ijpam.eu · International Journal of Pure and Applied Mathematics ————————————————————————– Volume 17 No. 3 2004, 351-379

International Journal of Pure and Applied Mathematics————————————————————————–Volume 17 No. 3 2004, 351-379

SUPERPROCESS OVER A STOCHASTIC

FLOW WITH SUPERPROCESS CATALYST

Jie Xiong1, Xiaowen Zhou2 §

1Department of MathematicsUniversity of Tennessee

Knoxville, TN 37996-1300, USAe-mail: [email protected]

and1Department of Mathematics

Hebei Normal UniversityShijiazhuang, 050016, P.R. CHINA

2Department of Mathematics and StatisticsConcordia University

7141 Sherbrooke Street West, Montreal, Quebec, H4B 1R6, CANADAe-mail: [email protected]

Abstract: In this paper, we study the catalytic superprocesses under a stocha-stic flow where the catalyst itself is a superprocess under the same flow. Com-paring with the study of the superprocess under a stochastic flow with deter-ministic catalyst, here we encounter a serious adaptability problem caused bythis common stochastic flow in proving the uniqueness by using the usual con-ditional log-Laplace transform approach. To overcome this difficulty, we find alimiting moment dual and show that the moments increase not too fast so thatthe moments determine the distribution. We also prove the state property bythe moment method.

AMS Subject Classification: 60J80, 60K25Key Words: catalytic superprocesses, random medium, branching particlesystem

Received: September 10, 2004 c© 2004, Academic Publications Ltd.

§Correspondence author

352 J. Xiong, X. Zhou

1. Introduction

Superprocess over a stochastic flow was studied by Skoulakis and Adler [15]using a moment duality argument. It is conjectured in [15] that the conditionallog-Laplace transform of the superprocess should be the unique solution to anonlinear stochastic partial differential equation (SPDE). This conjecture wasproved by Xiong [19]. As the first application of the conditional log-Laplaceapproach, Xiong [20] studied the long-term behavior of the superprocesses overa stochastic flow.

A related model has been studied by Wang in two earlier papers ([16], [17]),where the random medium is given by a space-time color-white noise using themoment duality argument. It was proved in Wang ([16], [18]) that the processis absolutely continuous for the uniformly elliptic case and purely-atomic forthe degenerate case. For the first case, a SPDE is derived for the density byDawson et al [7]. This model was later generalized by Dawson et al [6]. Forthis generalized model, Li et al ([12], [13], [14]) studied the conditional log-Laplace equation and various properties of the process, e.g., the scaling limit,the long-term behavior and the excursion representation.

In this paper, we study a catalyst-reactant pair of superprocesses over astochastic flow. Without the stochastic flow, this problem has been studiedby many authors (see the survey papers of Dawson and Fleischmann [4] andKlenke [11]) since the early work of Dawson and Fleischmann [5].

Now let us introduce our model in details. First, we introduce the catalyst.At independent exponential times (with exponent γ > 0), the catalyst particleseither die or split into 2 with equal probabilities. Between branching times,the motion of the ith particle is governed by an individual Brownian motionBci (t) and a common Brownian motion W (t) which applies to all particles in

the system:

dηit = bc(ηit)dt+ σc(η

it)dB

ci (t) + σc(η

it)dW (t) ,

where bc, σc, σc : R → R are continuous maps, W, Bc1, B

c2, · · · are independent

(standard) one dimensional Brownian motions, ηit is the position of the i-thcatalyst particle at time t. Skoulakis and Adler [15] proved that the high-density limit ρt exists as an MF (R)-valued process, where MF (R) denotesthe collection of all finite Borel measures on R. By a conditional log-Laplaceargument, Xiong [19] proved that ρt is the unique solution to the followingconditional martingale problem (CMP): ∀ φ ∈ C2

b (R),

Mc,φt ≡ 〈ρt, φ〉 − 〈ρ0, φ〉 −

∫ t

0

⟨

ρs, acφ′′ + bcφ

′⟩

ds

SUPERPROCESS OVER A STOCHASTIC... 353

−∫ t

0

⟨

ρs, σcφ′⟩

dWs (1)

is a continuous PW -martingale with quadratic variation process

⟨

M c,φ⟩

t=

∫ t

0

⟨

ρs, γφ2⟩

ds , (2)

where ac = 12

(

σ2c + σ2

c

)

, PW denotes the conditional probability with the wholepath of W as given.

Next, we introduce the reactant branching particle system. Let A = α =n1 · · ·nℓ(α) : ni ∈ N denote the collection of particles. We define an arborealorder in A as follows: m1 · · ·mp ≺ n1 · · ·nq iff p ≤ q andmi = ni, i = 1, 2, · · · , p.Let Bα(t) : t ≥ 0, α ∈ A be a family of independent one-dimensionalBrownian motions. Let Sα be i.i.d. with common exponential distribution withparameter θ. For t ≥ 0 and x ∈ R, let ηα(t, x), α ∈ A, be i.i.d. Z

+-valuedrandom variables such that

Eρ,Wηα(t, x) = 1 and E

ρ,W (ηα(t, x) − 1)2 = Pǫρt(x) ,

where Pǫ is the Brownian semigroup.For α ∈ A, let βα and ζα (= βα + Sα) be the birth and death times

of the particle α. Then βα = 0 if ℓ(α) = 1; βα = ζα−1 if ℓ(α) > 1, whereα− 1 = n1 · · · nℓ(α)−1 is the father of α. The trajectory xα(t) : βα ≤ t ≤ ζαis given by

xα(βα + t) = xα(βα) +

∫ βα+t

βα

br(xα(s))ds

+

∫ βα+t

βα

σr(xα(s))dBα(s) +

∫ βα+t

βα

σr(xα(s))dW (s) ,

where xα(βα) = xα−1(ζα−1), br, σr, σr : R → R are continuous maps. Let⟨

Xθt , φ⟩

= θ−1∑

α∈A

φ(xα(t))1[βα,ζα)(t).

