**E**l e c t ro nic
**J**

o f

**P**r

ob a bi l i t y

Electron. J. Probab.**19**(2014), no. 65, 1–25.

ISSN:1083-6489 DOI:10.1214/EJP.v19-2928

**Ergodic properties for** α **-CIR models and** **a class of generalized Fleming-Viot processes**

### Kenji Handa

^{∗}

*Dedicated to Professor Ken-iti Sato on the occasion of his 80th birthday*

**Abstract**

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 gen- eralized Fleming-Viot process obtained as the time-changed ratio process.

**Keywords:**measure-valued branching process ; CIR model ; spectral gap ; generalized Fleming-
Viot process.

**AMS MSC 2010:**Primary 60J75, Secondary 60G57.

Submitted to EJP on July 17, 2013, final version accepted on July 15, 2014.

SupersedesarXiv:1307.2407.

**1** **Introduction**

The study of ergodic behaviors of a Markov process is of quite interest for vari- ous 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-valuedα-CIR models, each of which is thought of as an infinite-dimensional ex- tension 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- valuedα-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-valuedα-CIR models, which enabled us to give an expression for stationary distributions of our gen- eralized Fleming-Viot processes in terms of those of the measure-valuedα-CIR models.

It should be mentioned that such links have been discussed in another context in [1] and

∗Saga University, Japan. E-mail:handa@ms.saga-u.ac.jp

[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

L1=z d^{2}

dz^{2} + (−bz+c) d

dz, z∈R+:= [0,∞), (1.1) whereb ∈ 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 continu- ous 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

LαF(z) = α+ 1 Γ(1−α)z

Z ∞ 0

[F(z+y)−F(z)−yF^{0}(z)] dy
y^{α+2}

−b

αzF^{0}(z) +c α
Γ(1−α)

Z ∞ 0

[F(z+y)−F(z)] dy

y^{α+1}, (1.2)
whereΓ(·)is the gamma function. The operatorL_{α} withb= 0is found in Example 1.1
of [5]. Observing that, asα↑ 1,L_{α}F(z)→L_{1}F(z)for anyz >0 and ‘nice’ functionsF
onR+, we call a Markov process associated withaLαfor some constanta >0anα-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 ergod-
icity 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 pro-
cesses with generators of the form (1.2), sayL^{0}_{α}andL^{00}_{α}, with commonαandb. On the
level of generators such a link can be reformulated as the identity

(L^{0}_{α}F(·, z2)) (z1) + (L^{00}_{α}F(z1,·)) (z2) =C(z1+z2)^{−α}AαG
z1

z_{1}+z_{2}

, z1, z2>0, (1.3) whereGis any smooth function on[0,1], F is defined byF(z1, z2) =G(z1/(z1+z2)),C is a positive constant independent ofGandAαis the generator of a jump-type version of the Wright-Fisher diffusion model. (See (1.3) in [4] for a concrete expression forAα

or (5.1) below for its generalization.) A significant consequence of (1.3) is that Dirichlet
form associated withA_{α}is, up to some multiplicative constant, a restriction of Dirichlet
form associated with the two independentα-CIR models. Therefore, ergodic properties
ofα-CIR models would be expected to help us obtain the same kind of results for the
process associated withAα.

Such an idea can extend naturally to the measure-valuedα-CIR model, which is re- garded roughly as ‘continuum direct sum’ ofα-CIR models with coefficients depending on a spatial parameter. More precisely, it is a MBI-process on a type spaceE, say, with zero mutation, branching mechanism

E×R+3(r, λ) 7→ a(r) α+ 1 Γ(1−α)

Z ∞ 0

e^{−λy}−1 +λy dy

y^{α+2} −b(r)
α λ

=−1

α(a(r)λ^{α+1}+b(r)λ) (1.4)

and nonlocal immigration mechanism f(·)7→ α

Γ(1−α) Z

E

m(dr) Z ∞

0

h

1−e^{−f(r)y}i dy
y^{α+2} =

Z

E

m(dr)f(r)^{α}, (1.5)

wherea(r) > 0, b(r) ∈ R andm is a finite non-null measure onE. An important fea- ture of such processes is that the transition semigroups (for fixed t) and stationary distributions form a convolution semigroup with respect tom. (See (2.15) and (2.17) below.) 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 theα-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-
valuedα-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 forL_{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.

**2** **The measure-valued** α **-CIR models**

To discuss in the setting of measure-valued processes, we need the following nota-
tion. LetE be a compact metric space and C(E)(resp. B+(E)) the set of continuous
(resp. nonnegative, bounded Borel) functions onE. Also, denote byC++(E)the set of
functions inC(E)which are uniformly positive. DefineM(E)to be the totality of finite
Borel measures onE, and we equipM(E)with the weak topology. Denote byM(E)^{◦}
the set of non-null elements ofM(E). The setM1(E)of Borel probability measures on
Eis regarded as a subspace ofM(E). We also use notationhη, fi:=R

Ef(r)η(dr). For
eachr∈E, letδrdenote the delta distribution atr. Given a probability measureQ, we
write alsoE^{Q}[·]for the expectation with respect toQ.

