Advances in Systems Science and Applications (2012) Vol.12 No.3 283-296

Central Limit Theorems for the Single Point
Catalytic Super-Brownian Motion with Immigration

Xu Yang and Mei Zhang
Laboratory of Mathematics and Complex Systems, School of Mathematical

Sciences, Beijing Normal University,
Beijing 100875, People’s Republic of China

Abstract

We establish the central limit theorems for the single point catalytic super-
Brownian motion with deterministic immigration and single point catalytic super-
Brownian motion immigration on the Schwartz space and a weighted Sobolev
space. For the catalytic immigration case, the weak convergence depends on the
branching rates of both immigration part and non-immigration part.
Keywords Super-Brownian motion, catalytic, conditional log-Laplace functional,
immigration, central limit theorem

1 Introduction and Main Results

Catalytic super-Brownian motion is the superprocess with Brownian motion as
underlying spatial motion,whose branching occurs only in the presence of some
catalysts. Dawson and Fleischmann [1] considered the case of single point catalyt-
ic super-Brownian motion, namely, the underlying particles move as independent
Brownian motions in R. The life time of each particle is exponentially distributed.
When it dies, it splits according to critical branching only if they pass 0, which is
called the (single) catalyst point. Fleischmann and Xiong [2], Yang and Zhang [3]
and Li and Wang [4] proved the large deviation, moderate deviation and central
limit theorem for the single point catalytic super-Brownian motion, respectively.
If we suppose the situation where there are additional particles added, we need
to consider the process with immigration. Superprocesses with immigration are
studied by many authors; see e.g. [5-10].

In the present paper,first we will prove the central limit theorem for the s-
ingle point catalytic super-Brownian motion with deterministic immigration on
Schwartz space, and then extend the result to a weighted Sobolev space. We shall
also investigate the processes with catalytic immigration. In this case, the weak
convergence depends not only on the branching rate ϱ of non-immigration part,
but also on the branching rate ϱ0 of immigration part.

1.1 Notations and Preliminaries

First we introduce some notations. Let p ≥ 2, hp(x) = (1 + x2)−
p
2 and Cp(R)

denote the set of all real-valued continuous functions φ on R such that φ(x)/hp(x)
has a finite limit as |x| → ∞. Equipped with the norm 9φ9p := sup{|φ(x)|/hp(x) :
x ∈ R}, Cp(R) is a Banach space. Let C+

p (R) denote all the positive functions



284 Xu Yang:Central Limit Theorems for the Single Point Catalytic...

of Cp(R) and Mp(R) be the set of all measures µ on R such that ⟨µ, hp⟩ < ∞.
Suppose that 9µ9p = ⟨µ, hp⟩. Let C∞(R) be the set of bounded infinitely differ-
entiable functions on R with bounded derivatives. Let S (R) ⊂ C∞(R) denote
the Schwartz space of rapidly decreasing functions on R and S+(R) be the col-
lection of non-negative elements of S (R). That is, each f ∈ S (R) is infinitely
differentiable and for each non-negative integer k and each non-negative integer
α we have

lim
|x|→∞

xk
dα

dxα
f(x) = 0.

Now we introduce some basic results in the following subsection, which can be
found in [11]. We define the Hilbertian norms {q0, q1, q2, · · · } on S (R) by

qn(f)
2 =

n∑
k=0

∫
R
(1 + x2)n(f (k)(x))2dx.

The Hermite polynomials on R are given by

gk(x) = (−1)kex
2 dk

dxk
e−x2

, k = 0, 1, 2, · · · .

Based on those we define the Hermite functions

hk(x) =
1

4
√
π
√
2kk!

e−x2/2gk(x), k = 0, 1, 2 · · · .

Then hk ∈ S (R) and {hk : k ≥ 0} is a complete orthonormal system in
L2(R). Let ⟨·, ·⟩ denote the inner product of L2(R). For f ∈ S (R) we write

f =
∞∑
k=0

⟨f, hk⟩hk and define

∥f∥2n =

∞∑
k=0

(2k + 1)2n⟨f, hk⟩2 (1)

for n = 0,±1,±2, · · · . Let Hn(R) be the completion of S (R) with respect to
∥ · ∥n. By approximation we can extend ⟨·, ·⟩ to a bilinear form between H−n(R)
and Hn(R). Let ⟨·, ·⟩n denote the inner product of Hn(R). For g, f ∈ Hn(R) we
have

⟨g, f⟩n =

∞∑
k=0

(2k + 1)2n⟨g, hk⟩⟨f, hk⟩ = ⟨πng, f⟩,

where

πng =

∞∑
k=0

(2k + 1)2n⟨g, hk⟩hk ∈ Hn(R).