In this paper, we study the limit of Xθ.

Theorem 1. Suppose that θ → ∞, ǫ → 0 and ǫ−2θ−1 → 0. ThenXθ is tight in C([0,∞),MF (R)) and the limit X solves the following CMP:∀ φ ∈ C2

b (R),

Mr,φt ≡ 〈Xt, φ〉 − 〈X0, φ〉 −

∫ t

0

⟨

Xs, arφ′′ + brφ

′⟩

ds


−∫ t

0

⟨

Xs, σrφ′⟩

dWs (3)

is a continuous PW,ρ-martingale with quadratic variation process

⟨

M r,φ⟩

t=⟨

L[X,ρ](t), φ2⟩

, (4)

where ar = 12

(

σ2r + σ2

r

)

,

⟨

L[X,ρ](t), φ2⟩

= limδ→0

∫ t

0ds

1

δ

∫ δ

0du

∫

Xs(dx)

∫

ρs(dy)

∫

dz

Prc(u, (x, y), (z, z))φ2(z)

is the collision local time between X and ρ, Prc is the transition probability

density of the 2-dimensional diffusion consists of one catalyst particle and onereactant particle, i.e., the diffusion with generator

Lcrf(x1, x2) = ac(x1)∂

2x1f + bc(x1)∂x1

f + ar(x2)∂2x2f + br(x2)∂x2

f

+σc(x1)σr(x2)∂2x1x2

f ,

for f ∈ C2b (R

2).

Remark 2. Usually, the collision local time is defined as (cf. Dawson etal [3] which generizes that of Barlow et al [1])

limδ→0

1

δ

∫ δ

0dǫ

∫ t

0ds

∫

R

dxpǫ ∗Xs(x)pǫ ∗ ρs(x)φ(x). (5)

If X and ρ have continuous density, these two definitions coincide and is givenby

⟨

L[X,ρ](t), φ2⟩

=

∫ t

0

⟨

Xs, ρsφ2⟩

ds.

However, they are different in general. For example, take

X(dx) = x−1/21x>0dx, ρ(dx) = (−x)−1/21x<0dx.

Then L[X,ρ](t) = ctδ0 where the constant depends on the approximate proce-dure.

Also, note that by [8] (Theorem 6.4.5, p. 141),

Prc(u, (x, y), (z, z)) ≤ cpu(x− z)pu(y − z) ,


and hence,

⟨

L[X,ρ](t), φ2⟩

≤ c limδ→0

1

δ

∫ δ

0dǫ

∫ t

0ds

∫

R

dxpǫ ∗Xs(x)pǫ ∗ ρs(x)φ2(x) ,

where pǫ is the heat kernel.In the special case that σr = σc = 1, br = bc = σr = σc = 0, we see that our

definition of the collision local time coincides with (5).

Throughout this paper, we make the following assumptions:

(BC) σc, σc, bc and σr, σr, br are bounded Lipschitz continuous functions.Further, σr and σc are bounded below away from 0.

(IC) X0, ρ0 ∈ MF (R).

In this paper, we first prove in Section 2 the tightness of Xθ and show thateach limit point will be a solution to the CMP (3)-(4).

As indicated by [15] in the study of a related model, it is natural to usethe conditional log-Laplace transform of Xt to derive the uniqueness for thesolution to the CMP (3-4). For deterministic branching rate case, this has beenstudied by [19] and [13]. It has been demonstrated in [20] and [13] that sucha log-Laplace transform is useful in deriving properties of the process. Mimic[19], we guess that

Eρ,Wµ e−〈Xt,φ〉 = e−〈µ,y0,t〉 ,

where ys,t is governed by the following stochastic partial differential equation(SPDE):

ys,t(x) = φ(x) +

∫ t

s(ar(x)∂

2xyu,t(x) + br(x)∂xyu,t(x) − ρu(x)yu,t(x)

2)du

+

∫ t

sσr(x)∂xyu,t(x)dWu, (6)

where dWu is the backward Ito integral. Note that ρu is Fu-measurable andfor the backward Ito integral, it requires the integrand to be F[u,t]-measurable.However, it is not clear whether yu,t(x) is F[u,t]-measurable. Therefore, weencounter serious adaptivity problem in solving the SPDE (6). To prove theuniqueness of the solution to the CMP (3-4), we construct a limit moment dualof the process. The following theorem will be proved in Section 3. Recall thatps is the heat kernel.

Theorem 3. Suppose that

sups>0

∫

R2

ps(x− y)X0(dx)ρ0(dy) <∞. (7)


Then the CMP (3-4) has a unique solution.

Finally, in Section 4, we derive a property of the process based on themoment formula.

Theorem 4. Suppose that conditions (BC), (IC) and (7) hold, then Xt

is absolutely continuous with respect to Lebesgue measure for almost all t > 0and Pµ-almost surely.

2. Existence of a Solution

In this section, we establish the existence of a solution to the CMP (3)-(4).

Lemma 5.⟨

Xθt , φ⟩

=⟨

Xθ0 , φ⟩

+Mθt (φ) +N θ

t (φ) (8)

+

∫ t

0

⟨

Xθs , σrφ

′⟩

dWs +

∫ t

0

⟨

Xθs , arφ

′′ + brφ′⟩

ds ,

where Mθt and N θ

t are orthogonal P ρ,W -martingales with quadratic variationprocesses

⟨

Mθ(φ)⟩

t= (2θ)−1

∫ t

0

⟨

Xθs , ar(φ

′)2⟩

ds (9)

and⟨

N θ(φ)⟩

t=

∫ t

0

⟨

Xθs , (Pǫρs)φ

2⟩

ds , (10)

where ar = σ2r .

Proof. By Ito’s formula, we see that (8) holds with

Mθt (φ) =

∑

α∈A

∫ t

0θ−1σrφ

′(xα(s))1[βα,ζα)(s)dBα(s)

andN θt (φ) =

∑

α∈A

(ηα(ζα, xα(ζα−)) − 1)θ−1φ(xα(ζα−))1(0,t](ζα).