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 onM(E)associated with

LαΨ(η) =L^{(1)}_{α} Ψ(η) +L^{(2)}_{α} Ψ(η) +L^{(3)}_{α} Ψ(η)
:= α+ 1

Γ(1−α) Z ∞

0

dz
z^{2+α}

Z

E

η(dr)a(r)

Ψ(η+zδr)−Ψ(η)−zδΨ δη(r)

− 1

αhη, bδΨ δηi

+ α

Γ(1−α) Z ∞

0

dz
z^{1+α}

Z

E

m(dr) [Ψ(η+zδr)−Ψ(η)], η∈ M(E), (2.1)
where ^{δΨ}_{δη}(r) = _{d}^{d}Ψ(η+δr)

_{=0}. The operatorL^{(3)}α describes the mechanism of immi-
gration. (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 beA≡0.)
SetΨf(η) = e^{−hη,fi} forf ∈ B+(E)and defineDto be the linear span of functionsΨf

withf ∈C++(E). It is immediate to see from (1.4) and (1.5) that for anyf ∈B+(E)
LαΨ_{f}(η) = Ψ_{f}(η)1

αhη, af^{α+1}+bfi −Ψ_{f}(η)hm, f^{α}i. (2.2)
Lαis well-defined also on the classFof functionsΨof the form

Ψ(η) =ϕ(hη, f_{1}i, . . . ,hη, f_{n}i) (2.3)
for someϕ∈C_{0}^{2}(R^{n}_{+}),fi ∈C++(E)and a positive integern. Our first result below not
only verifies this but also gives bounds for eachL^{(k)}α Ψ (k∈ {1,2,3})for a more general

class of functionsΨ. In what followsk · k∞denotes the sup norm. LetFebe 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 constantsC_{j}^{(i)}(1≤i, j≤n),
C_{k}^{(ij)}(1≤i, j, k≤n)and >0such that for eachi, j∈ {1, . . . , n}

|ϕi(x1, . . . , xn)| ≤

n

X

k=1

C_{k}^{(i)}

xk+ ^{for any}(x1, . . . , xn)∈R^{n}f ^{(2.4)}
and

|ϕij(x_{1}, . . . , x_{n})| ≤

n

X

k=1

C_{k}^{(ij)}

(xk+)^{2} ^{for any}(x_{1}, . . . , x_{n})∈R^{n}f, (2.5)
whereϕi = ∂ϕ

∂x_{i}^{,}ϕij = ∂^{2}ϕ

∂x_{i}∂x_{j} ^{and}R^{n}f is defined to be

(x1, . . . , xn)∈(0,∞)^{n}: inf_{x∈E}fi(x)
kf_{j}k_{∞} ≤ xi

x_{j} ≤ kfik_{∞}

inf_{x∈E}f_{j}(x) (1≤i, j≤n)

.

Note that hη,fi := (hη, f1i, . . . ,hη, fni) ∈ 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].

It will turn out in Section 5 that an important example of functions inF \ Fe is
Ψ(η) =hη, f_{1}i · · · hη, f_{n}i(hη, f_{n+1}i+)^{−n},

wheref_{i} ∈ C_{++}(E), > 0 and n is a positive integer. This function corresponds to
ϕ(x_{1}, . . . , x_{n+1}) =x_{1}· · ·x_{n}(x_{n+1}+)^{−n}, for which the following are verified to hold:

ϕ_{i}(x_{1}, . . . , x_{n+1}) =
(

x1 i

· · ·ˇ xn(xn+1+)^{−n} (i∈ {1, . . . , n})

−nx1· · ·xn(xn+1+)^{−(n+1)} (i=n+ 1)
and

ϕ_{ij}(x_{1}, . . . , x_{n+1}) =

0 (i=j∈ {1, . . . , n}) x1

i,j

· · ·ˇ xn(xn+1+)^{−n} (i, j∈ {1, . . . , n}, i6=j)

−nx1 i

· · ·ˇ x_{n}(x_{n+1}+)^{−(n+1)} (i∈ {1, . . . , n}, j=n+ 1)
n(n+ 1)x1· · ·xn(xn+1+)^{−(n+2)} (i=j=n+ 1).

Here, · · ·ˇ^{i} (resp.

i,j

· · ·ˇ) indicates deletion of the ith (resp. ith and jth) factor(s). These
equalities are sufficient to show inequalities of the form (2.4) and (2.5). We can take in
particularC_{k}^{(i)}= 0 =C_{k}^{(ij)}for anyi, j∈ {1, . . . , n+ 1}andk∈ {1, . . . , n}.

**Lemma 2.1.** (i) It holds thatF ⊂Fe.

(ii) LetΨ∈Fe be expressed as (2.3) withϕsatisfying (2.4) and (2.5) andfi ∈ C++(E). Then for anyη∈ M(E)

L^{(1)}_{α} Ψ(η) = 1
αΓ(1−α)

Z

E

η(dr)a(r) Z ∞

0

u^{−α}du

n

X

i,j=1

fi(r)fj(r)ϕij(hη+uδr,fi), (2.6)

L^{(2)}_{α} Ψ(η) =−1
α

n

X

i=1

hη, bfiiϕi(hη,fi) (2.7)

and

L^{(3)}_{α} Ψ(η) = 1
Γ(1−α)

Z

E

m(dr) Z ∞

0

w^{−α}dw

n

X

i=1

fi(r)ϕi(hη+wδr,fi). (2.8) Also, we have the bounds

|L^{(1)}_{α} Ψ(η)| ≤ Γ(α)

n

X

i,j,k=1

C_{k}^{(ij)} hη, af_{i}f_{j}f_{k}^{α−1}i
(hη, fki+)^{α+1}

≤ Γ(α)

n

X

i,j,k=1

C_{k}^{(ij)}kafif_{j}f_{k}^{α−1}k∞

inf_{x∈E}fk(x) (hη, fki+)^{−α}, (2.9)

|L^{(2)}_{α} Ψ(η)| ≤ 1
α

n

X

i,j=1

C_{j}^{(i)} hη,|b|fii
hη, fji+ ≤ 1

α

n

X

i,j=1

C_{j}^{(i)} kbfik∞

infx∈Efj(x) ^{(2.10)}
and

|L^{(3)}_{α} Ψ(η)| ≤Γ(α)

n

X

i,j=1

C_{j}^{(i)} hm, fif_{j}^{α−1}i

(hη, fji+)^{α}. (2.11)
In particular,L^{(1)}α Ψ,L^{(2)}α ΨandL^{(3)}α Ψare bounded.

