Ergodic properties for \alpha-CIR models and a class of generalized Fleming-Viot processes

We discuss a Markov jump process regarded as a variant of the CIR (Cox-Ingersoll-Ross) model and its infinite-dimensional extension. These models belong to a class of measure-valued branching processes with immigration, whose jump mechanisms are governed by certain stable laws. The main result gives a lower spectral gap estimate for the generator. As an application, a certain ergodic property is shown for the generalized Fleming-Viot process obtained as the time-changed ratio process.


Introduction
The study of ergodic behaviors of a Markov process is of quite interest for various reasons. For instance, it is typical that the analysis of such behaviors depends heavily on the mathematical structure of the model, so that resulting properties are expected to yield deep understanding for it. In this paper, we discuss two specific classes of measure-valued Markov jump processes. The one consists of what we will call measure-valued α-CIR models, each of which is thought of as an infinite-dimensional extension for a jump-type version of the CIR model, and the other generalizes naturally a class of Fleming-Viot processes with parent-independent mutation. As for the measure-valued α-CIR model, identification of a stationary distribution is easy thanks to its nice structure as a measure-valued branching process with immigration (henceforth MBI-process). For the latter class of models, stationary distributions are identified recently in [4]. A key idea there is to exploit a special relationship with measure-valued α-CIR models, which enabled us to give an expression for stationary distributions of our generalized Fleming-Viot processes in terms of those of the measure-valued α-CIR models. It should be mentioned that such links have been discussed in another context in [1] and [2]. Our attempt here is to rely still on that relationship to explore ergodic properties for the generalized Fleming-Viot process.
It is worth illustrating by taking up a one-dimensional model which is regarded as a 'prototype' of the above mentioned MBI-process. Consider the well-known CIR model governed by generator where b ∈ R and c > 0 are constants. Rather than its importance in the context of mathematical finance, we emphasize that this model belongs to the class of continuous state branching processes with immigration (CBI-processes in short). (See [5] for fundamental results regarding this class.) Let 0 < α < 1 be arbitrary. As a natural non-local version of (1.1) within the class of generators of conservative CBI-processes (cf. Theorem 1.2 in [5]), we will be concerned with where Γ(·) is the gamma function. The operator L α with b = 0 is found in Example 1.1 of [5]. Observing that, as α ↑ 1, L α F (z) → L 1 F (z) for any z > 0 and 'nice' functions F on R + , we call a Markov process associated with aL α for some constant a > 0 an α-CIR model. Although this class of models would be of interest in its own right especially in the mathematical finance context, our main motivation to study it is the analysis of ergodicity for a jump-type version of a Wright-Fisher diffusion model with mutation, which is obtained through normalization and random time-change from two independent processes with generators of the form (1.2), say L ′ α and L ′′ α , with common α and b. On the level of generators such a link can be reformulated as the identity (L ′ α F (·, z 2 )) (z 1 ) + (L ′′ α F (z 1 , ·)) (z 2 ) = C(z 1 + z 2 ) −α A α G z 1 z 1 + z 2 , z 1 , z 2 > 0, (1.3) where G is any smooth function on [0, 1], F is defined by F (z 1 , z 2 ) = G(z 1 /(z 1 + z 2 )), C is a positive constant independent of G and A α is the generator of a jump-type version of the Wright-Fisher diffusion model. (See (1.3) in [4] for a concrete expression for A α or (5.1) below for its generalization.) A significant consequence of (1.3) is that Dirichlet form associated with A α is, up to some multiplicative constant, a restriction of Dirichlet form associated with the two independent α-CIR models. Therefore, ergodic properties of α-CIR models would be expected to help us obtain the same kind of results for the process associated with A α . Such an idea can extend naturally to the measure-valued α-CIR model, which is regarded roughly as 'continuum direct sum' of α-CIR models with coefficients depending on a spatial parameter. Because of this structure studying ergodic properties of the extended model would be reduced to the one-dimensional case at least under the assumption of uniform bounds for the coefficients. In addition, as observed in [4], the relation (1.3) admits a generalization in the setting of measure-valued processes. (See also (5.2) below.) For this reason the above mentioned extension of the α-CIR model is considered to play an important role in studying the generalized Fleming-Viot process obtained as the time-changed ratio process.
