 3.4. Suppose that the process X satisfies Assumption (H) and (ODHK). We have the following two statements.
(2) Suppose that 0 < γ a (0) < 1 for any a ∈ M. Then N = ∅ and thus M 0 = M. Moreover, if the process X also satisfies (NDLHK) and I 1 (a) = ∞ for any a ∈ M, then (3.14) We will prove Theorem 3.4 by following the argument of [20, Theorem 1] (see also the proof of [25, Theorem 2.1]). To do so, we need two lemmas.
Proof. We split the proof into four steps.
(i) We show that the function s → γ a (s) is nonincreasing. By the change of variables formula with u = t 1/γ , for any γ > 0, we have thanks to the fact that φ is increasing on [0, 1] with φ(0) = 0 and φ(1) = 1.
Putting the arguments in (i)-(iv) together, we arrive at the desired assertion.
Proof. The first assertion is essentially taken from [17, Example 2.1.2], and we present the details here for the sake of completeness. Fix a ∈ M, and let δ a be the Dirac measure at a. We first assume that u 1 (a, a) < ∞. Then, by [ We next assume that u 1 (a, a) = ∞. Then, by [17, Exercise 4.2.2], the measure δ a is not of finite energy integral. Let us suppose that Cap({a}) > 0. Then, according to [17, Lemma 2.2.6 and the subsequent comment] again, it follows that for some c > 0, the measure cδ a would be the equilibrium potential of {a}, so that δ a is of finite energy integral. This is a contradiction, and so Cap({a}) = 0.
Let us prove the second assertion. By (ODHK), dt.
Note that the second term of the right hand side above is finite, because the function t → V (a, φ −1 (t)) is nondecreasing. Then, the proof is complete by the first assertion.
Here and in what follows, let Cap γ denote the 1-capacity relative to the subordinate process X γ . If 0 < γ < γ a (0), then and so Cap γ ({a}) = 0 by Lemma 3.6 applied to X γ , also thanks to Lemma 3.1(1). Therefore, the process X γ can not hit the point a by [ This implies that P γ (τ t ∈ {s > 0 : X s (ω) = a} for some t > 0) = 0, P x -a.s. ω ∈ Ω for any x ∈ M 0 .
If γ a (0) > 1, then Hence, by Lemma 3.6 applied to X, we have Cap({a}) = 0 and thus the process X can not hit the point a by [17, Theorems 4.1.2 and 4.2.1 (ii)] again. The proof of (1) is complete. We next prove (2). Assume that γ a (0) < 1 for any a ∈ M. Then for any γ ∈ (γ a (0), 1], since we have Cap γ ({a}) > 0 by Lemma 3.6 applied to X γ , also due to Lemma 3.1(1) again. In particular, it follows by [17, Theorems 4.1.3 and A.2.6 (i)] that the point a is regular relative to X γ for any γ ∈ (γ a (0), 1], i.e., (3.20) On the other hand, since (3.19) is valid with γ = 1, we have Cap({a}) > 0 for any a ∈ M, which implies that N = ∅ and P x (σ a < ∞) > 0 for any x ∈ M. Furthermore, the process X is irreducible and recurrent by Assumption (H), (NDLHK) and I 1 (a) = ∞ for any a ∈ M, with the comment just before Lemma 3.3. Hence by [17, Theorem 4.7.1 (iii) and Exercise 4.7.1], we obtain P x (σ a < ∞) = 1 for any x ∈ M. Note that X σa = a because {a} is closed in M. Therefore, by (3.20) and the strong Markov property of the process X, By using [20,Section 3] Letting γ ↓ γ a (0) along a sequence, we have Combining this with (3.13), we get (3.14).
Example 3.7. Let the process X satisfy Assumption (H), (ODHK) and (NDLHK). We impose the next conditions on the functions V (x, r) and φ(r) : • There exist positive constants d 1 , d 2 and c i , 1 ≤ i ≤ 4, such that and • There exist positive constants α, β, c i , 5 ≤ i ≤ 8, such that Then for any a ∈ M, γ a (s) = (d 1 − s)/α for any s ∈ [0, d 1 ], and γ a (0) ≤ 1 if and only if 0 < d 1 ≤ α. We also see that I 1 (a) = ∞ for any a ∈ M if and only if 0 < d 2 ≤ β. By the calculation above and Theorem 3.4, we have the following:

Hausdorff dimensions of inverse images
In this subsection, we determine the Hausdorff dimensions of the inverse images for the process X. For this purpose, we make a stronger assumption on the volume function.
Note that under the assumption above, the value γ a (u) defined by (3.12) is independent of the choice of a ∈ M. Hence we write γ(u) for γ a (u). In other words, We also define Then, by the proof of Lemma 3.5, the function s → γ(s) is Lipschitz continuous on [0, ∞) and s 0 defined above is positive; moreover, γ(s) is strictly decreasing on [0, s 0 ] such that γ(s) = 0 for s ≥ s 0 . We also introduce the next assumption on M in order for the validity of Proposition A.4 below.
Assumption 3.9. Any closed ball in M is compact.