It is clear that Mθt (φ) is a P ρ,W -martingale with quadratic variation process

given by (9). Similar to Li et al [13], we can prove that N θt (φ) is a P ρ,W -

martingale, orthogonal to Mθt (ψ), with quadratic variation process given by

(10).


To consider the limit of Xθ as θ → ∞ and ǫ = ǫ(θ) → 0, we need someestimates on the moments. To derive formulas for the moments of Xθ

t , weconsider the following backward SPDE

pWs,t(x) = φ(x) +

∫ t

s(ar∂

2xpWu,t(x) + br∂xp

Wu,t(x))du

+

∫ t

sσr∂xp

Wu,t(x)dWu , (11)

where dW denotes the backward Ito integral. For convenience, we denote thesolution of (11) formally by

pWs,t(x) =

∫

pW (s, x; t, du)φ(u).

Lemma 6.

Eρ,W

⟨

Xθt , φ⟩

=⟨

X0, pW0,t

⟩

=

∫

X0(dx)

∫

pW (0, x; t, du)φ(u). (12)

Proof. Take expectations on both sides of (8), we have

⟨

Eρ,WXθ

t , φ⟩

= 〈X0, φ〉 +

∫ t

0

⟨

Eρ,WXθ

s , σrφ′⟩

dWs

+

∫ t

0

⟨

Eρ,WXθ

s , arφ′′ + brφ

′⟩

ds. (13)

(12) follows from similar arguments as in Lemma 5.3 in Li et al [13] (takeZ = 0, σ = m = 0 there, and replace (5.1) and (5.2) there by current (13) and(11)).

For f ∈ C2b (R

2), let

Lf(x1, x2) = ar(x1)∂2x1f + br(x1)∂x1

f + ar(x2)∂2x2f + br(x2)∂x2

f

+ σr(x1)σr(x2)∂2x1x2

f

andGf(x1, x2) = σr(x1)∂x1

f + σr(x2)∂x2f.

Let PWs,t (x1, x2) be the unique solution to the following linear SPDE:

PWs,t = f +

∫ t

sLPWr,t dr +

∫ t

sGPWr,t dWt. (14)


Again, for convenience, we denote formally

PWs,t (x1, x2) =

∫

PW (s, (x1, x2); t, d(u1, u2))f(u1, u2).

Lemma 7.

Eρ,W

⟨

(Xθt )

⊗2, f⟩

=

∫ ∫

X0(dx1)X0(dx2)

∫

PW (0, (x1, x2); t, d(u1, u2))f(u1, u2)

+

∫ t

0ds

∫

X0(dx)

∫

pW (0, x; s, dy)(Pǫρs)(y)

×∫

PW (s, (y, y); t, d(u1, u2))f(u1, u2)

+

∫ t

0ds

∫

X0(dx)

∫

pW (0, x; s, dy)θ−1ar(y)

× ∂2y1y2

∣

∣

∣

∣

y1=y2=y

∫

PW (s, (y1, y2); t, d(u1, u2))f(u1, u2). (15)

Proof. Applying Ito’s formula to (8), we have

d(⟨

Xθt , φ1

⟩⟨

Xθt , φ2

⟩)

= d(P ρ,W -mart.)t +⟨

Xθt , φ1

⟩(⟨

Xθt , arφ

′′2 + brφ

′2

⟩

dt +⟨

Xθt , σrφ

′2

⟩

dWt

)

+⟨

Xθt , φ2

⟩(⟨

Xθt , arφ

′′1 + brφ

′1

⟩

dt +⟨

Xθt , σrφ

′1

⟩

dWt

)

+

(

⟨

Xθt , σrφ

′1

⟩⟨

Xθt , σrφ

′2

⟩

+⟨

Xθt , (Pǫρt)φ1φ2

⟩

+ θ−1⟨

Xθt , arφ

′1φ

′2

⟩

)

dt.

Therefore

d⟨

(Xθt )

⊗2, f⟩

=⟨

(Xθt )

⊗2,Lf⟩

dt +⟨

(Xθt )

⊗2,Gf⟩

dWt

+⟨

Xθt , (Pǫρt)f(x, x) + θ−1ar∂

2x1x2

|x1=x2=xf(x1, x2)⟩

dt

+ d(P ρ,W -mart.)t (16)

holds for f = φ1 ⊗ φ2. Take expectations on both sides of (16), we have

⟨

Eρ,W (Xθ

t )⊗2, f

⟩

=⟨

X⊗20 , f

⟩

+

∫ t

0

⟨

Eρ,W (Xθ

s )⊗2,Lf

⟩

ds


+

∫ t

0

⟨

Eρ,W (Xθ

s )⊗2,Gf

⟩

dWs +

∫ t

0ds

∫

X0(dx)pW (0, x; s, du)

×(

(Pǫρs)(u)f(u, u) + θ−1ar(u)∂2u1u2

∣

∣

∣

∣

u1=u2=u

f(u1, u2)

)

(17)

holds for f = φ1 ⊗ φ2. By approximation, it is easy to see that (17) holds forall f ∈ C2

b (R2). Similar to Lemma 5.3 in [13] again (take Z = 0, σ = 0,

m =

∫

X0(dx)pW (0, x; s, du)

×(

(Pǫρs)(u)f(u, u) + θ−1ar(u)∂2u1u2

∣

∣

∣

∣

u1=u2=u

f(u1, u2)

)

there, and replace (5.1) and (5.2) there by current (17) and (14)), we have that(15) holds.

The following corollary follows from the same arguments as (12) and (15).