Proof. (i) Letϕ∈C_{0}^{2}(R^{n}_{+})be given and takeR_{1}, . . . , R_{n} >0large enough so that
ϕ(x1, . . . , xn) = 0 whenever max{x1/R1, . . . , xn/Rn}>1.

Let >0be arbitrary. Then it is easy to see that for all(x1, . . . , xn)∈R^{n}_{+}

|ϕi(x1, . . . , xn)| ≤ 1 n

n

X

k=1

Rk+
x_{k}+kϕik_{∞}
and

|ϕij(x1, . . . , xn)| ≤ 1 n

n

X

k=1

(R_{k}+)^{2}

(xk+)^{2}kϕijk∞.

In view of (2.4) and (2.5), what we have just seen suffice to imply thatF ⊂Fe.

(ii) First, we consider L^{(2)}α Ψ(η), assuming that η ∈ M(E)^{◦}. (If η is the null measure,
(2.10) is trivial.) Observe that

δΨ δη(r) =

n

X

i=1

fi(r)ϕi(hη,fi), (2.12) from which (2.7) follows. Also, (2.10) is immediate from (2.4).

The next task is to prove the assertions forL^{(3)}α Ψ(η). Since_{dz}^{d}Ψ(η+zδr) = _{δ(η+zδ}^{δΨ}

r)(r), we have by Fubini’s theorem

Z ∞ 0

dz

z^{1+α}[Ψ(η+zδ_{r})−Ψ(η)] =
Z ∞

0

dz
z^{1+α}

Z z 0

dw δΨ

δ(η+wδ_{r})(r)

= 1

α Z ∞

0

w^{−α}dw δΨ

δ(η+wδr)(r). (2.13)
So (2.8) is deduced from (2.12). Noting thatη+wδr∈ M(E)^{◦} forw >0, apply (2.4) to
get

|L^{(3)}_{α} Ψ(η)| ≤ 1
Γ(1−α)

Z

E

m(dr) Z ∞

0

w^{−α}dw

n

X

i,j=1

fi(r)C_{j}^{(i)}
hη, fji+wf_{j}(r) +

= Γ(α)

n

X

i,j=1

C_{j}^{(i)}hm, fif_{j}^{α−1}i(hη, fji+)^{−α},

which proves (2.11). In the above equality we have used Z ∞

0

w^{−α} dw

sw+t = Γ(α)Γ(1−α)s^{α−1}t^{−α}, s, t >0. (2.14)
It remains to prove (2.6) and (2.9). Similarly to (2.13)

L^{(1)}_{α} Ψ(η) = α+ 1
Γ(1−α)

Z

E

η(dr)a(r) Z ∞

0

dz
z^{2+α}

Z z 0

dw

δΨ

δ(η+wδ_{r})(r)−δΨ
δη(r)

= 1

Γ(1−α) Z

E

η(dr)a(r) Z ∞

0

dw
w^{1+α}

δΨ

δ(η+wδ_{r})(r)−δΨ
δη(r)

and by (2.12) δΨ

δ(η+wδr)(r)−δΨ δη(r) =

Z w 0

du

n

X

i,j=1

f_{i}(r)f_{j}(r)ϕ_{ij}(hη+uδ_{r},fi).

Hence (2.6) is derived by Fubini’s theorem. (2.6) and (2.5) together yield

|L^{(1)}_{α} Ψ(η)| ≤ 1
αΓ(1−α)

Z

E

η(dr)a(r) Z ∞

0

u^{−α}du

n

X

i,j,k=1

fi(r)fj(r)C_{k}^{(ij)}
(hη, fki+uf_{k}(r) +)^{2}

= Γ(α)

n

X

i,j,k=1

C_{k}^{(ij)} hη, afifjf_{k}^{α−1}i
(hη, fki+)^{α+1}.
Here, the last equality is deduced from

Z ∞ 0

u^{−α} du

(su+t)^{2} =αΓ(α)Γ(1−α)s^{α−1}t^{−(α+1)}, s, t >0,
which is verified by differentiating (2.14) int.

Following [10], we consider the operator(Lα,F)as an operator onC_{∞}(M(E)), the
set of continuous functions on M(E) vanishing at infinity. In the theorem below we
collect basic properties ofLαand the associated transition semigroup.

**Theorem 2.2.** (i)(Lα,F)is closable inC_{∞}(M(E))and the closure(Lα, D(Lα))gener-
ates aC_{0}-semigroup(T(t))_{t≥0}. Moreover, Dis a core forL_{α}, and for eachf ∈B_{+}(E)
andη∈ M(E)

T(t)Ψ_{f}(η) = exp

−hη, V_{t}fi −
Z t

0

hm,(V_{s}f)^{α}ids

, t≥0, (2.15)

where

Vtf(r) = e^{−b(r)t/α}f(r)
h

1 +a(r)f(r)^{α}Rt

0e^{−b(r)s}dsi^{1/α}. (2.16)
(ii) Ifb ∈C_{++}(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, sayQα, ast→ ∞. Moreover, the Laplace
functional ofQαis given by

Z

M(E)^{◦}

Q_{α}(dη)Ψ_{f}(η) = exp

−hm, a^{−1}log(1 +ab^{−1}f^{α})i

, f ∈B_{+}(E). (2.17)

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, Vtf(r)was given there implicitly by

∂

∂tVtf(r) =−a(r)

α Vtf(r)^{1+α}−b(r)

α Vtf(r), V0f(r) =f(r), (2.18) from which (2.16) is obtained. (See Example 3.1 in [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 7→ hm,^{δΨ}_{δη}i. A (possibly unique) non-trivial
modification would be the step to construct, for eachη ∈ M(E)and t ≥ 0, qt(η,·) ∈
M1(M(E))with Laplace transform given by the right side of (2.15). By the observation
made in the last paragraph, we havept(η,·)∈ M1(M(E))such that

Z

M(E)

p_{t}(η, dη^{0})Ψ_{f}(η^{0}) = exp [−hη, V_{t}fi], f ∈B_{+}(E).