Then H−n(R) and Hn(R) are dual spaces with the duality ⟨·, ·⟩.



Advances in Systems Science and Applications (2012) Vol.12 No.3 285

Lemma 1.1 For every n ≥ 0 there is a constant c(n) > 0 such that

qn(f) ≤ c(n)∥f∥n and ∥f∥n ≤ c(n)q2n(f), f ∈ S (R).

The sequence of norms defined by (1) induces a topology on the set H∞ :=
∞∩
k=0

Hn(R), which is compatible with the metric ρ defined by

ρ(f, g) =

∞∑
k=0

∥f − g∥k
2k(1 + ∥f − g∥k)

.

Then (S (R), ρ) (written as S (R) for simplicity) is a nuclear space. This implies

S ′(R) =
∞∪
n=0

H−n(R) ⊃ · · · ⊃ H−2(R) ⊃ H−1(R) ⊃ H0(R)

⊃ H1(R) ⊃ H2(R) ⊃ · · · ⊃
∞∩
n=0

Hn(R) = S (R).

A subset B of the nuclear space S (R) is said to be bounded if it is bounded in
each norm ∥ · ∥n, that is, sup

x∈B
∥x∥n < ∞ for each n ≥ 0. For each bounded set

B ⊂ S (R) we define the semi-norm pB on S ′(R) by

pB(f) = sup{|f(x)| : x ∈ B}, f ∈ S ′(R).

We endow S ′(R) with the topology generated by the collection of semi-norms
{pB : B ⊂ S (R) is bounded}, which is called the strong topology. Then S ′(R)
is a nuclear space.

1.2 Models

For a process X taking its value in Mp(R), let Pr,ν denote its conditional law
given Xr = ν. Suppose that p is the heat kernel in R with constant ς > 0:

1√
2πςt

exp

{
− a2

2ςt

}
, t > 0, a ∈ R · · · . (2)

For φ ∈ C+
p (R) fixed, u(t, z; ϱ) denotes the unique non-negative solution to the

log-Laplace equation

u(t, z; ϱ) = Ptφ(z)− ϱ

∫ t

0
pt−r(z)u

2(r, 0; ϱ)dr, t ≥ 0, z ∈ R (3)



286 Xu Yang:Central Limit Theorems for the Single Point Catalytic...

and ω(t, z; ϱ, ϱ0) denotes the unique non-negative solution to the log-Laplace e-
quation

ω(t, z; ϱ, ϱ0)

=

∫ t

0
Pt−r [u(r, ·; ϱ)] (x)dr − ϱ0

∫ t

0
pt−r(z)ω

2(r, 0; ϱ, ϱ0)dr, t ≥ 0, z ∈ R · · · .
(4)

where ϱ and ϱ0 are positive constants and {Pt : t ≥ 0} denotes the Brownian
semigroup corresponding to (2). Let A be the generator of {Pt : t ≥ 0}. In the
following we suppose µ, η ∈ Mp(R). We say ξµ,ϱ = {ξµ,ϱt : t ≥ 0} is a single point
catalytic super-Brownian motion, if ξµ,ϱ0 = µ, and for r ≥ 0 and ν ∈ Mp(R),

− logPr,ν exp {−⟨ξt, φ⟩} = ⟨ν, u(t− r, ·; ϱ)⟩, 0 ≤ r ≤ t, φ ∈ C+
p (R) · · · . (5)

Suppose Z = {Zt : t ≥ 0} is the single point catalytic super-Brownian motion
with Z0 = µ, deterministic immigration controlled by η and log-Laplace func-
tional given by

− logPr,νe
−⟨Zt,φ⟩

=⟨ν, u(t− r, ·; ϱ)⟩+
∫ t

r
⟨η, u(t− s, ·; ϱ)⟩ds, 0 ≤ r ≤ t, φ ∈ C+

p (R) · · · .
(6)

Now we suppose that Xξ = {Xξ
t : t ≥ 0} is the single point catalytic super-

Brownian motion with single point catalytic immigration determined by ξη,ϱ0 .
Let P0,ν,η denote its conditional law given Xξ

0 = ν and ξ0 = η. By Theorem 3.2
of [12] we have the log-Laplace functional of Xξ:

− logP0,ν,η exp
{
−⟨Xξ

t , φ⟩
}

=− logP0,ν,η

[
P0,ν,η exp

{
−⟨Xξ

t , φ⟩
} ∣∣∣∣ {σ(ξs : 0 ≤ s ≤ t)}

]
=− logP0,η exp

{
−⟨ν, u(t, ·; ϱ)⟩ −

∫ t

0
⟨ξs, u(t− s, ·; ϱ)⟩ds

}
=⟨ν, u(t, ·)⟩+ ⟨η, ω(t, ·; ϱ, ϱ0)⟩, µ, η ∈ Mp(R), t ≥ 0, φ ∈ C+

p (R) · · · .