The organization of this paper is as follows. In Section 2, we introduce the measure-valued α-CIR model, and it is shown in Section 3 that a lower spectral gap estimate for the generator can reduce to the one-dimensional case in a suitable sense. In Section 4, we prove such an estimate for L α , establishing exponential convergence to equilibrium for the measure-valued α-CIR model. The latter result will be applied to a class of generalized Fleming-Viot processes in Section 5.

The measure-valued α-CIR models
To discuss in the setting of measure-valued processes, we need the following notation. Let E be a compact metric space and C(E) (resp. B + (E)) the set of continuous (resp. nonnegative, bounded Borel) functions on E. Also, denote by C ++ (E) the set of functions in C(E) which are uniformly positive. Define M(E) to be the totality of finite Borel measures on E, and we equip M(E) with the weak topology. Denote by M(E) • the set of non-null elements of M(E). The set M 1 (E) of Borel probability measures on E is regarded as a subspace of M(E). We also use notation η, f := E f (r)η(dr). For each r ∈ E, let δ r denote the delta distribution at r. Given a probability measure Q, we write also E Q [·] for the expectation with respect to Q.
Suppose that 0 < α < 1, a ∈ C ++ (E), b ∈ C(E) and m ∈ M(E) • are given. As a natural generalization of the α-CIR model generated by (1.2), we shall discuss in this section the Markov process on M(E) associated with . The operator L (3) α describes the mechanism of immigration. (See (9.25) in [6] for a general form of generators of MBI-processes. In our model, there is no 'motion process', whose generator is thus considered to be A ≡ 0.) Set Ψ f (η) = e − η,f for f ∈ B + (E) and define D to be the linear span of functions Ψ f with f ∈ C ++ (S). It is direct to see that for any f ∈ B + (E) L α is well-defined also on the class F of functions Ψ of the form for some ϕ ∈ C 2 0 (R n + ), f i ∈ C ++ (E) and a positive integer n. Our first result below not only verifies this but also gives bounds for each L (k) α Ψ (k ∈ {1, 2, 3}) for a more general class of functions Ψ. In what follows · ∞ denotes the sup norm. Let F be the totality of functions Ψ of the form (2.3) with ϕ ∈ C 2 (R n + ) and f := (f 1 , . . . , f n ) ∈ C ++ (E) n satisfying the following conditions; there exist nonnegative constants C and Note that η, f := ( η, f 1 , . . . , η, f n ) ∈ R n f for any η ∈ M(E) • . Intuitively, these conditions enable one to control the effect of long-range jumps governed by stable laws, and are inspired by the calculations in the proof of Proposition 3.4 in [4].
Following [10], we consider the operator (L α , F ) as an operator on C ∞ (M(E)), the set of continuous functions on M(E) vanishing at infinity. In the theorem below we collect basic properties of L α and the associated transition semigroup.
) and the closure (L α , D(L α )) generates a C 0 -semigroup (T (t)) t≥0 . Moreover, D is a core for L α , and for each f ∈ B + (E) and η ∈ M(E) , then Markov process with transition semigroup (T (t)) t≥0 is ergodic in the sense that for every initial state η ∈ M(E), the law of the process at time t converges to a unique stationary distribution, say Q α , as t → ∞. Moreover, the Laplace functional of Q α is given by Proof. (i) If m were the null measure, the assertions except (2.16) follow from more general Theorem 1.1 in [10], and also (2.16) is deduced from the proof of it. Indeed, V t f (r) was given there implicitly by  [6].) Based on these facts, the proof of the assertions for m ∈ M(E) • can be done by modifying suitably the proof of Corollary 1.3 in [10], which deals with the immigration mechanism described by the operator Ψ → m, δΨ δη . A (possibly unique) non-trivial modification would be the step to construct, for each η ∈ M(E) and t ≥ 0, q t (η, ·) ∈ M 1 (M(E)) with Laplace transform given by the right side of (2.15). By the observation made in the last paragraph, we have p t (η, ·) ∈ M 1 (M(E)) such that Additionally, for every η ∈ M(E), let s α (η, ·) be the law of an α-stable random measure with parameter measure η, i.e., and define p t,α (η, ·) ∈ M 1 (M(E)) to be the mixture It then follows that Therefore, for each N = 1, 2, . . ., the convolution which converges to the right side of (2.15) as N → ∞. Thus, the weak limit of q is verified by combining (2.2) with (2.18). Once (2.15) is in hand, the assertion that D is a core for L α follows as a direct consequence of Lemma 2.2 in [13]. (ii) As t → ∞ the right side of (2.15) converges to This proves ergodicity required and that the unique stationary distribution Q α has the Laplace functional given by the right side of (2.17). The fact that Q α is supported on M(E) • follows by observing that the right side of (2.17) with f ≡ β > 0 tends to 0 as β → ∞.