Corollary 8. Let qWs,t and QWs,t be defined similar to pWs,t and PWs,t (with r

replaced by c) in (11) and (14). Then

EW (〈ρs′ , φ〉 |ρs) =

∫

ρs(dx)

∫

qW (s, x; s′, du)φ(u) (18)

and

EW(⟨

ρ⊗2t , f

⟩

|ρs)

=

∫ ∫

ρs(dx1)ρs(dx2)

∫

QW (s, (x1, x2); t, d(u1, u2))f(u1, u2)

+

∫ t

sds′∫

ρs(dx)

∫

qW (s, x; s′, dy)

∫

QW (s′, (y, y); t, d(u1, u2))f(u1, u2).

Here is the key estimate in proving the tightness.

Theorem 9.

supθ>0,t≤T

E

⟨

Xθt , Pǫρt

⟩2<∞. (19)

Proof. Take

f(x1, x2) = (Pǫρt)(x1)(Pǫρt)(x2)


in (15), we have

Eρ,W

⟨

(Xθt )

⊗2, f⟩

≡ I1 + I2 + I3

where

I1 =

∫ ∫

X0(dx1)X0(dx2)

×∫

PW (0, (x1, x2); t, d(u1, u2))(Pǫρt)(u1)(Pǫρt)(u2),

I2 =

∫ t

0ds

∫

X0(dx)

∫

pW (0, x; s, dy)(Pǫρs)(y)

×∫

PW (s, (y, y); t, d(u1, u2))(Pǫρt)(u1)(Pǫρt)(u2),

and

I3 =

∫ t

0ds

∫

X0(dx)

∫


× ∂2y1y2 |y1=y2=y

∫

PW (s, (y1, y2); t, d(u1, u2))(Pǫρt)(u1)(Pǫρt)(u2).

By Corollary 8, we have

EW I3 = E

W I31 + EW I32 ,

where

I31 =

∫ t

0ds

∫

X0(dx)

∫


× ∂2y1y2 |y1=y2=y

∫

PW (s, (y1, y2); t, d(u1, u2))

∫ ∫

ρs(dx1)ρs(dx2)

×∫

QW (s, (x1, x2); t, d(z1, z2))pǫ(u1 − z1)pǫ(u2 − z2) ,

and

I32 =

∫ t

0ds

∫

X0(dx)

∫


× ∂2y1y2 |y1=y2=y

∫

PW (s, (y1, y2); t, d(u1, u2))


×∫ t

sds′∫

ρs(dx′)

∫

qW (s, x′; s′, dy′)

×∫

QW (s′, (y′, y′); t, d(z1, z2))pǫ(u1 − z1)pǫ(u2 − z2).

Note that

EW I31 =

∫ t

0ds

∫

X0(dx)

∫


× ∂2y1y2|y1=y2=y

∫

PW (s, (y1, y2); t, d(u1, u2))

∫ ∫

ρ0(dv1)ρ0(dv2)

×∫

QW (0, (v1, v2); s, d(x1, x2))

∫

QW (s, (x1, x2); t, d(z1, z2))

× pǫ(u1 − z1)pǫ(u2 − z2)

+

∫ t

0ds

∫

X0(dx)

∫


× ∂2y1y2 |y1=y2=y

∫

PW (s, (y1, y2); t, d(u1, u2))

∫ s

0ds′∫

ρ0(dx1)

×∫

qW (0, x1; s′, dv)

∫

QW (s′, (v, v); s, d(v1 , v2))

×∫

QW (s, (v1, v2); t, d(z1, z2))pǫ(u1 − z1)pǫ(u2 − z2) ≡ I311 + I312.

Then

EI312 = E

∫ t

0ds

∫ s

0ds′∫

X0(dx)

×∫ ∫

pW (0, x; s′, dz)pW (s′, z; s, dy)θ−1ar(y)

× ∂2y1y2|y1=y2=y

∫

PW (s, (y1, y2); t, d(u1, u2))

∫

ρ0(dx1)

×∫

qW (0, x1; s′, dv)

∫

QW (s′, (v, v); s, d(v1 , v2))

×∫

QW (s, (v1, v2); t, d(z1, z2))pǫ(u1 − z1)pǫ(u2 − z2).

It is easy to show that

EpW (0, x; s′, dz)qW (0, x1; s′, dv) = P

rc(s′, (x, x1), d(z, v)) ,


where the notation Prc represents the transition probability for the two-dimen-

sional diffusion consists of 1 catalyst point and 1 reactant particle.Similarly, we have

EpW (s′, z; s, dy)QW (s′, (v, v); s, d(v1 , v2))

= Prcc(s− s′, (z, v, v), d(y, v1 , v2))

and

EPW (s, (y1, y2); t, d(u1, u2))QW (s, (v1, v2); t, d(z1, z2))

= Prrcc(t− s, (y1, y2, v1, v2), d(u1, u2, z1, z2)).

Hence

EI312 =

∫ t

0ds

∫ s

0ds′∫

X0(dx)

∫

ρ0(dx1)

∫

Prc(s′, (x, x1), d(z, v))

×∫

Prcc(s− s′, (z, v, v), d(y, v1 , v2))θ

−1ar(y)

× ∂2y1y2 |y1=y2=y

∫

Prrcc(t− s, (y1, y2, v1, v2), d(u1, u2, z1, z2))

× pǫ(u1 − z1)pǫ(u2 − z2). (20)

Let ξt = ξt(y1, y2, v1, v2) be the 4-dimensional diffusion with transition prob-ability P

rrcc and initial location (y1, y2, v1, v2). We now want to estimate

∂2y1y2Epǫ(ξ

1t − ξ3t )pǫ(ξ

2t − ξ4t ) = E∂y2

(

p′ǫ(ξ1t − ξ3t )pǫ(ξ

2t − ξ4t )∂y1(ξ

1t − ξ3t )

+ pǫ(ξ1t − ξ3t )p

′ǫ(ξ

2t − ξ4t )∂y1(ξ

2t − ξ4t )

)

= Ep′′ǫ (ξ1t − ξ3t )pǫ(ξ

2t − ξ4t )∂y1(ξ

1t − ξ3t )

× ∂y2(ξ1t − ξ3t ) + 2Ep′ǫ(ξ

1t − ξ3t )p

′ǫ(ξ

2t − ξ4t )∂y1(ξ

1t − ξ3t )∂y2(ξ

1t − ξ3t )

+ Ep′ǫ(ξ1t − ξ3t )pǫ(ξ

2t − ξ4t )∂

2y1y2(ξ

1t − ξ3t ) + Epǫ(ξ

1t − ξ3t )p

′′ǫ (ξ

2t − ξ4t )∂y1(ξ

2t − ξ4t )

× ∂y2(ξ2t − ξ4t ) + Epǫ(ξ

1t − ξ3t )p

′ǫ(ξ

2t − ξ4t )∂

2y1y2(ξ

2t − ξ4t ).