(7)

In the following, we breviate u(t, ·; ϱ) and ω(t, ·; ϱ, ϱ0) by u(t, ·) and ω(t, ·), respec-
tively. We construct the superprocesses Z, ξη,ϱ0 and Xξ on the same probability
space (Ω,F ,P).



Advances in Systems Science and Applications (2012) Vol.12 No.3 287

1.3 Main results

Let {Z(k)
t : t ≥ 0} be the single point catalyst super-Brownian motion with

deterministic immigration characterized by(3) and (6), with ϱ replaced by 1/k,
Let

W
(k)
t = k

1
2

(
Z

(k)
t − µPt −

∫ t

0
ηPsds

)
.

Theorem 1.2 As k → ∞, the sequence {W (k)
t : t ≥ 0} converges weakly to the

Gaussian process {Wt : t ≥ 0} in C([0,∞),S ′(R)) with W0 = 0 and Laplace
functional given by

P exp

{
− ⟨Wt, f⟩

}
=exp

{∫ t

0
⟨µ, pt−r(·)⟩P 2

r f(0)dr +

∫ t

0
ds

∫ s

0
⟨η, ps−r(·)⟩P 2

r f(0)dr

}
,

where f ∈ S+(R).

Remark 1.3 If η ≡ 0 in Theorem ??, the result can be found in [4].

Theorem 1.4 For any n ≥ 3, C([0,∞),S ′(R)) can be replaced by C([0,∞),
H−n(R)) in Theorem ??.

Similarly, we can establish the central limit theorem for the single point cat-
alytic super-Brownian motion with immigration controlled by another single
point catalytic super-Brownian motion. The weak convergence depends on ρ
and ρ0, which are the branching rates of non-immigration and immigration part-
s. To specify the effects of ϱ and ϱ0 on the convergence of Xξ, we suppose that
ϱ = γ1k

−1, ϱ0 = γ2k
−β(β > 0), where k ∈ N, γ1, γ2 are positive constants. Let

{ξ(k)t : t ≥ 0} be defined by (3) and (5) with ϱ0 replaced by γ2k
−β(β > 0). Let

{X(k)
t : t ≥ 0} be defined accordingly by (7), (3) and(4) with ϱ, ϱ0 replaced by

γ1k
−1, γ2k

−β(β > 0), respectively. Define

Y k
t = kα

(
X

(k)
t − µPt − tηPt

)
(8)

Theorem 1.5 As k → ∞, the sequence {Y (k)
t : t ≥ 0} converges weakly to the

Gaussian process {Yt : t ≥ 0} in C([0,∞),S ′(R)) with Y0 = 0 and Laplace
functional given by

P exp

{
− ⟨Yt, f⟩

}
= exp

{∫ t

0
Fs(f)ds

}
, f ∈ S+(R) (9)



288 Xu Yang:Central Limit Theorems for the Single Point Catalytic...

where Fs(f) is determined by the following table:

β α Fs(f)

(0, 1) 1
2β γ2⟨η, pt−s(·)⟩ (sPsf(0))

2

1 1
2 γ1⟨µ, pt−s(·)⟩ (Psf(0))

2 + γ1
∫ s
0 ⟨η, ps−r(·)⟩ (Prf(0))

2 dr

+γ2⟨η, pt−s(·)⟩ (sPsf(0))
2

(1,∞) 1
2 γ1⟨µ, pt−s(·)⟩ (Psf(0))

2 + γ1
∫ s
0 ⟨η, ps−r(·)⟩ (Prf(0))

2 dr

Theorem 1.6 For any n ≥ 3, C([0,∞),S ′(R)) can be replaced by C([0,∞),
H−n(R)) in Theorem ??.

Since the proofs are similar, we only show the case β = 1 of Theorems ??–??.

2 Proofs of of Theorem ?? and Theorem ?? (case β = 1)

In the proof of Theorem ??, we need the following proposition.