Suppose that the process X satisfies Assumption (H), and that Assumption 3.8 holds. Suppose also that for any s ≥ 0 with γ(s) > 0 and for any γ ∈ (0, γ(s)), there exists a constant c 1 > 0 such that for any T ∈ (0, 1/2), (2) Suppose further that M satisfies Assumption 3.9, and the process X satisfies Assumption To prove Theorem 3.10, we need the following lemma.
Lemma 3.11. Suppose that the process X satisfies Assumption (H). If A is a subset of M, We also need the notation for the energy of a Borel measure. Let ψ : Then, I ψ (ν) is called the ψ-energy of ν. If ψ(t) = t s for some s > 0, then we write I ψ as I s .
Proof of Theorem 3.10. We first prove (1) under the condition that γ(s F ) ≤ 1. Let F be a Borel subset of M. Without loss of generality, we assume that γ(s F ) > 0. Then, by Lemma 3.5 and its proof, there exists δ ∈ (0, s F /2) such that γ(u) > 0 for any u ∈ (s F , s F + δ). If we fix u ∈ (s F , s F + δ), then γ(u) < γ(s) < γ(s F ) for any s ∈ (s F , u), thanks to Lemma 3.5 again. Therefore, for any C > 0, there exists T 0 ∈ (0, 1/2) such that In particular, for any x, y ∈ M with d(x, y) ≤ φ −1 (T 0 ), it follows by (3.21) that This implies that for any compact subset K of F , there exists a constant C 0 : Let X γ(u) be the γ(u)-stable subordinate process of the process X, and (E γ(u) , F γ(u) ) the associated regular Dirichlet form. We now assume that there exists a finite and nontrivial Borel measure ν on M such that it is compactly supported in K and charges no set of zero capacity relative to (E γ(u) , F γ(u) ). Then for any s ∈ (s F , u), since H s (K) = 0, Proposition A.2 yields I s (ν) = ∞. Combining this with (3.23), we obtain Let u γ(u) 1 (x, y) be the 1-resolvent kernel for X γ(u) . According to Lemma 3.1(2), under (NDLHK), (3.25) where the constant c 3 > 0 may depend on the set K. In particular, (3.24) and (3.25) yield On the other hand, if ν K . This is a contradiction so that we get Cap γ(u) (K) = ν Then, by [20,Section 3] Letting u ↓ s F along a sequence, we arrive at the assertion (1) provided that γ(s F ) ≤ 1.
If γ(s F ) > 1, then, by the proof of Lemma 3.5, we can take u > s F so that γ(u) = 1. Hence the same argument as before implies that Cap(F ) = 0, and thus P x ({t > 0 : X t ∈ F } = ∅) = 1 for any x ∈ M 0 . The proof of (1) is complete.
We next prove (2). Without loss of generality, we assume that s F > 0 and 0 < γ(s F ) < 1. Then, by Lemma 3.5, there exists a constant ε > 0 such that for all s ∈ (s F − ε, s F ), γ(s F ) < γ(s) < 1. We now fix such s ∈ (s F − ε, s F ). Then the regularity of the Hausdorff measure yields H s (K) > 0 for some compact subset K of F . Under Assumption 3.9, we can further use Proposition A.4 to show that there exists a finite and nontrivial Borel measure ν s K on M such that supp[ν s K ] ⊂ K and I s (ν s K ) < ∞. On the other hand, Lemma 3.5 implies again that γ(s) < γ(v) < 1 for any v ∈ (s F − ε, s). Then for any v ∈ (s F − ε, s) and T ∈ (0, 1/2), which implies that for some c 2 > 0, Let X γ(v) be the γ(v)-stable subordinate process of the process X. Since γ(v) < γ(0), it follows by Lemma 3.1(3) and (3.21) that under (WUHK), there exists a constant c 3 > 0 such that for all x, y ∈ K, Combining this with (3.27), we have for some c 4 > 0, Therefore, Then, by [17,Exercise 4.2.2], the measure ν s K is of finite energy integral relative to We now follow the argument of [26,Theorem 1]. Let σ F be the hitting time of Namely, F r γ(s) is the totality of regular points of F relative to the process X γ(s) . By Lemma 3.11 applied to the process  1], we have P x (σ K < ∞) = 1 for any x ∈ M. Noting that X σ K ∈ K and K ⊂ F γ(s) , we further obtain by the strong Markov property of X, Then, by [20,Section 3] Letting s ↑ s F along a sequence, we have by Lemma 3.5, Combining this with Theorem 3.10, we complete the proof.
With Examples 3.7 and 3.12, one can easily get the first assertion (1) in Theorem 1.1, also thanks to Remark 2.7.
(E, F ) is also conservative by [17, Exercise 4.5.1]. The heat kernel of X is given by For γ ∈ (0, 1), let X γ := ((X γ t ) t≥0 , {P γ x } x∈M ×M ) be a subordinate process of X with respect to the γ-stable subordinator S γ = ({τ t } t≥0 , P γ ), that is, for any t ≥ 0 and x ∈ M × M, Let (E γ , F γ ) be the associated Dirichlet form on L 2 (M × M; µ ⊗ µ). Then, by [30, Theorems 2.1(ii) and 3.1(i)(ii)], (E γ , F γ ) is also regular, irreducible and conservative. The heat kernel of X γ is given by where π t (s) is the density function of S γ t . For λ ≥ 0, let u γ λ (x, y) be the λ-resolvent density of X γ , i.e., for x, y ∈ M 1 0 × M 2 0 , In the following, we will assume that the processes X 1 and X 2 satisfy the common one of the conditions in Definition 2.4. Under this assumption, we use the notations φ i , α i1 and α i2 , respectively, to denote the corresponding scaling function φ and the associated indices α 1 , α 2 . For x = (x 1 , x 2 ) ∈ M × M and y = (y 1 , y 2 ) ∈ M × M, let We first show the lower bound for the resolvent density of the process X γ .