Additionally, for everyη∈ M(E), lets_{α}(η,·)be the law of anα-stable random measure
with parameter measureη, i.e.,

Z

M(E)

sα(η, dη^{0})Ψf(η^{0}) = exp [−hη, f^{α}i], f ∈B+(E)
and definept,α(η,·)∈ M1(M(E))to be the mixture

p_{t,α}(η,·) =
Z

M(E)

s_{α}(η, dη^{0})p_{t}(η^{0},·).

It then follows that Z

M(E)

p_{t,α}(η, dη^{0})Ψ_{f}(η^{0}) = exp [−hη,(V_{t}f)^{α}i], f ∈B_{+}(E).

Therefore, for eachN = 1,2, . . ., the convolution
q^{(N)}_{t} (η,·) :=pt(η,·)∗

## ∗

Nk=1p_{tk/N,α}
t

Nm,· !

has Laplace transform Z

M(E)

q_{t}^{(N}^{)}(η, dη^{0})Ψf(η^{0}) = exp

"

−hη, Vtfi −

N

X

k=1

t

Nhm,(V_{tk/N}f)^{α}i

# ,

which converges to the right side of (2.15) asN → ∞. Thus, the weak limit ofq^{(N)}_{t} (η,·)
asN → ∞is identified with the desired probability measureq_{t}(η,·)onM(E). Hence
the semigroup(T(t))_{t≥0}defined by

T(t)Ψ(η) = Z

M(E)

q_{t}(η, dη^{0})Ψ(η^{0}), Ψ∈B(M(E))
satisfies (2.15). The identity _{dt}^{d}T(t)Ψ_{f}

_{t=0} =L_{α}Ψ_{f} forf ∈C_{++}(E)is verified by com-
bining (2.2) with (2.18). Once (2.15) is in hand, the assertion thatD is a core forLα

follows as a direct consequence of Lemma 2.2 in [13].

(ii) Ast→ ∞the right side of (2.15) converges to exp

− Z ∞

0

hm,(Vtf)^{α}idt

= exp

−hm, a^{−1}log(1 +ab^{−1}f^{α})i

since by (2.18) d dtlog

1 +a(r)b(r)^{−1}(V_{t}f(r))^{α}

=−a(r)(Vtf(r))^{α}.

This proves the required ergodicity and that the unique stationary distributionQα has
the Laplace functional given by the right side of (2.17). The fact thatQ_{α} is supported
onM(E)^{◦}follows by observing that the right side of (2.17) withf ≡β >0tends 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).

**Remark 2.3.** (i) A random measure with law Q_{α} in Theorem 2.2 (ii) is an infinite-
dimensional analogue of the random variable with law sometimes referred to as a (non-
symmetric) Linnik distribution, whose Laplace exponent is of the formλ7→clog(1+dλ^{α})
for somec, d >0. Observe from (2.17) that, asα↑ 1,Qα converges toQ1, the law of a
generalized gamma process such that

Z

M(E)^{◦}

Q1(dη)Ψf(η) = exp

−hm, a^{−1}log(1 +ab^{−1}f)i

, f ∈B+(E).

In addition, one can see that lim

α↑1L_{α}Ψ(η) =hη, aδ^{2}Ψ

δη^{2}i − hη, bδΨ

δηi+hm,δΨ

δηi=:L_{1}Ψ(η)
for ‘nice’ functionsΨ, where ^{δ}_{δη}^{2}^{Ψ}_{2}(r) = _{d}^{d}^{2}_{2}Ψ(η+δ_{r})

_{=0}. (For instance, this is immedi-
ate forΨ = Ψf from (2.2).) L1is the generator of an MBI-process discussed in Section
4 of [11] and in Section 3 of [10], whereQ_{1} was shown to be a reversible stationary
distribution of the process associated withL_{1}.

(ii) In contrast,Qα(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-
valuedα-CIR models. Namely, one can show, by a proof by contradiction, that the for-
mal symmetryE^{Q}^{α}[(−L_{α})Ψ_{f}·Ψ_{g}] =E^{Q}^{α}[(−L_{α})Ψ_{g}·Ψ_{f}] fails for some f, g ∈ C_{++}(E).
For this purpose, an expression for the Dirichlet form E^{Q}^{α}[(−Lα)Ψf·Ψg] given after
Lemma 3.1 below is helpful.

**3** **Associated Dirichlet forms**

From now on, we suppose additionally thatb∈C++(E). Thus, only ergodic measure-
valuedα-CIR models will be discussed. To study the speed of convergence to equilib-
rium in theL^{2}-sense, we consider the symmetric part of Dirichlet form associated with
Lα in (2.1). It is a bilinear form on F × F defined byE(Ψ,e Ψ^{0}) := E^{Q}^{α}[Γ(Ψ,Ψ^{0})]with
Γ(·,∗)being the ‘carré du champ’:

Γ(Ψ,Ψ^{0})(η) := 1

2[−Ψ(η)LαΨ^{0}(η)−Ψ^{0}(η)LαΨ(η) +Lα(ΨΨ^{0})(η)]

= 1

2 Z ∞

0

nB(dz) Z

E

η(dr)a(r) [Ψ(η+zδr)−Ψ(η)] [Ψ^{0}(η+zδr)−Ψ^{0}(η)]

+1 2

Z ∞ 0

n_{I}(dz)
Z

E

m(dr) [Ψ(η+zδ_{r})−Ψ(η)] [Ψ^{0}(η+zδ_{r})−Ψ^{0}(η)],
wherenB(dz) = (α+ 1)z^{−α−2}dz/Γ(1−α)andnI(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 (Lα,F) is closable in
L^{2}(Qα)and that the closure (Lα

(2), D(Lα

(2)))generates a C0-semigroup(T^{2}(t))_{t≥0} on
L^{2}(Qα)which coincides with (T(t))_{t≥0} when restricted to C_{∞}(M(E)). We set E(Ψ) =
E^{Q}^{α}h

(−Lα

(2))Ψ·Ψi

for anyΨ∈D(Lα

(2)), remarking thatE(Ψ) =Ee(Ψ,Ψ)forΨ∈ F.
Letvar(Ψ)stand for the variance ofΨ∈L^{2}(Q_{α})with respect toQ_{α}, namely,

var(Ψ) =E^{Q}^{α}h

Ψ−E^{Q}^{α}[Ψ]^{2}i
.