Proposition 2.1 As k → ∞, the finite dimensional distributions of {Y (k)
t : t ≥

0} converge weakly to a S ′(R)-valued Gaussian process {Yt : t ≥ 0} with Y0 = 0
and Laplace functional determined by (9).

Proof. To simplify the notations, we consider the two dimensional distributions

of Y (k). First we calculate the Laplace transform of (⟨Y (k)
t1
, f⟩, ⟨Y (k)

t2
, f⟩). By the

Markov property, (7), (3), (8), [10] and [12], for f ∈ C+
p (R), t2 > t1 > 0, we get

the Laplace transform of (⟨Y (k)
t1
, f⟩, ⟨Y (k)

t2
, f⟩):

logP0,µ,η exp

{
− θ1⟨Y (k)

t1
, f⟩ − θ2⟨Y (k)

t2
, f⟩

}
= logP0,µ,η

[
P0,µ,η

(
exp

{
− ⟨X(k)

t1
, θ1k

1
2 f + v(k)(t2 − t1, ·; θ2)⟩

−
∫ t2

t1

⟨ξ(k)s , v(k)(t2 − s, ·; θ2)⟩ds
}∣∣∣∣ {σ(ξ(k)s : s ≤ t2)

})]
+ k

1
2

(
⟨µPt1 , θ1f⟩+ ⟨µPt2 , θ2f⟩+ t1⟨ηPt1 , θ1f⟩+ t2⟨ηPt2 , θ2f⟩

)
= logP0,η exp

{
− ⟨µ, u(k)(t1, ·; θ1, θ2)−

∫ t1

0
⟨ξ(k)s , u(k)(t1 − s, ·; θ1, θ2)⟩ds

−
∫ t2

t1

⟨ξ(k)s , v(k)(t2 − s, ·; θ2)⟩ds
}
+ ⟨µPt1 , k

1
2 θ1f⟩

+ ⟨µPt2 , k
1
2 θ2f⟩+ t1⟨ηPt1 , k

1
2 θ1f⟩+ t2⟨ηPt2 , k

1
2 θ2f⟩.



Advances in Systems Science and Applications (2012) Vol.12 No.3 289

Using Markov property, (3) and (8), we get

logP0,ν,η exp

{
− θ1⟨Y (k)

t1
, f⟩ − θ2⟨Y (k)

t2
, f⟩

}
= logP0,η exp

{
− ⟨µ, u(k)(t1, ·; θ1, θ2)−

∫ t1

0
⟨ξ(k)s , u(k)(t1 − s, ·; θ1, θ2)⟩ds

− ⟨ξ(k)t1
, ω(k)(t2 − t1, ·; θ2)⟩

}
+ ⟨µPt1 , k

1
2 θ1f⟩+ ⟨µPt2 , k

1
2 θ2f⟩+ t1⟨ηPt1 , k

1
2 θ1f⟩+ t2⟨ηPt2 , k

1
2 θ2f⟩

=k−1γ1

∫ t2−t1

0
⟨µ, pt2−r(·)⟩

[
v(k)(r, 0; θ2)

]2
dr

+ k−1γ1

∫ t1

0
⟨µ, pt1−r(·)⟩

[
u(k)(r, 0; θ1, θ2)

]2
dr

+ k−1γ2

∫ t2−t1

0
⟨η, pt2−l(·)⟩[ω(k)(l, 0; θ1, θ2)]

2dl

+ k−1γ2

∫ t1

0
⟨η, pt1−r(·)⟩[s(k)(r, 0; θ1, θ2)]2dr

+ k−1γ1

∫ t2−t1

0
dl

∫ l

0
⟨η, pt2−r(·)⟩[v(k)(r, 0; θ2)]2dr

+ k−1γ1t1

∫ t2−t1

0
⟨η, pt2−r(·)⟩[v(k)(r, 0; θ2)]2dr

+ k−1γ1

∫ t1

0
dl

∫ l

0
⟨η, pt1−l(·)⟩[u(k)(r, 0; θ1, θ2)]2dr,

where θ1, θ2 ≥ 0, v(k)(·, ·; θ2) is the non-negative solution to

v(k)(r, x; θ2) = k
1
2 θ2Prf(x)− γ1k

−1

∫ r

0
pr−l(x)

[
v(k)(l, 0; θ2)