Now we only consider the first term. Note that

p′′ǫ ≤ cǫ−2.

So1st ≤ cǫ−2

E|∂y1(ξ1t − ξ3t )∂y2(ξ1t − ξ3t )| ≤ cǫ−2 ,


where the last inequality follows from Theorem 5.4 in [8], p. 122. The otherterms can be treated similarly. Now we can continue (20) with

EI312 ≤ cǫ−2θ−1

∫ t

0ds

∫ s

0ds′∫

X0(dx)

∫

ρ(dx1) = cǫ−2θ−1.

Take ǫ−2θ−1 → 0. We have EI312 → 0. I311 and I32 can be treated similarly.Therefore, EI3 → 0. Similar calculations for I1 and I2 yield that

supθ>0,t≤T

E(I1 + I2 + I3) <∞.

This proves (19).

Proof of Theorem 1. From (19), we easily get the tightness of(

Xθ,⟨

Mθ(φ)⟩

,⟨

N θ(φ)⟩)

.

Further, it is easy to show that for t > 0 fixed, Xθt (φ)2 and

⟨

N θ(φ)⟩

are uni-formly integrable.

It is clear that Mθ(φ) → 0. Suppose that(

Xθ,⟨

N θ(φ)⟩)

→ (X,Λ). Weonly need to show that Λ(t) =

⟨

L[X,ρ](t), φ2⟩

.Now we adapt the argument in the proof of Theorem 3.4 in Xiong and Zhou

[21]. Note that

E(Λ(t) − Λ(s)|Fs) = limθ→∞

E(⟨

N θ(φ)⟩

t−⟨

N θ(φ)⟩

s|Fs)

= limθ→∞

E

(∫ t

s

⟨

Xθu, Pǫ(θ)ρuφ

2⟩

du

∣

∣

∣

∣

Fs)

= limθ→∞

E

(∫ t

sdu

∫

Xθs (dx)

∫

pW (s, x;u, dy)Pǫ(θ)ρu(y)φ2(y)

∣

∣

∣

∣

Fs)

where the last equality follows from (12). By (18), we can continue with

E(Λ(t) − Λ(s)|Fs)

= limθ→∞

E

(

∫ t

sdu

∫

Xθs (dx)

∫

pW (s, x;u, dy)

∫

dzpǫ(y − z)

×∫

ρs(dw)qW (s,w;u, z)φ2(y)

∣

∣

∣

∣

ρs,Xθs

)

=

∫ t

sdu

∫

Xs(dx)

∫

ρs(dw)

∫

dyPrc(u− s, (x,w), (y, y))φ2(y).


Therefore,

∫ t

0ds

1

δE(Λ(s+ δ) − Λ(s)|Fs)

=

∫ t

0ds

1

δ

∫ s+δ

sdu

∫

Xs(dx)

∫

ρs(dy)

∫

dzPrc(u− s, (x, y), (z, z))φ2(z).

Take δ → 0, we have

Λ(t) = limδ→0

∫ t

0ds

1

δ

∫ δ

0du

∫

Xs(dx)

∫

ρs(dy)

×∫

dzPrc(u, (x, y), (z, z))φ2(z) =⟨

L[X,ρ](t), φ2⟩

.

3. Moment Duality

In this section, we first establish a moment duality for the process (Xt, ρt).Then we verify Carleman’s condition to show that the distribution of (Xt, ρt)is determined by the moments.

First, we need the following lemma.

Lemma 10. If (ρ,X) is a solution to the CMP (1-4), then it solves thefollowing martingale problem (MP):

Nc,φt ≡ 〈ρt, φ〉 − 〈ρ0, φ〉 −

∫ t

0

⟨

ρs, acφ′′ + bcφ

′⟩

ds

and

Nr,ψt ≡ 〈Xt, ψ〉 − 〈X0, ψ〉 −

∫ t

0

⟨

Xs, arψ′′ + brψ

′⟩

ds

are continuous martingales with quadratic covariation processes

⟨

N c,φ⟩

t=

∫ t

0

⟨

ρs, σcφ′⟩2ds+

∫ t

0

⟨

ρs, γφ2⟩

ds,

⟨

N r,ψ⟩

t=

∫ t

0

⟨

Xs, σrψ′⟩2ds +

⟨

L[X,ρ](t), ψ2⟩

,

and⟨

N c,φ, N r,ψ⟩

t=

∫ t

0

⟨

ρs, σcφ′⟩ ⟨

Xs, σrψ′⟩

ds.


Proof. Since M c,φt is a P ρ,W -martingale, it is clearly a martingale. For

s < t, we have

E(M c,φt M

r,ψt |Fs) = E(E(M r,ψ

t |σ(ρ,W ) ∨ Fs)M c,φt |Fs)

= E(M r,ψs M

c,φt |Fs)

= M r,ψs M c,φ

s .

Hence⟨

M c,φ,M r,ψ⟩

t= 0. Similarly,

⟨

M r,ψ,W⟩

t=⟨

M c,φ,W⟩

t= 0.