We call the Markov process on M(E) associated with (2.1) in the sense of Theorem 2.2 the measure-valued α-CIR model with triplet (a, b, m). It is said to be ergodic if b ∈ C ++ (E).
Remarks. (i) A random measure with law Q α in Theorem 2.2 (ii) is an infinitedimensional analogue of the random variable with law sometimes referred to as a (non-symmetric) Linnik distribution, whose Laplace exponent is of the form λ → c log(1 + dλ α ) for some c, d > 0. Observe from (2.17) that, as α ↑ 1, Q α converges to Q 1 , the law of a generalized gamma process such that In addition, one can see that . (For instance, this is immediate for Ψ = Ψ f from (2.2).) L 1 is the generator of an MBI-process discussed in Section 4 of [11] and in Section 3 of [10], where Q 1 was shown to be a reversible stationary distribution of the process associated with L 1 .
(ii) In contrast, Q α (0 < α < 1) is not a reversible stationary distribution of the measure-valued α-CIR model. See Theorem 2.3 in [3] for an assertion of this type regarding CBI-processes. Essentially the same proof works at least in the case of ergodic measure-valued α-CIR models. Namely, one can show, by a proof by contradiction, that the formal symmetry ] fails for some f, g ∈ C ++ (E). For this purpose, an expression for the Dirichlet form E Qα [(−L α )Ψ f · Ψ g ] given Remark after Lemma 3.1 below is helpful.

Expressions for Dirichlet form
From now on, we suppose additionally that b ∈ C ++ (E). Thus, only ergodic measurevalued α-CIR models will be discussed. To study the speed of convergence to equilibrium in the L 2 -sense, we consider the symmetric part of Dirichlet form associated with Γ(·, * ) being the 'carré du champ': where n B (dz) = (α + 1)z −α−2 dz/Γ(1 − α) and n I (dz) = αz −α−1 dz/Γ(1 − α) govern the jump mechanisms associated with branching and immigration, respectively. The same argument as in the proof of Proposition 1.6 in [10] shows that (L α , F ) is closable in L 2 (Q α ) and that the closure (L α (2) , D(L α (2) )) generates a C 0 -semigroup (T 2 (t)) t≥0 on L 2 (Q α ) which coincides with (T (t)) t≥0 when restricted to C ∞ (M(E)). We set It is known that the largest κ ≥ 0 such that We refer the reader to e.g. Theorem 2.3 in [7] for the proof of this fact in a general setting. Besides, an estimate of the form gap(L α (2) ) ≥ κ implies that L α (2) has a spectral gap below 0 of size larger than or equal to κ. (See Remark 1.13 in [10].) In calculating Dirichlet form and the variance functional with respect to Q α , we will make an essential use of the following expression for the 'log-Laplace functional' in (2.17): where m a (dr) = a(r) −1 m(dr), Λ is the Lévy measure of the infinite divisible distribution on (0, ∞) with Laplace exponent λ → log(1 + λ α ) and f * = (a/b) 1/α f . In what follows the domain of integration is understood to be (0, ∞) when suppressed. Define nonnegative functions K B and K I on R 2 + by and respectively. The above identities are verified easily by differentiating in s and t.
which is finite.
Proof. It follows that Note that the function we need only to show that and that this is finite. The second equality can be verified to hold by (2.17) and (3.1) together: For the proof of the last equality in (3.4), we make use of another expression for I(f + g; h) deduced from (2.17) only: Here, by (3.2) and so I(f + g; h) is finite.
Remark. Noting that (3.4) is clearly valid for every h ∈ B + (E) and combining (2.2) with (3.4), we get for any f, g ∈ B + (E) from which the last expression in Lemma 3.1 for the symmetric part can be recovered.