]2
dl,

u(k)(·, ·; θ1, θ2) is the non-negative solution to

u(k)(r, x; θ1, θ2) = Pr

[
k

1
2 θ1f + v(k)(t2 − t1, ·; θ2)

]
(x)

− γ1k
−1

∫ r

0
pr−l(x)

[
u(k)(l, 0; θ1, θ2)

]2
dl,

ω(k)(·, ·; θ2) is the non-negative solution to

ω(k)(r, x; θ2) =

∫ r

0
Pr−l

[
v(k)(l, ·; θ2)

]
(x)dl − γ2k

−1

∫ r

0
pr−l(x)

[
ω(k)(l, 0; θ2)

]2
dl,



290 Xu Yang:Central Limit Theorems for the Single Point Catalytic...

and s(k)(·, ·; θ1, θ2) is the non-negative solution to

s(k)(r, x; θ1, θ2) = Pr

[
ω(k)(t2 − t1, ·; θ2)

]
(x) +

∫ r

0
Pr−l

[
u(k)(l, ·; θ1, θ2)

]
(x)dl

−γ2k−1

∫ r

0
pr−l(x)

[
s(k)(l, 0; θ1, θ2)

]2
dl.

It is easy to show that k−1/2v(k), k−1/2u(k), k−1/2ω(k) and k−1/2s(k) are all con-
vergent as k → ∞. Then

lim
k→∞

logP0,µ,η exp

{
− θ1⟨Y (k)

t1
, f⟩ − θ2⟨Y (k)

t2
, f⟩

}
=γ1θ

2
1

∫ t1

0
⟨µ, pt1−r(·)⟩

[
Prf(0)

]2
dr + γ1θ

2
2

∫ t2

0
⟨µ, pt2−r(·)⟩

[
Prf(0)

]2
dr

+ 2γ1θ1θ2

∫ t1

0
⟨µ, pt1−r(·)⟩Prf(0)Pt2−t1+rf(0)dr

+ γ1θ
2
1

∫ t1

0
dl

∫ l

0
⟨η, pt1−r(·)⟩

[
Prf(0)

]2
dr

+ γ1θ
2
2

∫ t2

0
dl

∫ l

0
⟨η, pt2−r(·)⟩

[
Prf(0)

]2
dr

+ 2γ1θ1θ2

∫ t1

0
dl

∫ l

0
⟨η, pt1−r(·)⟩Prf(0)Pt2−t1+rf(0)dr

+ γ2θ
2
1

∫ t1

0
⟨η, pt1−r(·)⟩

[
rPrf(0)

]2
dr + γ2θ

2
2

∫ t2

0
⟨η, pt2−r(·)⟩

[
rPrf(0)

]2
dr

+ 2γ2θ1θ2

∫ t1

0
dl

∫ l

0
⟨η, pt1−r(·)⟩r(t2 − t1 + r)Prf(0)Pt2−t1+rf(0)dr.

Recalling (9), we can obtain the result by the method of [12] Page 110. 2

In the following, for fixed interval I := [0, T ], T > 0, we introduce the Banach
space CI

p (R) of all continuous maps u of I into Cp(R) equipped with the norm

∥u∥Ip := sup{9u(t)9p : t ∈ I}. The proofs of the following Lemmas ??, ?? and
?? are essentially similar to those of [1, Lemma 2.5.2], [1, Lemma 2.6.2] and [1,
Lemma 3.2.1], so we only present the results and omit the proofs here:

Lemma 2.2 There are two positive constants ε1, ε2, such that for |θ| < ε1, there
is a unique solution u = uθ ∈ CI

p (R) to the following equation

u(t, x) = θPtf(x)−
∫ t

0
pt−r(x) [u(r, 0)]

2 dr, 0 ≤ t ≤ T, x ∈ R

satisfying ∥u∥Ip < ε2.



Advances in Systems Science and Applications (2012) Vol.12 No.3 291

Set

v(t, x) := θPtf(x)− u(t, x), 0 ≤ t ≤ T, x ∈ R, |θ| < ε1.

Let v(n) denote the nth derivative of v with respect to θ, taken at θ = 0. Put

∥Sf∥T := sup

{
|Ptf(0)| : t ∈ (0, T ]

}
, f ∈ Cp(R), T > 0.

Lemma 2.3 For each f ∈ Cp(R), 0 ≤ t ≤ T, n ≥ 2,

v(2)(t, x) = 2

∫ t

0
pt−r(x) [Prf(0)]

2 dr, x ∈ R

and there is a constant cn such that

9v(n)(t)9p ≤ n!cn∥Sf∥nTα(t)n−1,

where α(t) = t
1
2 + t

p+1
2 .