Hence

Nc,φt = M

c,φt +

∫ t

0

⟨

ρs, σcφ′⟩

dWs

is a martingale. Similarly, N r,ψt is a martingale. Further, we have

⟨

N c,φ, N r,ψ⟩

t=

∫ t

0

⟨

Xs, σrψ′⟩ ⟨

ρs, σcφ′⟩

ds.

The other statement can be proved similarly.

Appling Ito’s formula to (3) and (1), we have

dΠmi=1 〈Xt, φi〉

=m∑

j=1

(Πi6=j 〈Xt, φi〉)(

⟨

Xt, arφ′′j + brφ

′j

⟩

dt+⟨

Xt, σrφ′j

⟩

dWt + dMr,φj

t

)

+1

2

∑

1≤j 6=k≤m

(Πi6=j,k 〈Xt, φi〉)

(⟨

Xt, σrφ′j

⟩ ⟨

Xt, σrφ′k

⟩

dt +⟨

L[X,ρ](dt), φjφk⟩)

(21)

and

dΠni=1 〈ρt, ψi〉 =

n∑

j=1

(Πi6=j 〈ρt, ψi〉) ×(⟨

ρt, acψ′′j + bcψ

′j

⟩

dt

+⟨

ρt, σcψ′j

⟩

dWt + dMc,ψj

t

)

+1

2

∑

1≤j 6=k≤m

(Πi6=j,k 〈ρt, ψi〉)

×(⟨

ρt, σcψ′j

⟩ ⟨

ρt, σcψ′k

⟩

+ 〈ρt, γψjψk〉)

dt. (22)


Let

f(x1, · · · , xm; y1, · · · , yn) = Πmi=1φi(xi)Π

nj=1ψj(yj)

and

Fm,n,f (X, ρ) =⟨

X⊗m ⊗ ρ⊗n, f⟩

.

Define the generator for m+ n-points motion as

Am,nf(x1, · · · , xm; y1, · · · , yn)

=

m∑

j=1

(

ar(xj)∂2f

∂x2j

+ br(xj)∂f

∂xj

)

+1

2

∑

1≤j 6=k≤m

σr(xj)σr(xk)∂2f

∂xj∂xk

+

n∑

j=1

(

ac(yj)∂2f

∂y2j

+ bc(yj)∂f

∂yj

)

+1

2

∑

1≤j 6=k≤n

σc(yj)σc(yk)∂2f

∂yj∂yk

+1

2

m∑

j=1

n∑

k=1

σr(xj)σc(yk)∂2f

∂xj∂yk.

Appling Ito’s formula to (21) and (22), we see that

Fm,n,f (Xt, ρt) −∫ t

0

1

2

∑

1≤j 6=k≤m

⟨

X⊗(m−2)s ⊗ L[X,ρ](ds) ⊗ ρ⊗ns , Grjkf

⟩

−∫ t

0

Fm,n,Am,nf (Xs, ρs) +1

2

∑

1≤j 6=k≤n

Fm,n−1,Gcjkf (Xs, ρs)

ds (23)

is a martingale, where

Grjkf(x1, · · · , xm−2, x; y1, · · · , yn)= f(x1, · · · , x, · · · , x, · · · , xm−2; y1, · · · , yn)

and

Gcjkf(x1, · · · , xm; y1, · · · , yn−2, y)

= γf(x1, · · · , xm; y1, · · · , y, · · · , y, · · · , yn−2),

where the x and y are at their respective jth and kth places. By approximation,we can show that (23) holds for other bounded smooth function f (instead ofa product). Note that


∫ t

0

1

2

∑

1≤j 6=k≤m

⟨

Grjkf,X⊗(m−2)s ⊗ L[X,ρ](ds) ⊗ ρ⊗ns

⟩

= limδ→0

∫ t

0

1

2

∑

1≤j 6=k≤m

Fm−1,n+1,Gr,δ

jkf(Xs, ρs)ds,

where

Gr,δjk f(x1, · · · , xm−2, x; y, y1, · · · , yn)

=1

δ

∫ δ

0dǫ

∫

R

dzPrc(ǫ, (x, y), (z, z))

× f(x1, · · · , z, · · · , z, · · · , xm−2; y1, · · · , yn).

Now we construct an approximate dual process (mt, nt, fδt ) as follows: (mt, nt)

is a birth-death process with a rate 12m(m − 1) to jump from (m,n) to (m −

1, n + 1) and a rate 12n(n − 1) from (m,n) to (m,n − 1). Namely, the second

component is Kingman’s coalescent process and the first is itself a Kingman’scoalescent process but gives a birth to the second type during its coalescentevent.

Let τ0 = 0 and τm0+n0−1 = ∞, and let τk : 1 ≤ k ≤ m0 + n0 − 2 be thesequence of jump times of (mt, nt) : t ≥ 0. Let Γδk : 1 ≤ k ≤ m0 + n0 − 2be a sequence of random operators which are conditionally independent given(mt, nt) : t ≥ 0 and satisfy

P

(

Γδk = Gcij |nτk− = ℓ, nτk = ℓ− 1)

=1

ℓ(ℓ− 1), 1 ≤ i 6= j ≤ ℓ

and

P

(

Γδk = Gr,δij |mτk− = ℓ,mτk = ℓ− 1

)

=1

ℓ(ℓ− 1), 1 ≤ i 6= j ≤ ℓ.

For τk ≤ t < τk+1, we define

f δt = P(mτk

,nτk)

t−τkΓδkP

(mτk−1,nτk−1

)

τk−τk−1Γδk−1 · · ·P

(mτ1,nτ1

)τ2−τ1 Γδ1P

(mτ0,nτ0

)τ1 f0 , (24)

where P(m,n)t is the semigroup generated by Am,n of an m + n-dimensional

diffusion process. Then (mt, nt, fδt ) is a Markov process taking values on N

2×C

where C = ∪m≥1C(Rm).


Theorem 11.