It is known that the largestκ≥0such that

var(T^{2}(t)Ψ)≤e^{−κt}var(Ψ) for allΨ∈L^{2}(Q_{α})andt >0
is identified with

gap(Lα

(2)) := infn

E(Ψ) : var(Ψ) = 1, Ψ∈D(Lα (2))o

= supn

κ≥0 : κ·var(Ψ)≤ E(Ψ)for allΨ∈D(Lα (2))o

.

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 formgap(Lα

(2))≥κimplies thatLα

(2)has a spectral gap below 0 of size larger than or equal toκ. (See Remark 1.13 in [10].) In calculating Dirichlet form and the variance functional with respect toQα, we will make an essential use of the following expression for the ‘log-Laplace functional’ in (2.17):

ψ(f) := hm, a^{−1}log(1 +ab^{−1}f^{α})i

= Z

E

ma(dr) Z ∞

0

Λ(dz)

1−e^{−f}^{∗}^{(r)z}

, (3.1)

wherema(dr) =a(r)^{−1}m(dr),Λis the Lévy measure of the infinite divisible distribution
on(0,∞)with Laplace exponentλ7→log(1 +λ^{α})andf^{∗}= (a/b)^{1/α}f. In what follows the
domain of integration is understood to be(0,∞)when suppressed. Define nonnegative
functionsKB andKI onR^{2}_{+}by

KB(s, t) :=

Z

nB(dy)(1−e^{−sy})(1−e^{−ty}) =α^{−1}

(s+t)^{α+1}−s^{α+1}−t^{α+1}

(3.2) and

K_{I}(s, t) :=

Z

n_{I}(dy)(1−e^{−sy})(1−e^{−ty}) =s^{α}+t^{α}−(s+t)^{α}, (3.3)
respectively. The above identities are verified easily by differentiating insandt.
**Lemma 3.1.** For anyf, g∈B+(E)

E(Ψe f,Ψg) = 1

2e^{−ψ(f+g)}
Z

E

m(dr)a(r)^{1/α}
b(r)^{1/α}

Z

nB(dy)(1−e^{−f(r)y})(1−e^{−g(r)y})
Z

Λ(dz)ze^{−(f}^{∗}^{(r)+g}^{∗}^{(r))z}
+1

2e^{−ψ(f+g)}
Z

E

m(dr) Z

nI(dy)(1−e^{−f(r)y})(1−e^{−g(r)y})

= 1

2e^{−ψ(f+g)}

hm,α(f +g)^{α−1}aK_{B}(f, g)

b+a(f+g)^{α} i+hm, KI(f, g)i

, which is finite.

Proof. It follows that
Γ(Ψ_{f},Ψ_{g}) = 1

2e^{−hη,f+gi}
Z

E

η(dr)a(r) Z

n_{B}(dy)(1−e^{−f(r)y})(1−e^{−g(r)y})
+1

2e^{−hη,f+gi}
Z

E

m(dr) Z

n_{I}(dy)(1−e^{−f(r)y})(1−e^{−g(r)y})

= 1

2e^{−hη,f+gi}(hη, aKB(f, g)i+hm, KI(f, g)i).

Note that the functionr 7→K_{I}(f(r), g(r))is an element ofB_{+}(E). Defining h∈B_{+}(E)
by h(r) = a(r)K_{B}(f(r), g(r))and recalling thatEe(Ψ_{f},Ψ_{g}) = E^{Q}^{α}[Γ(Ψ_{f},Ψ_{g})], we need
only to show that

I(f +g;h) := E^{Q}^{α}h

e^{−hη,f+gi}hη, hii

= e^{−ψ(f+g)}
Z

E

ma(dr)a(r)^{1/α}
b(r)^{1/α}h(r)

Z

Λ(dz)ze^{−(f}^{∗}^{(r)+g}^{∗}^{(r))z}

= e^{−ψ(f+g)}hm,α(f+g)^{α−1}h

b+a(f+g)^{α}i (3.4)

and that this is finite. The second equality can be verified to hold by (2.17) and (3.1) together:

I(f +g;h) = − d
dE^{Q}^{α}h

e^{−hη,f+g+hi}i
_{=0}

= − d

de^{−ψ(f+g+h)}
_{=0}

= e^{−ψ(f+g)}
Z

E

ma(dr)h^{∗}(r)
Z

Λ(dz)ze^{−(f}^{∗}^{(r)+g}^{∗}^{(r))z}

= e^{−ψ(f+g)}
Z

E

ma(dr)a(r)^{1/α}
b(r)^{1/α}h(r)

Z

Λ(dz)ze^{−(f}^{∗}^{(r)+g}^{∗}^{(r))z}.

For the proof of the last equality in (3.4), we make use of another expression forI(f+ g;h)deduced from (2.17) only:

I(f +g;h) = − d dexp

−hm, a^{−1}log(1 +ab^{−1}(f+g+h)^{α})i
_{=0}

= e^{−ψ(f+g)}hm,α(f+g)^{α−1}h
b+a(f+g)^{α}i.

Here, by (3.2)

0≤α(f+g)^{α−1}h≤(f+g)^{α−1}a(f+g)^{α+1}=a(f+g)^{2α}
and soI(f+g;h)is finite.