Lemma 2.4 For each k ≥ 1, n ≥ 2, f ∈ Cp(R),

P0,µ,η

[∣∣∣⟨Y (k)
t , f⟩

∣∣∣2] = 2

∫ t

0
⟨µ, pt−s(·)⟩(s2 + 1) [Psf(0)]

2 ds

+2

∫ t

0
ds

∫ s

0
⟨η, pt−r(·)⟩ [Prf(0)]

2 dr,

and there exists a constant Cn such that∣∣∣P0,µ,η

[
⟨Y (k)

t , f⟩
]n∣∣∣ ≤ Cnt

n
4 ∥Sf∥n1

n−1∑
i=1

( 9 µ 9p + 9 η 9p

)i
.

By the proofs of Proposition ?? and [1, Lemma 3.2.1], for all t2 > t1 > 0 and
φ,ψ ∈ Cp(R) we have

P0,µ,η

[
⟨Y (k)

t1
, φ⟩+ ⟨Y (k)

t2
, ψ⟩

]
= 0 (10)

and

(−1)nP0,µ,η

[
⟨Y (k)

t1
, φ⟩+ ⟨Y (k)

t2
, ψ⟩

]n
=⟨kµ, ū(n)k (t1, ·, φ, ψ)⟩+ ⟨kη, s̄(n)k (t1, ·, φ, ψ)⟩+

∑
2≤j≤n−2

(
n− 1

j

)[
⟨kµ, ū(n−j)

k

(t1, ·, φ, ψ)⟩+ ⟨kη, s̄(n−j)
k (t1, ·, φ, ψ)⟩

]
(−1)jP0,µ,η

[
⟨Y (k)

t1
, φ⟩+ ⟨Y (k)

t2
, ψ⟩

]j
,

(11)



292 Xu Yang:Central Limit Theorems for the Single Point Catalytic...

where n ≥ 2, v̄k(·, ·, ψ) is the non-negative solution to

v̄k(r, x, ψ) = k−
1
2Prψ(x)−

∫ r

0
pr−l(x)

[
v̄k(l, 0, ψ)

]2
dl,

ūk(·, ·, φ, ψ) is the non-negative solution to

ūk(r, x, φ, ψ) = Pr

[
k−

1
2φ+ v̄k(t2 − t1, ·, ψ)

]
(x)−

∫ r

0
pr−l(x)

[
ūk(l, 0, φ, ψ)

]2
dl,

ω̄k(·, ·, ψ) is the non-negative solution to

ω̄k(r, x, ψ) =

∫ r

0
Pr−l

[
v̄k(l, ·, ψ)

]
(x)dl −

∫ r

0
pr−l(x)

[
ω̄k(l, 0, ψ)

]2
dl,

and s̄k(·, ·, φ, ψ) is the non-negative solution to

s̄k(r, x, φ, ψ) = Pr

[
ω̄k(t2 − t1, ·, ψ)

]
(x) +

∫ r

0
Pr−l

[
ūk(l, ·, φ, ψ)

]
(x)dl

−
∫ r

0
pr−l(x)

[
s̄k(l, 0, φ, ψ)

]2
dl.

Then by (10), (11) and calculations, we get the following estimate.

Lemma 2.5 There exists a positive constant c0 such that

P0,µ,η

[
⟨Y (k)

t , Phf⟩ − ⟨Y (k)
t+h, f⟩

]6
≤ c0h

3
2 ∥Sf∥61, 0 ≤ t < t+h ≤ 1, k ≥ 1, f ∈ S (R).

Similarly to [1, Lemma 3.2.2], we have the sixth moment of the increments of

the process {Y (k)
t : t ≥ 0}:

Lemma 2.6 There is a positive constant c0 such that

P0,µ,η

[
⟨Y (k)

t − Y (k)
s , f⟩

]6
≤ c0(t− s)

3
2 , 0 ≤ s < t ≤ 1, k ≥ 1, f ∈ S+(R).