E⟨

X⊗mt ⊗ ρ⊗nt , f

⟩

= limδ→0

E

[⟨

X⊗mt

0 ⊗ ρ⊗nt

0 , f δt

⟩

exp

1

2

∫ t

0(ms(ms − 1) + ns(ns − 1)) ds

]

. (25)

Proof. When m = 0, this theorem has been proved by [6] for all n. Inthat case, the process (nt, ft) does not depend on δ and is the dual process ofρt. For m = 1, there is no coalescence for the first component and hence, thetheorem follows from the case of m = 0 directly. Now we prove our theorem byinduction in m.

We assume that (25) holds for m and prove it for m + 1. To this end, wefirst verify that

E

(

〈Xt, 1〉m+1 〈ρt, 1〉n)

<∞. (26)

By (23),

〈Xt, 1〉m+1 〈ρt, 1〉n −∫ t

0

m(m+ 1)

2〈Xs, 1〉m−1 〈ρs, 1〉n

⟨

L[X,ρ](ds), 1⟩

−∫ t

0

n(n− 1)

2〈Xs, 1〉m+1 〈ρs, 1〉n−1 ds (27)

is a martingale. We estimate

E

∫ t

0〈Xs, 1〉m−1 〈ρs, 1〉n

⟨

L[X,ρ](ds), 1⟩

≤ c lim infδ→0

E

∫ t

0〈Xs, 1〉m−1 〈ρs, 1〉n

⟨

Xs ⊗ ρs,1

δ

∫ δ

0dǫp2ǫ(x− y)

⟩

ds.

Take

f0(x1, · · · , xm; y1, · · · , yn+1) =1

δ

∫ δ

0dǫp2ǫ(xm − yn+1).

Now we estimate the process defined by (24). By Friedman [8] (Theorem 6.4.5,

p. 141), we know that P(m,n)t has a density dominated by cp

⊗(m+n)t . Hence, for

τ1 ≤ t,

P (m0,n0)τ1 f0 ≤ c

δ

∫ δ

0dǫp2τ1+ǫ(x− y) ≤ c√

τ1

(note that the constant c depends on t but not on τ1). Hence

f δt ≤ cΠkj=1(τj − τj−1)

−1/2. (28)


Therefore

E

(

〈Xs, 1〉m−1 〈ρs, 1〉n⟨

Xs ⊗ ρs,1

δ

∫ δ

0dǫpǫ(x− y)

⟩)

≤ cE(

Πkj=1(τj − τj−1)

−1/2)

<∞,

where the finiteness above follows from the fact that τj − τj−1, j ≥ 1 areindependent exponential random variables. Applying induction to (27) withrespect to n, we can show that (26) holds. Now we proceed to proving (25) form+ 1.

Let 0 = t0 < t1 < · · · < tk = t be a partition of [0, t] such that

maxti − ti−1 : 1 ≤ i ≤ k → 0.

Then

E⟨


⟩

− E

[

⟨

X⊗mt0 ⊗ ρ⊗nt

0 , f δt

⟩

exp

1

2

∫ t

0(ms(ms − 1) + ns(ns − 1)) ds

]

=

k∑

i=1

E

[

⟨

X⊗mt−titi

⊗ ρ⊗nt−titi

, f δt−ti

⟩

exp

1

2

∫ t−ti

0(ms(ms − 1) + ns(ns − 1)) ds

]

− E

[

⟨

X⊗mt−ti−1

ti−1⊗ ρ

⊗nt−ti−1

ti−1, f δt−ti−1

⟩

exp

1

2

∫ t−ti−1

0(ms(ms − 1) + ns(ns − 1)) ds

]

.

Note that

limk→∞

k∑

i=1

E

[

⟨

X⊗mt−titi

⊗ ρ⊗nt−titi

, f δt−ti

⟩

exp

1

2

∫ t−ti

0(ms(ms − 1) + ns(ns − 1)) ds

]

− E

[

⟨

X⊗mt−ti−1

ti⊗ ρ

⊗nt−ti−1

ti, f δt−ti−1

⟩

exp

1

2

∫ t−ti

0(ms(ms − 1) + ns(ns − 1)) ds

]


= −∫ t

0E

(

exp

1

2

∫ t−u

0(ms(ms − 1) + ns(ns − 1)) ds

(

Fmt−u,nt−u,Amt−u,nt−ufδ

t−u(Xu, ρu)

+1

2

nt−u∑

j,k=1

Fmt−u,nt−u−1,Gcjkfδ

t−u(Xu, ρu)

− mt−u(mt−u − 1) + nt−u(nt−u − 1)

2Fmt−u,nt−u,fδ

t−u(Xu, ρu)

)

du

−∫ t

0E

(

exp

1

2

∫ t−u

0(ms(ms − 1) + ns(ns − 1)) ds

1

2

mt−u∑

j,k=1

⟨

X⊗(mt−u−2)u ⊗ Lδ[X,ρ](du) ⊗ ρ⊗nt−u

u , Grjkfδt−u

⟩

)

du,

limk→∞

k∑

i=1

E

[

⟨

X⊗mt−ti−1

ti⊗ ρ

⊗nt−ti−1

ti, f δt−ti−1

⟩

exp

1

2

∫ t−ti

0(ms(ms − 1) + ns(ns − 1)) ds

]

− E

[

⟨

X⊗mt−ti−1

ti−1⊗ ρ

⊗nt−ti−1


⟩

exp

1

2

∫ t−ti

0(ms(ms − 1) + ns(ns − 1)) ds

]

=

∫ t

0E

(

exp

1

2

∫ t−u

0(ms(ms − 1) + ns(ns − 1)) ds

(

Fmt−u,nt−u,Amt−u,nt−ufδ

t−u(Xu, ρu)

+1

2

nt−u∑

j,k=1

Fmt−u,nt−u−1,Gcjkfδ

t−u(Xu, ρu)

)