Our objective is to show the positivity of gap(L α (2) ). The contribution here in this direction is the reduction to a certain estimate regarding the one-dimensional model. For a measurable function f on E, the essential supremum (resp. the essential infimum) of f with respect to m is denoted by ess sup the linear span of functions on R + of the form F λ (z) := e −λz for some λ > 0.
then for any Ψ ∈ D var(Ψ) ≤ γ ess sup and it holds that gap(L α (2) ) ≥ γ −1 ess inf This kind of reduction was discovered by Stannat [12] (Theorem 1.2) for a lower estimate for the quadratic form of gradient type. In particular, for the process associated with L 1 in Remark at the end of Section 2, the condition corresponding to (3.5) reads where Λ 1 (dz) = z −1 e −z dz is the Lévy measure of a gamma distribution. While (3.7) with γ = 1 is verified easily by applying Schwarz's inequality to F (z) − F (0) = z 0 F ′ (w)dw, showing an inequality of the form (3.5) is more difficult and we postpone it until the next section. However, as will be seen below, the reduction itself is proved in a similar way to [12].

Proof of Theorem 3.2. Consider a function Ψ expressed as a finite sum Ψ =
Rewrite in terms of the N-fold product measures m ⊗N a and Λ ⊗N to obtain the following disintegration formula for the variance functional: where r N = (r 1 , . . . , r N ) and z N = (z 1 , . . . , z N ). Given r N = (r 1 , . . . , r N ) ∈ E N and z 1 , . . . , z N −1 ∈ R + arbitrarily, apply (3.5) to the function (3.10) Here, a suitable change of variable has been made for each integral with respect to n B (dy) and n I (dy) in order to replace f * i (r N )y by f i (r N )y. Set C = ess sup in distributional sense. Combining (3.9) with (3.10), we can dominate 2var(Ψ)/γ by where the last two equalities are seen by similar calculations to (3.8) and (3.9). Since the symmetric part E of Dirichlet form is bilinear, (3.6) for Ψ ∈ D follows from Lemma 3.1. It remains to prove that (3.6) extends to Ψ ∈ D(L α (2) ). Since (L α (2) , D(L α (2) )) is the closure of (L α , F ) in L 2 (Q α ), we need only to show that (3.6) extends to Ψ ∈ F . Given Ψ ∈ F , we see from Theorem 2.2 (i) that there exists a sequence This implies that (3.6) holds for any Ψ ∈ F and we complete the proof of Theorem 3.2.

Spectral gap for the α-CIR model
This section is devoted to the proof of (3.5) for some 0 < γ < ∞. The strategy should be different from the one already mentioned for (3.7) with Λ 1 (dz) = z −1 e −z dz at least because no informative expression for the density of Λ in (3.1) appears to be available. Let us illustrate another approach we will take and call 'the method of intrinsic kernel' by revisiting (3.7). Suppose that F ∈ D is a finite sum F = i c i F λ i . We will use the notation 1 S standing for the indicator function of a set S and ∂ t = ∂/∂t for simplicity. Letting ψ 1 (λ) = log(1 + λ) = Λ 1 (dz)(1 − e −λz ), observe that where F (s) = i c i 1 [0,λ i ] (s). On the other hand, by putting K 1 (s, t) = stψ ′ 1 (s + t) It would be reasonable that ∂ s ∂ t K 1 is called the intrinsic kernel of the quadratic form V 1 . Similarly, (4.1) shows that the intrinsic kernel of U 1 is the function (s, t) → −ψ ′′ 1 (s + t).
Given two symmetric measurable functions J and K on R 2 + , we write K ≫ J if K − J is nonnegative definite in the sense that ds dtG(s)G(t)(K(s, t) − J(s, t)) ≥ 0 for any bounded Borel function G on R + with compact support. By virtue of Fubini's theorem, for some measure space (S, M) and measurable function σ on R + × S. In view of (4.1) and (4.2), it is clear that the inequality γV 1 For γ = 1, this holds true since by direct calculations which is of canonical form. Furthermore, this expression makes it possible to identify the associated 'remainder form': It should be emphasized that the above calculations require only an explicit form of the Laplace exponent ψ 1 .
Turning to the case 0 < α < 1, we adopt the method of intrinsic kernels to show (We continue to adopt this notation as it is a one-dimensional version of (3.1).) Namely, we shall (I) calculate the intrinsic kernels of U(F ) := Λ(dz)(F (z) − F (0)) 2 and of and then (II) compare the two kernels as nonnegative definite functions.