Noting that (3.4) is clearly valid for everyh∈B_{+}(E)and combining (2.2) with (3.4),
we get for anyf, g∈B+(E)

E^{Q}^{α}[(−Lα)Ψf·Ψg] = −E^{Q}^{α}

Ψf+g(η)· 1

αhη, af^{α+1}+bfi −Ψf+g(η)hm, f^{α}i

= −e^{−ψ(f+g)}

hm,(f+g)^{α−1}(af^{α+1}+bf)

b+a(f+g)^{α} i − hm, f^{α}i

,

from which the last expression in Lemma 3.1 for the symmetric part E(Ψe f,Ψg) = 1

2 E^{Q}^{α}[(−Lα)Ψf·Ψg] +E^{Q}^{α}[(−Lα)Ψg·Ψf]

can be recovered.

Our objective is to show the positivity ofgap(Lα

(2)). The contribution here in this direction is the reduction to a certain estimate regarding the one-dimensional model.

For a measurable functionf onE, the essential supremum (resp. the essential infimum) off with respect tomis denoted byess sup

(E,m)

f (resp. ess inf

(E,m)f). LetDbe the linear span
of functions onR+of the formFλ(z) :=e^{−λz}for someλ >0.

**Theorem 3.2.** Suppose thatb∈C_{++}(E). Letγ >0be a constant. If for everyF ∈D
Z

Λ(dz)(F(z)−F(0))^{2}

≤ γ 2

Z

Λ(dz)z Z

nB(dy)(F(z+y)−F(z))^{2}+
Z

nI(dy)(F(y)−F(0))^{2}

, (3.5)

then for anyΨ∈ D

var(Ψ)≤γess sup

(E,m)

(b^{−1})E(Ψ) (3.6)

and it holds thatgap(Lα

(2))≥γ^{−1}ess inf

(E,m)b.

This kind of reduction was discovered by Stannat [12] (Theorem 1.2) for a lower es- timate for the quadratic form of gradient type. In particular, for the process associated withL1in Remark 2.3 (i), the condition corresponding to (3.5) reads

Z

Λ1(dz)(F(z)−F(0))^{2}≤γ
Z

Λ1(dz)z(F^{0}(z))^{2}, (3.7)
whereΛ1(dz) =z^{−1}e^{−z}dzis the Lévy measure of a gamma distribution. While (3.7) with
γ = 1is verified easily by applying Schwarz’s inequality toF(z)−F(0) =Rz

0 F^{0}(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 Ψexpressed as a finite sum Ψ = P

iciΨf_{i},
whereci∈Randfi∈B+(E). Puttingdi=cie^{−ψ(f}^{i}^{)}, observe from (3.1) that

var(Ψ) = X

i,j

c_{i}c_{j}

e^{−ψ(f}^{i}^{+f}^{j}^{)}−e^{−ψ(f}^{i}^{)}e^{−ψ(f}^{j}^{)}

= X

i,j

didj

e^{ψ(f}^{i}^{)+ψ(f}^{j}^{)−ψ(f}^{i}^{+f}^{j}^{)}−1

= X

i,j

didj

exp

Z

E

Z

ma(dr)Λ(dz)(1−e^{−f}^{i}^{∗}^{(r)z})(1−e^{−f}^{j}^{∗}^{(r)z})

−1

= X

i,j

didj

∞

X

N=1

1 N!

Z

E

Z

ma(dr)Λ(dz)(1−e^{−f}^{i}^{∗}^{(r)z})(1−e^{−f}^{j}^{∗}^{(r)z})
^{N}

.(3.8)

Rewrite in terms of theN-fold product measuresm^{⊗N}_{a} andΛ^{⊗N} to obtain the following

disintegration formula for the variance functional:

var(Ψ) =

∞

X

N=1

1 N!

X

i,j

didj

Z

E^{N}

Z

R^{N}_{+}

m^{⊗N}_{a} (drN)Λ^{⊗N}(dzN)

N

Y

k=1

(1−e^{−f}^{i}^{∗}^{(r}^{k}^{)z}^{k})

N

Y

l=1

(1−e^{−f}^{j}^{∗}^{(r}^{l}^{)z}^{l})

=

∞

X

N=1

1 N!

Z

E^{N}

Z

R^{N}_{+}

m^{⊗N}_{a} (drN)Λ^{⊗N}(dzN)

"

X

i

di N

Y

k=1

(1−e^{−f}^{i}^{∗}^{(r}^{k}^{)z}^{k})

#^{2}
,(3.9)

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

z_{N} 7→X

i

d_{i}
(_{N}_{−1}

Y

k=1

(1−e^{−f}^{i}^{∗}^{(r}^{k}^{)z}^{k})
)

e^{−f}^{i}^{∗}^{(r}^{N}^{)z}^{N}

to get

2 γ

Z

Λ(dzN)

"

X

i

di N

Y

k=1

(1−e^{−f}^{i}^{∗}^{(r}^{k}^{)z}^{k})

#^{2}

≤ Z

Λ(dz)z Z

n_{B}(dy)

"

X

i

d_{i}

N−1

Y

k=1

(1−e^{−f}^{i}^{∗}^{(r}^{k}^{)z}^{k})(e^{−f}^{i}^{∗}^{(r}^{N}^{)(z+y)}−e^{−f}^{i}^{∗}^{(r}^{N}^{)z})

#^{2}

+ Z

n_{I}(dy)

"

X

i

d_{i}

N−1

Y

k=1

(1−e^{−f}^{i}^{∗}^{(r}^{k}^{)z}^{k})(e^{−f}^{i}^{∗}^{(r}^{N}^{)y}−1)

#^{2}

= a(rN)^{1+1/α}
b(rN)^{1+1/α}

Z Λ(dz)z

Z

nB(dy)

"

X

i

di N−1

Y

k=1

(1−e^{−f}^{i}^{∗}^{(r}^{k}^{)z}^{k})e^{−f}^{i}^{∗}^{(r}^{N}^{)z}(1−e^{−f}^{i}^{(r}^{N}^{)y})

#^{2}

+a(rN) b(rN) Z

nI(dy)

"

X

i

di N−1

Y

k=1

(1−e^{−f}^{i}^{∗}^{(r}^{k}^{)z}^{k})(1−e^{−f}^{i}^{(r}^{N}^{)y})

#^{2}

. (3.10)

Here, a suitable change of variable has been made for each integral with respect to
nB(dy)andnI(dy)in order to replacef_{i}^{∗}(rN)ybyfi(rN)y.