Proof. Applying the elementary inequality

|x+ y|n ≤ 2n−1(|x|n + |y|n), x, y ∈ R, n ≥ 0,

there are positive constants D1, D2, D3 such that

P0,µ,η

[
⟨Y (k)

t − Y (k)
s , f⟩

]6
≤ D1

(
P0,µ,η

[
⟨Y (k)

s , Pt−sf − f⟩
]6

+P0,µ,η

[
⟨Y (k)

s Pt−s − Y
(k)
t , f⟩

]6)
≤ D2

(
∥S(Pt−sf − f)∥61 + (t− s)

3
2 ∥Sf∥61

)
≤ D3

(
∥Pt−sf − f∥6∞ + (t− s)

3
2 ∥f∥6∞

)



Advances in Systems Science and Applications (2012) Vol.12 No.3 293

by Lemmas ?? and ??. Since Phf−f
h −→ ς

2f
′′ = Af as h → 0, h 7→ Phf−f

h is
strongly continuous on h ∈ [0, 1]. This implies that there exists a positive constant
D4 such that ∥Phf − f∥∞ ≤ hD4 for h ∈ [0, 1]. Let c0 = D3(D

6
4 + ∥f∥6∞). We

finish the proof. 2

Lemma ?? and Kolmogorov’s criterion lead to

Lemma 2.7 For any f ∈ S (R), the sequence {⟨Y (k)
t , f⟩ : t ≥ 0; k ≥ 1} is tight

in C([0,∞),R).

Proof of Theorem ??. By Lemma ?? and Proposition ??, the result follows from
[13, Theorem 6.15]. 2

Proposition 2.8 For each k ≥ 1 and f ∈ Cp(R),

M
(k)
t (f) := ⟨Y (k)

t , f⟩−
∫ t

0
⟨Y (k)

s , Af⟩ds−
∫ t

0

(
⟨ξ(k)r , k

1
2 f⟩ − ⟨η, Prk

1
2 f⟩

)
dr, t ≥ 0

is a continuous martingale.

Proof. Fix k ≥ 1. By the Markov property of {X(k)
t : t ≥ 0}, for t > s ≥ 0,

P0,µ,η

(
⟨X(k)

t , f⟩ −
∫ t

s
⟨X(k)

r , Af⟩dr
∣∣Fs

)
=⟨X(k)

s , Pt−sf⟩+ (t− s)⟨ξ(k)s , Pt−sf⟩

−
∫ t

s

(
⟨X(k)

s , Pr−sAf⟩+ (r − s)⟨ξ(k)s , Pr−sAf⟩
)
dr.

Then by [14, Proposition 1.5], we have

P0,µ,η

(
⟨X(k)

t , f⟩ −
∫ t

0
⟨X(k)

r , Af⟩dr
∣∣Fs

)
=⟨X(k)

s , f⟩ −
∫ s

0
⟨X(k)

r , Af⟩dr +
∫ t−s

0
⟨ξ(k)s , Prf⟩dr

and

⟨µ, Ptf⟩+ t⟨η, Ptf⟩ −
∫ t

0

(
⟨µ, PrAf⟩+ r⟨η, PrAf⟩

)
dr = ⟨µ, f⟩+

∫ t

0
⟨η, Prf⟩dr.

Recalling Y
(k)
t = k

1
2 (X

(k)
t − µPt − tηPt), we have

P0,µ,η

[
M

(k)
t (f)

∣∣Fs

]
=M (k)

s (f).

2



294 Xu Yang:Central Limit Theorems for the Single Point Catalytic...

Lemma 2.9 There is a locally bounded function t→ C(t) on [0,∞) such that

sup
k≥1

P0,µ,η

{
sup

0≤t≤T

∣∣∣⟨Y (k)
t , f⟩

∣∣∣2} ≤ C(T )∥f∥22, T ≥ 0, f ∈ H2(R).

Proof. We only consider f ∈ S (R). For each k ≥ 1,

P0,µ,η

{
sup

0≤t≤T

∣∣∣⟨Y (k)
t , f⟩

∣∣∣2}

≤ 3 P0,µ,η

{
sup

0≤t≤T
M

(k)
t (f)2

}
+ 3 P0,µ,η

{[∫ T

0

∣∣∣⟨Y (k)
s , Af⟩

∣∣∣ ds]2}

+3 P0,η

{[∫ T

0

∣∣∣⟨ξ(k)s , k
1
2 f⟩ − ⟨η, Psk

1
2 f⟩

∣∣∣ ds]2} .

Using Hölder inequality, Fubini theorem and (5), we have

P0,η

{[∫ T

0

∣∣∣⟨ξ(k)s , k
1
2 f⟩ − ⟨η, Psk

1
2 f⟩

∣∣∣ ds]2}

≤2 T

∫ T

0
ds

∫ s

0
⟨η, ps−r(·)⟩ (Prf(0))

2 dr.