The following lemma concerns the step (I).
Proof. The intrinsic kernel of U is deduced in the same manner as (4.1). The derivation of (4.4) is similar to the calculations at the beginning of the proof of where the last equality is seen from Λ(dz)ze −λz = ψ ′ (λ), (4.3), (3.2) and (3.3). The rest of the proof is the same as (4.2) with K in place of K 1 .
Remark that, as α ↑ 1, the right side of (4.4) tends to 2st 1+s+t = 2K 1 (s, t). The main result of this section is obtained by accomplishing not only the step (II) but also identification of the remainder form.
Proof. Let J, K and K be as in Lemma 4.1. Recalling (3.2) and (3.3), we will exploit the following expression for 2K in (4.4): Here, it is direct to see that the sum of the last three terms on the right side equals By the above coincidence and (4.9) together where the last equality follows from (4.7). Consequently 2 K(s, t) − J(s, t) Each of the terms on the right side is nonnegative definite because of and Therefore, 2 K ≫ J and so 2V(F ) ≥ U(F ) for any F ∈ D. We further proceed to identify the remainder form F → 2V(F ) − U(F ). With the help of the canonical representations in the above, we deduce 2V Proof. For the aforementioned reason, it suffices to show the upper estimate To this end, we use (a variant of) a characterization due to Liggett ([7], (2.5)): This implies that for any Ψ such that 0 < var(Ψ) < ∞. We now take Ψ(η) = exp(−η(E)), for which T 2 (t)Ψ(η) is given by the right side of (2.15) with f = 1 E =: 1. Recalling that the log-Laplace functional of Q α is ψ(f ) = m a , log(1+ab −1 f α ) , one can derive by (2.15) where the last equality is deduced from Since by (2.16) (V t 1(r)) α ≤ e −tb(r) , ∆(t) := 2ψ(V t 1) − ψ(2V t 1) → 0 as t → ∞ and so lim inf By virtue of (4.10), the proof of Theorem 4.3 reduces to showing that − lim sup By straightforward calculations Further, with the help of the inequality log(1 + z) ≥ z/(1 + z) for z ≥ 0, we get Here, again by (2.16) and therefore lim sup This establishes (4.11) and completes the proof of Theorem 4.3.
Remarks. (i) The same argument may apply to the case α = 1. But the resulting bound exhibits discontinuity at α = 1. Indeed, for α = 1 (4.12) becomes where V t 1(r) is given by the right side of (2.16) with α = 1 and f (r) = 1. (See the formula for ψ t (f )(x) on p.1380 in [11].) As a result we can show that lim sup and accordingly gap(L 1 (2) ) ≤ ess inf (E,m) b. In fact, the equality is valid because the opposite inequality is implied by (3.7) with γ = 1 with the help of Theorem 1.2 in [12].
(ii) Theorem 4.3 would be regarded as a sort of continuous analogue of Theorem 2.6 in [7], which concerns a vector Markov process whose (countably many) components are independent Markov processes.

An application to generalized Fleming-Viot processes
Throughout this section, we assume that E is a compact metric space containing at least two distinct points. As mentioned in Introduction, the previous results will be applied to study convergence to equilibrium for a class of generalized Fleming-Viot processes, whose state space is M 1 (E). These models have been discussed in [4], where their stationary distributions were identified by exploiting connection with suitable measure-valued α-CIR models. To be more precise, given 0 < α < 1 and m ∈ M(E) • , we consider in this section the process associated with L α in (2.1) with a ≡ 1 ≡ b and the generalized Fleming-Viot process associated with where B c 1 ,c 2 denotes the beta distribution with parameter (c 1 , c 2 ) and Φ belongs to the class F 1 of functions of the form Φ f (µ) := µ ⊗n , f for some positive integer n and f ∈ C(E n ). It has been proved in [4] (Proposition 3.1) that the closure of (A α , F 1 ) generates a Feller semigroup (S(t)) t≥0 on C(M 1 (E)). Also, as observed in [4] (Proposition 3.3), the generators L α and A α together enjoy the following identity: where Ψ(η) = Φ(η(E) −1 η) and Φ is in the linear span F 0 of functions of the form µ → µ, f 1 · · · µ, f n with f i ∈ C ++ (E) and n being a positive integer. Notice that in one dimension (5.2) takes the form (1.3). Under the assumption that m(E) > 1, it was proved in Proposition 3.4 of [4] that is a unique stationary distribution of the process associated with A α , where Q α is the stationary distribution of processes associated with (2.1) with a ≡ 1 ≡ b. The pre-factor Γ(α + 1)(m(E) − 1) on the right side arises as the normalizing constant. More generally, the following moment formula holds for the random variable η(E) under Q α .