Lemma 4.1. Suppose that the independent processes X 1 and X 2 satisfy Assumption (H) and (NDLHK). Then, for any γ ∈ (0, 1], there exist positive constants c 1 and c 2 such that for any Proof. Without loss of generality, we assume that the processes X 1 and X 2 satisfy (NDLHK) with the constant c 0 = 1 involved in. Then, by (3.2) and (4.1), dt.
Hence, the proof is complete.
We next show the upper bound of the resolvent of X γ .
Before the proof of the Green function estimates of X γ , we give a criterion for recurrence or transience. For γ ∈ (0, 1], let Then, by the change of variables formula with s = t 1/γ , we have dt.
Proof. We first prove (1). Suppose that the processes X 1 and X 2 satisfy (NDLHK) and J γ (x) = ∞ for any x ∈ M × M. We can then follow the calculations of (4.2) and (3.8) to show that, for any Hence by Remark 2.3 (ii), X γ is recurrent. We next prove (2). Suppose that the processes X 1 and X 2 satisfy (WUHK) and J γ (x) < ∞ for any x ∈ M × M. Then for any x ∈ M 1 0 × M 2 0 , we follow the calculation as in the proof of Lemma 4.2 to see that s))s 1+γ ds dt =: c 3 I. Then, by the Fubini theorem, ds.
The first term above is convergent by (2.15) with φ = φ 1 and φ = φ 2 , and so is the second one by assumption. Hence by Remark 2.3 (i), X γ is transient. dt. (4.7) In particular, by the proof of Lemma 4.3 and Remark 2.3 (iii), we see that if the independent processes X 1 and X 2 satisfy (ODHK) and J γ < ∞, then X γ is transient.
By following the proofs of Lemmas 4.1 and 4.2, we also get the Green function estimates.
Lemma 4.5. Suppose that the independent processes X 1 and X 2 satisfy Assumption (H).

Hausdorff dimensions of collision time sets
In this subsection, we will determine the Hausdorff dimensions of collision time sets of two independent processes X 1 and X 2 on a given set in terms of the associated scale functions. In what follows, we impose Assumption 3.
We also let s 0 = inf s > 0 : Then, by the proof of Lemma 3.5, the function s → γ(s) is Lipschitz continuous on [0, ∞), and s 0 defined above is positive; moreover, γ(s) is positive and strictly decreasing on [0, s 0 ) and γ(s) = 0 for s ≥ s 0 .
Theorem 4.6. Let Assumption 3.8 hold. Suppose that the independent processes X 1 and X 2 satisfy Assumption (H) and (NDLHK). Let F ⊂ M be a Borel set with s F = dim H (F ) > 0. Assume that for any s ∈ [0, s 0 ) and γ ∈ (0, γ(s)), there exist constants c 1 > 0 and T 0 ∈ (0, 1) such that for any T ∈ (0, T 0 ), On the other hand, if γ(s F ) > 1, then {v > 0 : (2) Suppose further that M satisfies Assumption 3.9, and that the processes X 1 and Proof. Let F ⊂ M be a Borel set with s F = dim H (F ) > 0. We first prove (1). Let us now assume that γ(s F ) ≤ 1. Without loss of generality, we may and do assume that γ(s F ) > 0. Then, by the proof of Lemma 3.5, there exists δ ∈ (0, s F /2) such that γ(u) > 0 for any u ∈ (s F , s F + δ). If we fix u ∈ (s F , s F + δ), then for any s ∈ (s F , u), γ(u) < γ(s) < γ(s F ). Therefore, it follows by (4.11) that for any C 0 > 0, there exists T 0 ∈ (0, 1) such that for any T ∈ (0, T 0 ), Here c 1 is a positive constant depending on the choices of u ∈ (s F , s F + δ) and s ∈ (s F , u). Note that for any x, z ∈ M × M, , z)). (4.14) Hence, if φ γ(u) (d(x, z)) ≤ T 0 , then Lemma 4.1 and (4. We now assume that there exists a finite and nontrivial Borel measure ν on M × M such that it is compactly supported in K and charges no set of zero capacity relative to (E γ(u) , F γ(u) ). Then, by (4.15), (4.16) On the other hand, by Lemma 4.1, there exists c 5 := c 5 (T 0 ) > 0 such that for any x, z ∈ K with φ γ(u) (x, z) ≥ T 0 , we obtain u γ(u) 1 (x, z) ≥ c 5 . Hence, by (4.16), (4.17) We also note that H s (K) = 0 because Then Proposition A.2 yields I s (ν) = ∞. Therefore, by (4.17), be the equilibrium measure of K for X γ(u) . Since this measure is of finite energy integral relative to (E γ(u) , F γ(u) ), it charges no set of zero capacity ([17, Theorems 2.1.5(ii) and 2.2.3]). If we assume that ν γ(u) K is nontrivial, then (4.18) with ν = ν Namely, for any x ∈ M 1 0 × M 2 0 , we have