By Proposition ??, M
(k)
t (f) is a continuous martingale. Then by Doob’s martin-

gale inequality, Hölder inequality and Fubini theorem, we have

P0,µ,η

{
sup

0≤t≤T
M

(k)
t (f)2

}

≤ 12 P0,µ,η

{∣∣∣⟨Y (k)
T , f⟩

∣∣∣2}+ 12 T

{∫ T

0
P0,µ,η

[∣∣∣⟨Y (k)
s , Af⟩

∣∣∣2] ds}
+24 T

∫ T

0
ds

∫ s

0
⟨η, ps−r(·)⟩ (Prf(0))

2 dr.

By Lemma ??, there is a constant C such that

[Prf(0)]
2 ≤ 1√

2πςr
q0(f)

2 ≤ C√
2πςr

∥f∥20.

It is easy to show that ⟨µ, ps−r(·)⟩ ≤ 1√
2πς(s−r)

(9µ 9p +µ([−
√
2pπςs,

√
2pπςs])

)
.

Then by Lemmas ?? and ??, there is a locally bounded function t → C1(t) on



Advances in Systems Science and Applications (2012) Vol.12 No.3 295

[0,∞) such that P0,µ,η

[∣∣∣⟨Y (k)
s , f⟩

∣∣∣2] ≤ C2(s)∥f∥20 and P0,µ,η

[∣∣∣⟨Y (k)
s , Af⟩

∣∣∣2] ≤

C2(s)∥f∥22 for any all s ≥ 0. This finishes the proof. 2

Proof of Theorem ??. Using [13, Corollary 6.16], the result follows from Propo-
sition ??, Lemmas ?? and ??. 2

Acknowledgements
Mathematics Subject Classifications (2000):Supported by NSFC (No.11071021),Primary
60J80; Secondary 60G20, 60J68.

References

[1] Dawson, D.A, Fleischmann, K. (1994), “A super-brownian motion with a
single point catalyst”, Stoch. Proc. Appl, Vol.49, pp.3-40.

[2] Fleischmann K, Xiong J. (2005), “Large deviation principle for the single
point catalytic super-brownian motion”, Markov Processes Related Fields,
Vol.11, pp.519-533.

[3] Yang, X, Zhang, M. (2010),“ Moderate deviation for the single point cat-
alytic super-brownian motion”, Acta Mathematica Sinica, English Series,
Vol.28, pp.1799–1808.

[4] Li, Z.H, Wang, L. (2010), “Fluctuation limits of the super-brownian motion
with a single point catalyst” (Submitted).

[5] Hong W.M. (2002), “Longtime behavior for the occupation time processes
of a super-brownian motion withrandom immigration”, Stoch. Proc. Appl,
Vol.102, pp.43-62.

[6] Hong W.M. (2003), “Large deviations for the super-brownian motion with
super-brownian immigration”, Theoret. Probab, Vol.16, pp.899-922.

[7] Hong W.M, Li Z.H. (1999), “A central limit theorem for super-brownian
immigraion”, Appl. Prob, Vol.36, pp.1218-1224.

[8] Li Z. H, Shiga T. (1995), “Measure-valued branching diffusions: immi-
graiton, excursions and limit theorems”, Math. Kyoto Univ, Vol.35, pp.233-
274.

[9] Zhang, M. (2004), “Large deviation for super-brownian motion with immi-
gration”, J. Appl. Prob, Vol.41, pp.187-201.

[10] Zhang, M. (2005), “Functional central limit theorem for the super-brownian
motion with super-brownian immigration”, J. Theoret. Probab, Vol.18,
pp.665-685.



296 Xu Yang:Central Limit Theorems for the Single Point Catalytic...

[11] Li, Z.H. (2011), Measure-Valued Branching Markov Process, Springer,
Berlin.

[12] Iscoe I. (1986), “A weighted occupation time for a class of measure-valued
critical branching brownian motion”, Probab.Th. Rel. Fields, Vol.71, pp.85-
116.

[13] Walsh, J.B. (1986), “An introduction to stochastic partial differential equa-
tions”, In: Ecole D’Eté de Probabilités de Saint-Flour XIV -1984, pp.265-439,
Lecture Notes Math, pp.1180, Springer, Berlin.

[14] Ethier, S.N, Kurtz, T.G. (1986), Markov Processes: Characterization and
Convergenc, Wiley, New York.

Corresponding author
Xu Yang can be contacted at:xuyang@mail.bnu.edu.cn