SetC= ess sup

(E,m)

(b^{−1})so that

ma(drN) =a(rN)^{−1}m(drN)≤C·b(rN)a(rN)^{−1}m(drN)

in distributional sense. Combining (3.9) with (3.10), we can dominate2var(Ψ)/γby

∞

X

N=1

C N!

Z

E^{N−1}

Z

R^{N−1}_{+}

m^{⊗N−1}_{a} (drN−1)Λ^{⊗N}^{−1}(dzN−1)
Z

E

m(dr)a(r)^{1/α}
b(r)^{1/α}
Z

Λ(dz)z Z

nB(dy)

"

X

i

di N−1

Y

k=1

(1−e^{−f}^{i}^{∗}^{(r}^{k}^{)z}^{k})e^{−f}^{i}^{∗}^{(r)z}(1−e^{−f}^{i}^{(r)y})

#^{2}

+

∞

X

N=1

C N!

Z

E^{N−1}

Z

R^{N−1}_{+}

m^{⊗N−1}_{a} (drN−1)Λ^{⊗N}^{−1}(dzN−1)
Z

E

m(dr)

Z

n_{I}(dy)

"

X

i

d_{i}

N−1

Y

k=1

(1−e^{−f}^{i}^{∗}^{(r}^{k}^{)z}^{k})(1−e^{−f}^{i}^{(r)y})

#^{2}

≤

∞

X

N=0

C N!

Z

E^{N}

Z

R^{N}_{+}

m^{⊗N}_{a} (drN)Λ^{⊗N}(dzN)
Z

E

m(dr)a(r)^{1/α}
b(r)^{1/α}
Z

Λ(dz)z Z

n_{B}(dy)

"

X

i

d_{i}

N

Y

k=1

(1−e^{−f}^{i}^{∗}^{(r}^{k}^{)z}^{k})e^{−f}^{i}^{∗}^{(r)z}(1−e^{−f}^{i}^{(r)y})

#^{2}

+

∞

X

N=0

C N!

Z

E^{N}

Z

R^{N}_{+}

m^{⊗N}_{a} (dr_{N})Λ^{⊗N}(dz_{N})
Z

E

m(dr)

Z

nI(dy)

"

X

i

di N

Y

k=1

(1−e^{−f}^{i}^{∗}^{(r}^{k}^{)z}^{k})(1−e^{−f}^{i}^{(r)y})

#^{2}

= X

i,j

d_{i}d_{j}

∞

X

N=0

C N!

Z

E

Z

m_{a}(dr_{1})Λ(dz_{1})(1−e^{−f}^{i}^{∗}^{(r}^{1}^{)z}^{1})(1−e^{−f}^{j}^{∗}^{(r}^{1}^{)z}^{1})
^{N}

Z

E

m(dr)a(r)^{1/α}
b(r)^{1/α}

Z

nB(dy)(1−e^{−f}^{i}^{(r)y})(1−e^{−f}^{j}^{(r)y})
Z

Λ(dz)ze^{−(f}^{i}^{∗}^{(r)+f}^{j}^{∗}^{(r))z}

+X

i,j

d_{i}d_{j}

∞

X

N=0

C N!

Z

E

Z

m_{a}(dr_{1})Λ(dz_{1})(1−e^{−f}^{i}^{∗}^{(r}^{1}^{)z}^{1})(1−e^{−f}^{j}^{∗}^{(r}^{1}^{)z}^{1})
^{N}

Z

E

m(dr) Z

nI(dy)(1−e^{−f}^{i}^{(r)y})(1−e^{−f}^{j}^{(r)y})

= CX

i,j

cicje^{−ψ(f}^{i}^{+f}^{j}^{)}
Z

E

m(dr)a(r)^{1/α}
b(r)^{1/α}

Z

nB(dy)(1−e^{−f}^{i}^{(r)y})(1−e^{−f}^{j}^{(r)y})
Z

Λ(dz)ze^{−(f}^{i}^{∗}^{(r)+f}^{j}^{∗}^{(r))z}
+CX

i,j

c_{i}c_{j}e^{−ψ(f}^{i}^{+f}^{j}^{)}
Z

E

m(dr) Z

n_{I}(dy)(1−e^{−f}^{i}^{(r)y})(1−e^{−f}^{j}^{(r)y}),

where the last two equalities are seen by similar calculations to (3.8) and (3.9). Since the symmetric partEeof Dirichlet form is bilinear, (3.6) forΨ∈ D follows from Lemma 3.1.

It remains to prove that (3.6) extends toΨ∈D(Lα

(2)). Since(Lα

(2), D(Lα

(2)))is the
closure of(L_{α},F)inL^{2}(Q_{α}), we need only to show that (3.6) extends toΨ∈ F. Given
Ψ∈ F, we see from Theorem 2.2 (i) that there exists a sequence{ΨN}^{∞}_{N=1} ⊂ Dsuch
that

kΨN −Ψk∞+kLαΨN − LαΨk∞→0 as N → ∞.

Hence

kΨ_{N} −Ψk_{L}2(Q_{α})+E(Ψ_{N}−Ψ)→0 as N → ∞.