for P x -a.s. ω ∈ Ω, Then, by [20,Section 3] Letting s ↓ s F and then u ↓ s F along some sequences, we arrive at (4.12). If γ(s F ) > 1, then, by the proof of Lemma 3.5 again, there exists a constant u > s F such that γ(u) = 1. Then the same argument as above yields Cap(diag(F )) = 0 and thus We next prove (2). We assume that 0 ≤ γ(s F ) < 1. Then, by Lemma 3.5, there exists a constant ε > 0 such that for all s ∈ (s F − ε, s F ), γ(s F ) < γ(s) < 1. We now fix such s ∈ (s F − ε, s F ). Since s F = dim H (diag(F )), the regularity of the Hausdorff measure yields H s (K) > 0 for some compact subset K of diag(F ). We also note that any closed ball in M × M is compact by Assumption 3.9. Hence, as a consequence of Proposition A.4, there exists a finite and nontrivial Borel measure ν s K on M × M such that supp[ν s K ] ⊂ K and I s (ν s K ) < ∞. On the other hand, by the proof of Lemma 3.5 again, we have γ(s) < γ(v) < 1 for any v ∈ (s F − ε, s). Then, for any v ∈ (s F − ε, s) and T ∈ (0, 1/2), , which implies that for some c 2 > 0, Let X γ(v) be the γ(v)-stable subordinate process of the process X. Since γ(v) < γ(0), it follows by Lemma 4.2 and (4.11) with (4.3) that under (WUHK) pointwisely, there exists a constant c 3 > 0 such that for all x, y ∈ K, Here we note that x 1 = x 2 and y 1 = y 2 and thus (x, y)).
In particular, since J 1 = ∞ and (NDLHK) hold, X is recurrent. We can then have an inequality corresponding to (3.28) with some compact set K ⊂ (diag(F )) γ(s) . We further follow the proof of Theorem 3.10(2) to obtain The proof is complete.
The assumption in Theorem 4.6 (2) implies that the process X is recurrent. For instance, the assumption is fulfilled by a class of α-stable-like symmetric jump processes on ultra-metric spaces with any α > 0 (see, e.g., [3,18] for details). On the other hand, it is natural to allow X to be transient. For instance, if X 1 and X 2 are independent symmetric stable process on R with index α ∈ (1, 2), then their direct product process is transient. Here we utilize two types of the Wiener tests in Propositions B.6 and B.9. The price is to assume that the collision place F is closed, and to make the next assumption on M in addition to Assumption 3.9. Note that under Assumption 4.8, M × M is also connected. Theorem 4.9. Suppose that Assumptions 3.8, 3.9 and 4.8 hold. Let the processes X 1 and X 2 satisfy Assumption (H), (WUHK) and (HR), so that X 1 and X 2 are independent. Let F ⊂ M be an (s F , t F )-set for some s F ∈ (0, s 0 ) and t F > 0 with γ(s F ) < 1. Assume the following conditions on X 1 and X 2 : • For any γ ∈ (γ(s F ), 1], (4.3) holds, and there exists a constant c 1 > 0 such that for any T ∈ (0, 1/2), dt. (4.22) • J 1 < ∞, and (4.9) holds with γ = 1. Furthermore, there exists a constant c 2 > 0 such that for any T ≥ 1, dt. (4.23) Proof. For γ ∈ (0, 1], we use the same notations σ diag(F ) , (diag(F )) γ(s) and (diag(F )) r γ(s) as in the proof of Theorem 4.6 (2). For any s ∈ (0, s F ) with γ(s F ) < γ(s) < 1, and J γ(s) ≤ J 1 < ∞ by assumption. Since (4.3) and (4.22) are also valid by assumption, we apply Proposition B.9 for X γ(s) and thus diag(F ) = (diag(F )) r γ(s) ⊂ (diag(F )) γ(s) . Then, for any y ∈ diag(F ), (4.25) Note that (4.3) with γ = 1 is valid by assumption. Since (4.9) and (4.23) are also valid by assumption, we apply Proposition B.6 with γ = 1 to show that P x (σ diag(F ) < ∞) = 1 for any x ∈ M × M. We also see that X diag(F ) ∈ diag(F ) because diag(F ) is closed. Therefore, by (4.25) and the strong Markov property of the process X, we obtain for any x ∈ M × M, , P x -a.s. for any x ∈ M × M. Letting s ↑ s F along a sequence, we get , P x -a.s. for any x ∈ M × M. The proof is complete.
(i) Let F ⊂ M be a Borel subset with s F = dim H (F ) > 0.
It immediately follows from Example 4.11 and Remark 2.7 that the second assertion (2) in Theorem 1.1 holds.

A Hausdorff measure and dimension A.1 Frostman lemma
Here we follow the arguments in [16,22,23] to give a proof of the Frostman lemma on the complete separable metric space.  (ii) If t ≥ s > 0, then ϕ(t) ≥ ϕ(s).