In the rest of the paper, we assume that m(E) > 1. A key observation for the subsequent argument is that (5.2) yields the following connection between Dirichlet forms: Here, the left side calls for some explanation since it can not necessarily be written as E(Ψ). Instead, one can approximate such special Ψ's suitably by functions in D(L α (2) ). This rather technical point is the content of the next lemma.
In fact, Ψ N ∈ D(L α (2) ) and L α (2) Ψ N = L α Ψ N as will be shown in the last half of the proof, and we temporarily suppose the validity of them. Obviously Ψ N → Ψ boundedly and pointwise on M(E) • and so Ψ N → Ψ in L 2 (Q α ). In view of Example preceding to Lemma 2.1 and the calculations in the proof of it, one can verify that L α Ψ N (η) → L α Ψ(η) for each η ∈ M(E) • . Moreover, by virtue of Lemma 2.1 (ii) for some constants C 1 and C 2 independent of η ∈ M(E) • and N. Therefore, by Lebesgue's dominated convergence theorem since η(E) −α is integrable with respect to Q α by Lemma 5.1 together with m(E) > 1.
It follows from (5.5) and Lemma 5.2 that for any Φ ∈ F 0 where Ψ(η) = Φ(η(E) −1 η). Moreover, noting that by the Stone-Wierstrass theorem the linear span of functions f on E n of the form f (r 1 , . . . , r n ) = f 1 (r 1 ) · · · f n (r n ) with f i ∈ C + (E) is dense in C(E n ), one can show, with the help of the expression (3.2) in [4] for A α Φ f with f ∈ C(E n ), that (5.7) extends to any Φ ∈ F 1 . Indeed, that expression takes the form for some nonnegative bounded operators Θ (n) , Ξ (n) : C(E n ) → C(E n ) and some positive constant c n independent of f and µ, and so if {g k } ⊂ C(E n ), g k (r 1 , . . . , r n ) → f (r 1 , . . . , r n ) uniformly on E n as k → ∞, then A α Φ g k → A α Φ f and Φ g k → Φ f uniformly on M 1 (E) as k → ∞.
In contrast, the variance functionals of P α and of Q α , denoted by var Pα and var Qα respectively, do not seem to enjoy any nice relation with each other. Although it is not clear if exponential convergence to equilibrium occurs for the process associated with A α , we are going to discuss a weaker ergodic property by introducing another functional osc 2 (Φ) for Φ ∈ L 2 (P α ), which is defined to be the essential supremum of the function Since by (5.3) P α and Q α • Z −1 are mutually absolutely continuous, osc 2 (Φ) coincides with the essential supremum of the function
Proof. It follows from (5.3) that Our strategy is to carry out the well-known procedure to show algebraic convergence to equilibrium. More specifically, our ingredient for this is Theorem 2.2 in [8]. (See also [9].) Let {S 2 (t)} t≥0 be the strongly continuous semigroup on L 2 (P α ) of the process associated with A α . To be more precise, {S 2 (t)} t≥0 is defined to be the C 0 -semigroup on L 2 (P α ) generated by the closure (A α (2) , D(A α (2) )) of (A α , F 1 ) as an operator on L 2 (P α ). As in the case of (T 2 (t)) t≥0 , (S 2 (t)) t≥0 on L 2 (P α ) coincides with (S(t)) t≥0 when restricted to C(M 1 (E)). The following property of (S 2 (t)) t≥0 is needed for the abovementioned strategy.
This upper bound for var Pα (S 2 (t)Φ) immediately gives (5.9). The proof of Theorem 5.5 is complete.
What is unpleasant to us is that we do not know whether gap(A α (2) ) = 0 or not.
One difficulty is that any useful expression for the variance functional with respect to P α nor Dirichlet form associated with A α does not seem available at least for conventional choice of test functions. Besides, our argument in this section does not work in the case where 0 < m(E) ≤ 1 although the process associated with A α still has a unique stationary distribution. (See Theorem 3.2 in [4].)