This implies that (3.6) holds for any Ψ ∈ F and we complete the proof of Theorem 3.2.

**Remark 3.3.** In view of calculations in the proof of Theorem 3.2, it is clear that under
the same assumption an appropriate version of the inequality (3.6) holds for a more
general class of functionsaand b. To be more precise, suppose that a, b∈ B+(E)are
uniformly positive and thatQ_{α}has Laplace transform (2.17). Then, assuming that (3.5)
is valid for allF ∈D, we have

E^{Q}^{α}h

Ψ−E^{Q}^{α}[Ψ]^{2}i

≤γess sup

(E,m)

(b^{−1})E^{Q}^{α}[Γ(Ψ,Ψ)]

for anyΨ∈ D, where

Γ(Ψ,Ψ)(η) = 1 2

Z

nB(dz) Z

E

η(dr)a(r) [Ψ(η+zδr)−Ψ(η)]^{2}
+1

2 Z

nI(dz) Z

E

m(dr) [Ψ(η+zδr)−Ψ(η)]^{2}.

This fact reflects the convolution property with respect tommentioned in the Introduc- tion. (Note that the condition (3.5) is independent ofm.)

**4** **Spectral gap for the** α **-CIR model**

This section is devoted to the proof of (3.5) for some0< γ <∞. The strategy should
be different from the one already mentioned for (3.7) withΛ1(dz) = z^{−1}e^{−z}dzat least
because no informative expression for the density ofΛin (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 thatF ∈ D is a finite sumF =P

iciFλ_{i}. We will use the
notation1_{S} standing for the indicator function of a setS and∂_{t} =∂/∂t for simplicity.

Lettingψ_{1}(λ) = log(1 +λ) =R

Λ_{1}(dz)(1−e^{−λz}), observe that
U1(F) :=

Z

Λ1(dz)(F(z)−F(0))^{2} = X

i,j

cicj

Z

Λ1(dz)(1−e^{−λ}^{i}^{z})(1−e^{−λ}^{j}^{z})

= X

i,j

cicj(−ψ1(λi+λj) +ψ1(λi) +ψ1(λj))

= X

i,j

cicj

Z λ_{i}
0

ds
Z λ_{j}

0

dt(−ψ^{00}_{1}(s+t))

= Z

ds Z

dtF(s)F(t)(−ψ_{1}^{00}(s+t)), (4.1)
whereF(s) =P

ic_{i}1_{[0,λ}_{i}_{]}(s). On the other hand, by puttingK_{1}(s, t) =stψ_{1}^{0}(s+t)
V1(F) :=

Z

Λ1(dz)z(F^{0}(z))^{2} = X

i,j

cicjλiλj

Z

Λ1(dz)ze^{−λ}^{i}^{z}e^{−λ}^{j}^{z}

= X

i,j

cicjK1(λi, λj)

= X

i,j

cicj

Z λ_{i}
0

ds
Z λ_{j}

0

dt∂s∂tK1(s, t)

= Z

ds Z

dtF(s)F(t)∂_{s}∂_{t}K_{1}(s, t). (4.2)

It is reasonable to call∂_{s}∂_{t}K_{1} the intrinsic kernel of the quadratic formV_{1}. Similarly,
(4.1) shows that the intrinsic kernel ofU_{1}is the function(s, t)7→ −ψ_{1}^{00}(s+t).

Given two symmetric measurable functionsJandKonR^{2}_{+}, we writeKJ ifK−J
is nonnegative definite in the sense that

Z ds

Z

dtG(s)G(t)(K(s, t)−J(s, t))≥0

for any bounded Borel functionGonR+ with compact support. By virtue of Fubini’s theorem,K0ifKis of ‘canonical form’

K(s, t) = Z

S

M(dω)σ(s, ω)σ(t, ω), s, t∈R+

for some measure space(S, M)and measurable functionσonR+×S. In view of (4.1) and (4.2), it is clear that the inequalityγV1(F)≥U1(F)is implied by

γ∂s∂tK1(s, t) +ψ^{00}_{1}(s+t)0.

Forγ= 1, this holds true since by direct calculations

∂s∂tK1(s, t) +ψ^{00}_{1}(s+t) = 2st
(1 +s+t)^{3} =

Z

dzz^{2}e^{−z}se^{−sz}te^{−tz},

which is of canonical form. Furthermore, this expression makes it possible to identify the associated ‘remainder form’:

V1(F)−U1(F) = Z

ds Z

dtF(s)F(t) Z

dzz^{2}e^{−z}se^{−sz}te^{−tz}

= Z

dze^{−z}
Z

dsF(s)sze^{−sz}
^{2}

= Z

dze^{−z} X

i

ci

Z λ_{i}
0

dssze^{−sz}

!^{2}

= Z

dze^{−z} X

i

ci

e^{−λ}^{i}^{z}−1 +λize^{−λ}^{i}^{z}
z

!2

= Z

Λ_{1}(dz)z^{−1}(F(z)−F(0)−zF^{0}(z))^{2}.

It should be emphasized that the above calculations require only an explicit form of the
Laplace exponentψ_{1}.

Turning to the case 0 < α < 1, we adopt the method of intrinsic kernels to show (3.5) forΛsuch that

ψ(λ) := log(1 +λ^{α}) =
Z

Λ(dz)(1−e^{−λz}), λ≥0. (4.3)
(We continue to adopt this notation as it is a one-dimensional version of (3.1).) Namely,
we shall

(I) calculate the intrinsic kernels ofU(F) :=R

Λ(dz)(F(z)−F(0))^{2}and of
V(F) :=1

2 Z

Λ(dz)z Z

nB(dy)(F(z+y)−F(z))^{2}+
Z

nI(dy)(F(y)−F(0))^{2}

, and then

(II) compare the two kernels as nonnegative definite functions.

The following lemma concerns the step (I).