(2) A Hausdorff function ϕ is of finite order, if there exists a constant η > 0 such that Let (M, d) be a complete separable metric space. Let ϕ be a continuous Hausdorff function of finite order such that ϕ(0) = 0. For any subset F of M and δ > 0, define In the rest of this part, we always assume that ϕ is a continuous Hausdorff function of finite order such that ϕ(0) = 0. For any Borel measure ν on M, define the ϕ-energy of ν as Then ν(F ) ≤ cH ϕ (F ).
Proof. Let ν be a Borel measure on M, and let ν * be the associated outer measure. Then any Borel subset B ⊂ M is ν * -measurable and ν * (B) = ν(B). Suppose that (A.2) holds for some F ∈ B(M) and c > 0. For n, m ∈ N, define a Borel subset F n,m = x ∈ F : ν(B(x, r)) ≤ c + 1 n ϕ(r) for any r ∈ 0, 1 m , If F n,m ∩ U k = ∅, then, for any x ∈ F n,m ∩ U k , Therefore, Since the covering {U k } ∞ k=1 is taken arbitrary, we have for any n, m ∈ N, Letting m → ∞ and then n → ∞, we obtain ν(F ) ≤ cH ϕ (F ).
Proof of Proposition A.2. Let F be a Borel subset of M. Suppose that there exists a finite and nontrivial Borel measure ν on M such that supp[ν] ⊂ F and I ϕ (ν) < ∞. Let Then, for any x ∈ F 1 , there exist ε > 0 and a decreasing positive sequence {r n } ∞ n=1 such that r n ↓ 0 as n → ∞ and ν(B(x, r n )) ϕ(r n ) ≥ ε, n = 1, 2, 3, . . .
We also have ν({a}) = 0 for any a ∈ M because I ϕ (ν) < ∞. Hence, for each r n , there exists q n ∈ (0, r n ) such that Moreover, we may and do assume that q n > r n+1 for all n ≥ 1 by taking subsequences of {r n } ∞ n=1 and {q n } ∞ n=1 respectively, if necessary. Under this assumption, the annuli B(x, r n ) \ B(x, q n ), n = 1, 2, 3, . . . , are disjoint.
In the following, we present a criterion for a Borel set to be of zero Hausdorff measure in terms of the potential.
Proposition A.4. Let M satisfy Assumption 3.9, and let F be a Borel subset of M such that H ϕ (F ) > 0. Then for any ε ∈ (0, 1), there exists a finite and nontrivial Borel measure ν on M such that supp[ν] ⊂ F and I ϕ ε (ν) < ∞.
The proof of Proposition A.4 needs three lemmas. The first two lemmas concern the upper bound of the Hausdorff measure.
We refer to the next key lemma for the regularity of the Hausdorff measure. Let c * > 0 be the same constant as in Lemma A.5 and Since Lemma A.5 yields Then h n (x) → h(x) as n → ∞ for any x ∈ E \ E 1 . Hence, by the Egorov theorem and (A.1), there exists a compact subset K of E \ E 1 such that H ϕ (K) > 0 and Since h(x) ≤ 2c * for any x ∈ E \ E 1 , (A.6) implies that for some r 0 > 0, As the function ϕ is nondecreasing, we also have Hence if we let b = (4c * ) ∨ c 1 , then (A.7) and (A.8) yield ν(B(x, r)) ≤ bϕ(r) for any x ∈ K and r > 0. Moreover, by noting that K ⊂ E, we obtain The proof is complete. Fix a point x in K and let m(r) = ν(B(x, r)) for r > 0. Then, for any ε ∈ (0, 1),

Proof of Proposition
Moreover, we obtain by (A.9) that Therefore, there exists a constant c 1 = c 1 (ε, K) > 0 such that The proof is complete.

A.2 Locally s-set and s-measure
Let (M, d) be a locally compact separable metric space. In this subsection, for x ∈ M and r > 0, we still use the notation B(x, r) for the closed ball with radius r centered at x, i.e., B(x, r) = {y ∈ M : d(x, y) ≤ r}. We recall the notions of locally s-sets and s-measures.
Definition A.8. Let s and t be positive constants.
(i) A subset F of M is called a locally s-set, if F is a closed set and there exists a positive Borel measure η on M such that supp[η] ⊂ F , and, for some positive constants r 0 , c 1 (F ) and c 2 (F ), c 1 (F )r s ≤ η(B(x, r)) ≤ c 2 (F )r s , x ∈ F, r ∈ (0, r 0 ).
The measure η is called the locally s-measure of F .
(ii) A subset F of M is called a globally t-set, if F is a closed set and there exists a positive Borel measure η on M such that supp[η] ⊂ F , and, for some positive constants r 0 , c 3 (F ) and c 4 (F ), The measure η is called the globally t-measure of F .
(iii) A subset F of M is called an (s, t)-set, if F is a locally s-set and globally t-set such that the corresponding locally s-measure and globally t-measure are the same. In particular, an (s, s)-set is called the s-set.
Let F be a globally t-set with t-measure η. Then we also have This fact is already observed in [27,Section 1]. Fix x ∈ M and x 0 ∈ F . If r ≥ 2d(x, x 0 ), then, for any y ∈ B(x 0 , r/2),

B Wiener tests
In this appendix, we establish the Wiener tests for recurrence and regularity of the stable subordinate process of the direct product process. Hereafter, (M, d) is a locally compact separable metric space and µ is a positive Radon measure on M with full support.

B.1 Transience and regularity
where {F t } t≥0 is a minimum completed admissible filtration, and θ t : Ω → Ω is the shift of paths such that X t • θ s = X t+s for s, t ≥ 0. In this subsection, we will present equivalent conditions for the transience and regularity of sets relative to the process X. Let and define the tail σ-field T by We say that T is trivial, if for any A ∈ T , P x (A) = 1 for any x ∈ M or P x (A) = 0 for any x ∈ M. For B ∈ B(M), let σ B = inf{t > 0 : X t ∈ B} be the first hitting time of X to B, and let L B = sup{t > 0 : X t ∈ B} be the last exit time of X from B. Then, {L B < ∞} ∈ T . Below, for x ∈ M, n ≥ 1 and B ∈ B(M), define when λ > 1, and B x,λ n = y ∈ B : λ n+1 ≤ d(x, y) ≤ λ n when 0 < λ ≤ 1.
We first give equivalent conditions for the transience.
Lemma B.1. Assume that the process X is conservative and transient, and that T is trivial. Then, for any x ∈ M, λ > 1 and B ∈ B(M), the following assertions are equivalent to each other.
Proof. Since T is trivial and {L B < ∞} ∈ T , we obtain the equivalence between (i) and (ii). We now prove the equivalence between (i) and (iii). We simply write B n for B x,λ n . Suppose first that (i) holds. Then for P x -a.s. ω ∈ Ω, we have X t (ω) / ∈ B for all t > L B (ω). Since X is conservative, it follows by [14, p.95, Corollary] that P x (X t− ∈ M and X t ∈ M for any t > 0) = 1.
Then, by (i), which implies that Therefore, (iii) follows. Suppose next that (iii) holds. Since X is transient, we see that which yields (i).
We next show the equivalent conditions for the regularity of points.
Lemma B.2. Assume that the process X is transient and that the single point set {x} is polar relative to the process X. If the Blumenthal zero-one law holds for the process X, then, for any λ ∈ (0, 1) and B ∈ B(M), the following three conditions are equivalent to each other.
Proof. The equivalence between (i) and (ii) follows by the Blumenthal zero-one law. We now prove the equivalence between (i) and (iii). Assume first that (i) is valid. Then for P x -a.s. ω ∈ Ω, there exists a sequence {t n (ω)} such that t n (ω) ↓ 0 as n → ∞ and X tn(ω) (ω) ∈ B for all n ≥ 1. Since d(x, X tn(ω) (ω)) → 0 as n → ∞, we have (iii).

B.2 Zero-one law for the tail event
Let X 1 and X 2 be the independent µ-symmetric Hunt processes on M, and let X be the direct product of X 1 and X 2 on M × M. For γ ∈ (0, 1], let X γ be the γ-subordinate process of X.
We will present the zero-one law for the tail event of X γ . Let {F γ t } t≥0 denote the minimum completed admissible filtration of the process X γ , and set Let T γ be the tail σ-field of X γ , i.e., We then have Let us prove Proposition B.3 by following the proof of [24, Theorem 2.10] (see also the references therein for the original proofs). For i = 1, 2, let p i (s, x i , y i ) be the heat kernel of the process X i , and q γ (t, x, y) the heat kernel of X γ , i.e., q γ (t, x, y) = ∞ 0 p 1 (s, x 1 , y 1 )p 2 (s, x 2 , y 2 )π t (s) ds.
We also let q 1,γ (t, u, w) and q 2,γ (t, u, w) be the heat kernels of the subordinate processes of X 1 and X 2 , respectively, i.e., Since for i = 1, 2, any t > 0 and u ∈ M, we have for any A, B ∈ B(M), For x = (x 1 , x 2 ) ∈ M × M and r > 0, let B(x, r) be an open ball with radius r centered at x with respect to the product metric, i.e., B(x, r) = {y = (y 1 , y 2 ) ∈ M × M : d(x 1 , y 1 ) + d(x 2 , y 2 ) < r} .
Let τ B(x,r) = inf{t > 0 : X γ t ∈ B(x, r)} be the exit time from B(x, r) of the process X γ .
Lemma B.4. If the independent processes X 1 and X 2 satisfy (WUHK), then there exists a constant c 1 > 0 such that for any x ∈ M 1 0 × M 2 0 , t ≥ 0 and r > 0, .
Proof. Suppose that the processes X 1 and X 2 satisfy (WUHK). For x = (x 1 , x 2 ) ∈ M and r > 0, we write τ = τ B(x,r) for simplicity. Then Since X 1 and X 2 satisfy (WUHK), it follows by (3.5) that there exist positive constants c 1 and c 2 such that for each i = 1, 2, and for any t > 0 and r > 0, .

(B.4)
Then, by the triangle inequality and the strong Markov property, we get where the last inequality follows from the argument of (B.4). We thus complete the proof.
Proof of Proposition B.3. We split the proof into three parts.
(i) Throughout the proof, we will fix ε > 0 small enough. By Lemma B.4 and (2.13), there exist positive constants c 1 and c 2 so that for all x ∈ M × M, t 0 > 0 and c ⋆ ≥ 1, Take c ⋆ large enough so that c 2 /c γα 1 ⋆ < ε. Let c * > 1 and t 1 > 0 be constants which will be fixed later in this order. We first fix c * > 1. Then, by (WUHK), (3.5) and (2.13), Here the positive constants c 3 , c 4 , c 5 above are independent of the choices of c * , t 1 , x 1 and y 1 . Similarly, we have (B.7) Note that, by (B.3), Hence it follows by (B.6) and (B.7) that, if we take c * > 1 so large that (c 5 + c 6 )/c γα 1 * < ε/4, then for any x ∈ M × M and t 1 > 0, (B.8) In the same way, we can take and fix c * > 1 so large that for any z ∈ M × M and t 1 > 0, (B.8) holds and Next we assume that d( , then, by the triangle inequality and (B.10), Therefore, it follows by (B.9) that Combining this with (B.8), we obtain (B.11) Since the processes X 1 and X 2 satisfy (HR) by assumption, for each i = 1, 2, there exist constants θ i ∈ (0, 1] and C i > 0 such that for any t > 0 and u, v, w ∈ M, Therefore, as in the proof of Lemma 3.1 (1), we can show that ), then there exist positive constants c 7 and η such that where in the second inequality we used (2.15). In particular, if we take t 1 > 0 so large that Set I 1 (t, x, y, z) = ∞ 0 p 1 (s, x 1 , y 1 )(p 2 (s, x 2 , y 2 ) − p 2 (s, z 2 , y 2 )) π t (s) ds, and I 2 (t, x, y, z) = ∞ 0 p 2 (s, z 2 , y 2 )(p 1 (s, x 1 , y 1 ) − p 1 (s, z 1 , y 1 )) π t (s) ds. and Let t 0 and t 1 be the positive constants which are fixed in the argument in part (i). Then, for A ∈ T γ , there exists an event C ∈ F γ ∞ such that A = C • θ t 0 +t 1 . Let g(x) = P γ x (C) for x ∈ M × M. Since Y is F γ t 0 -measurable and the Markov property yields and ) . Since g ∞ ≤ 1, we get, by (B.17),

B.3 Wiener test for recurrence
In this subsection, we establish the Wiener test for the recurrence relative to stable-subordinate direct-product processes by using Proposition B.3. Let X 1 and X 2 be two independent µsymmetric Hunt processes on M satisfying Assumption (H), (WUHK) and (HR), and let X be the direct product of X 1 and X 2 on M × M. For γ ∈ (0, 1], X γ denotes the γ-subordinate process of X.
Suppose that Assumptions 4.8 is satisfied. Then, by Proposition B.3, one can apply the argument of Lemma B.1 to the process X γ and obtain that, if the process X γ is transient, then Proposition B.5. Let M satisfy Assumptions 3.8, 3.9 and 4.8. Suppose that the two independent processes X 1 and X 2 satisfy Assumption (H), (WUHK) and (HR). Fix a constant λ > 1 so that φ −1 i (λt) ≥ 2φ −1 i (t) for i = 1, 2 and all t > 0. Assume that for some γ ∈ (0, 1], J γ < ∞, Proof. We first note that, by Remark 2.6 (iii), under the assumption of this proposition, the processes X 1 and X 2 satisfy (NDLHK). We also note that X γ is transient by Lemma 4.3. Then, we take an approach similar to the proof of [28, p.67, Theorem 3.3]. In what follows, we simply write B n for B x,λ,φ n . Let m, n be positive integers such that |m − n| > 1. Without loss of generality, we suppose that m > n + 1. For any z ∈ B m and y ∈ B n , if φ 1 (d(x 1 , z 1 )) ≥ φ 2 (d(x 2 , z 2 )), then Noting that φ −1 1 (λφ d (x, y)) ≥ 2φ −1 1 (φ d (x, y)) by assumption, we have, by the triangle inequality, Since (B.24) also implies that we have, by the triangle inequality, Hence, by (2.13), there exists a constant c 1 ∈ (0, 1) such that Combining this with (B.25), we get In the same way, one can see that the inequality above is valid also when φ 1 (d(x 1 , z 1 )) ≤ φ 2 (d(x 2 , z 2 )). Since (4.3) and (4.9) hold by assumption, Lemma 4.5 with (2.15) and (B.26) implies that for any positive integers m, n with |m − n| > 1, and for any z ∈ B m and y ∈ B n , z)).

(B.27)
On the other hand, since B n is compact by Assumption 3.9, there exists a positive Radon measure ν n on M such that supp[ν n ] ⊂ B n , and, for any z ∈ B m , Note that if σ Bm < ∞, then X γ σ Bm ∈ B m because B m is closed. Therefore, by the strong Markov property of the process X γ , (B.28) By the same argument as before, we also see that for any y ∈ B n , and thus P γ Noting that we obtain, by (B.28) and (B.29), Hence, by combining (B.23) with Lemma B.10 below, we get the following equivalence: The proof is complete.
• There exists a constant c 1 > 0 such that for any T ≥ 1, dt.
For any n ≥ 1, define Hence, by the 0-order version of [17, Exercise 4.2.2], ν n is of finite 0-order energy integral relative to the process X γ , and the function g(z) := Bn u γ 0 (z, y) ν n (dy) is a quasi-continuous and excessive µ-version of the 0-potential of ν n . Since B n is compact, it follows by (2.4) and (B.32) that for all sufficiently large n ≥ 1, where Cap γ (0) is the 0-order capacity relative to (E γ , F γ ). Furthermore, if ν γ n denotes the equilibrium measure of B n , then, by Lemma 4.5, P γ x (σ Bn < ∞) = Bn u γ 0 (x, y) ν γ n (dy) ≥ c 10 ν γ n (B n ) Therefore, Proposition B.5 yields the desired assertion.

B.4 Wiener test for regularity
In this subsection, we show the Wiener test for the regularity of points relative to the process X γ . Let X 1 and X 2 be two independent µ-symmetric Hunt processes on M satisfying Assumption (H), (WUHK) and (HR), and let X be the direct product of X 1 and X 2 on M × M. For γ ∈ (0, 1], X γ denotes the γ-subordinate process of X. It follows from the proof of Lemma B.2 that the following equivalence holds: if the process X γ is transient and {x} is polar relative to X γ , and the Blumenthal zero-one law holds for the process X γ , then Using this equivalence, we can prove Proposition B.8. Let M satisfy Assumptions 3.8, 3.9 and 4.8. Suppose that the independent processes X 1 and X 2 satisfy Assumption (H), (WUHK) and (HR). Take λ ∈ (0, 1) so that φ −1 i (t/λ) ≥ 2φ −1 i (t) for i = 1, 2 and any t > 0. If for some γ ∈ (0, 1], the process X γ is transient and {x} is polar relative to X γ , then, for any B ∈ B(M × M), Proof. It follows from Proposition B.3 that, under the assumptions of this proposition, the Blumenthal zero-one law holds for the process X γ . We then take an approach similar to the proof of [28, p.67, Theorem 3.3]. To simplify the notation, we write B n for B x,λ,φ n . Let m and n be positive integers such that |m − n| > 1. Without loss of generality, we assume that n > m + 1. For any z ∈ B m and y ∈ B n , if φ 1 (d(x 1 , z 1 )) ≥ φ 2 (d(x 2 , z 2 )), then Hence, by the triangle inequality and φ −1 which yields φ d (y, z) ≥ φ d (x, y).
Let F ⊂ M be a locally s F -set, and η the corresponding s F -measure. Fix x ∈ diag(F ) and λ ∈ (0, 1). Let B = diag(F ) and B x,λ,φ n as in (B.37). Then, there exists a constant c 1 > 0 such that for all n ≥ 1, Furthermore, we can also follow the argument of (B.31) to show that there exist c 2 > 0 and n 0 ≥ 1 such that for all n ≥ n 0 , For B ∈ B(M × M), let B r γ be the totality of regular points for B relative to the process X γ , i.e., B r γ = y ∈ M × M : P γ y (σ B = 0) = 1 . If B is closed, then B r γ ⊂ B by the right continuity of sample paths of X γ . Proposition B.9. Let M satisfy Assumptions 3.8, 3.9 and 4.8. Suppose that the independent processes X 1 and X 2 satisfy Assumption (H), (WUHK) and (HR). Let F ⊂ M be a locally s F -set for some s F > 0 with γ(s F ) < 1. Assume that the following conditions hold for some γ ∈ (γ(s F ), 1]: • J γ < ∞ and • There exists a constant c 1 > 0 such that for any T ∈ (0, 1/2), dt. (B.42) Then, for any x ∈ diag(F ), P γ x (σ diag(F ) = 0) = 1, that is, (diag(F )) r γ = diag(F ).
Proof. We prove this proposition by applying Proposition B.8 to the process X γ . To do so, we first verify that X γ is transient and any one point set is polar relative to X γ . Since J γ < ∞ by assumption, X γ is transient by Lemma 4.3 (2). By Lemma 4.1 and (B.41) with Remark 2.6 (iii), there exists a constant c 0 > 0 such that for any x ∈ M, Hence, by Lemma 3.6, any one point set is polar relative to X γ . Let F be a locally s F -set, and η the corresponding s F -measure. We simply write ν for ν η . We take ε 0 ∈ (0, 1) so small that φ −1 (ε 0 r)/φ −1 (r) ≤ 1/2 for any r > 0. For fixed x ∈ diag(F ) and λ ∈ (0, 1), let B = diag(F ) and B n = B x,λ,φ n . Let z ∈ M × M. We first assume that φ d (z, x) ≥ λ n /ε 0 . Then, by the triangle inequality, we have for any y ∈ B n , φ d (z, y) ≥ λ n . Combining this with Lemma 4.5 and (B.39), we get Bn u γ 0 (z, y) ν(dy) ≤ c 1 dt. (B.43) We next assume that φ d (z, x) < λ n /ε 0 . Since φ d (z, y) ≤ c 3 λ n for any y ∈ B n and (B.42) holds, we can follow the calculation in (B.35) and its subsequent argument to prove that where c * = c 2 ∨ c 5 and dt.
Hence, by following the proof of Proposition B.6, there exists a constant c 6 ∈ (0, 1] such that P γ x (σ Bn < ∞) ≥ c 6 for any n ≥ 1. Then, by Proposition B.8, the proof is complete.

B.5 Generalized Borel-Cantelli lemma
We state the following generalized Borel-Cantelli lemma for the readers' convenience.