)

du

+

∫ t

0E

(

exp

1

2

∫ t−u

0(ms(ms − 1) + ns(ns − 1)) ds

1

2

mt−u∑

j,k=1

⟨

X⊗(mt−u−2)u ⊗ L[X,ρ](du) ⊗ ρ⊗nt−u

u , Grjkfδt−u

⟩

)

du


and

limk→∞

k∑

i=1

E

[

⟨

X⊗mt−ti−1

ti−1⊗ ρ

⊗nt−ti−1


⟩

exp

1

2

∫ t−ti

0(ms(ms − 1) + ns(ns − 1)) ds

]

− E

[

⟨

X⊗mt−ti−1

ti−1⊗ ρ

⊗nt−ti−1


⟩

exp

1

2

∫ t−ti−1

0(ms(ms − 1) + ns(ns − 1)) ds

]

)

= −∫ t

0E

(

exp

1

2

∫ t−u

0(ms(ms − 1) + ns(ns − 1)) ds

mt−u(mt−u − 1) + nt−u(nt−u − 1)

2Fmt−u,nt−u,fδ

t−u(Xu, ρu)

)

du.

Therefore, we have

E⟨


⟩

− E

[

⟨

X⊗mt0 ⊗ ρ⊗nt

0 , f δt

⟩

exp

1

2

∫ t

0(ms(ms − 1) + ns(ns − 1)) ds

]

= −1

2

∫ t

0E

(

exp

1

2

∫ t−u

0(ms(ms − 1) + ns(ns − 1)) ds

mt−u∑

j,k=1

(

⟨


u , Grjkfδt−u

⟩

−⟨


u , Grjkfδt−u

⟩

)

)

du. (29)

Take a subsequence if necessary, we may and will assume thatLδ[X,ρ] → L[X,ρ] in C([0, t],MF (R)) a.s. as δ → 0. Note that ms(ms − 1) +

ns(ns − 1) is bounded. The other factor on the right hand side of (29) is dom-inated by the right hand side of (28). By the dominated convergence theorem,we have

limδ→0

E

∫ t

0

(

exp

1

2

∫ t−u

0(ms(ms − 1) + ns(ns − 1)) ds


(

⟨


u , Grjkfδt−u

⟩

−⟨


u , Grjkfδt−u

⟩

)

)

= E limδ→0

∫ t

0

(

exp

1

2

∫ t−u

0(ms(ms − 1) + ns(ns − 1)) ds

(

⟨


u , Grjkfδt−u

⟩

−⟨


u , Grjkfδt−u

⟩

)

)

.

Therefore, we only need to prove that

limδ→0

∫ t

0

(

exp

1

2

∫ t−u

0(ms(ms − 1) + ns(ns − 1)) ds

(

⟨


u , Grjkfδt−u

⟩

−⟨


u , Grjkfδt−u

⟩

)

)

= 0 a.s. (30)

Now we fix ω. Since there are only finite many jumps for the process(mt−u, nt−u, f

δt−u), we estimate the limit for the integral over subinterval [a, b]

such that no jump occurs in it. In this case, f δt−u is bounded and equi-continuousin u ∈ [a, b]. Therefore

limδ→0

∫ b

a

(

exp

1

2

∫ t−u

0(ms(ms − 1) + ns(ns − 1)) ds

(

⟨


u , Grjkfδt−u

⟩

−⟨


u , Grjkfδt−u

⟩

)

)

= 0.

Sum up over all such subintervals, we see that (30) holds.

Now we show that the moments grow not too fast so that the momentproblem determines the distribution.


Proof of Theorem 3. First we estimate E (〈Xt, φ〉m 〈ρt, ψ〉n). To this end,let f0 = φ⊗m ⊗ ψ⊗n. Suppose that φ and ψ are bounded by c. Then

P (m,n)τ1 f0 ≤ cm+n. (31)

Now we seek the bound for Pτ2−τ1Γδ1P

(m,n)τ1 f0. At τ1, if two catalyst particle

coalesce, then the bound in (31) remains valid for Pτ2−τ1Γδ1P

(m,n)τ1

f0. Suppose that two reactant particles coalesce at time τ1, then

Γδ1P(m,n)τ1 f0(x1, · · · , xm−1; y1, · · · , yn+1)

≤ 1

δ

∫ δ

0dǫ

∫

R

dzpǫ(z − xm−1)pǫ(z − y1)cm+n

=1

δ

∫ δ

0dǫp2ǫ(xm−1 − y1)c

m+n.

Note that the value of c may have been changed. In fact, we always use cto denote a constant when its specific value is of no concern to us. Hence

Pτ2−τ1Γδ1P

(m,n)τ1 f0 ≤ 1

δ

∫ δ

0dǫp2ǫ+2(τ2−τ1)(xm−1 − yn+1)c

m+n. (32)

At τ2, there are three cases: two catalyst particles coalesce, two reactant parti-cles (not include xm−1), or xm−1 coalesces with another reactant particle. Forthe first case, there is no change for the bound. For the second case, it is easyto see that the bound is

cm+n 1

δ

∫ δ

0dǫ2

1

δ

∫ δ

0dǫ1p2ǫ2(xm−1 − yn+2)p2(ǫ1+τ2−τ1)(xm−2 − yn+1).

For the third case,

Γδ2Pτ2−τ1Γδ1P

(m,n)τ1 f0

≤ cm+n 1

δ

∫ δ

0dǫ2

∫

R

dz1

δ

∫ δ

0dǫ1p2ǫ1+2(τ2−τ1)(z − yn+1)

pǫ2(z − xm−2)pǫ2(z − yn+2).

Hence

Pτ3−τ2Γδ2Pτ2−τ1Γ

δ1P

(m,n)τ1 f0

≤ cm+n 1

δ

∫ δ

0dǫ2

∫

R

dz1

δ

∫ δ

0dǫ1p2ǫ1+2(τ3−τ1)(z − yn+1)


pǫ2+τ2−τ1(z − xm−2)pǫ2+τ2−τ1(z − yn+2)

≤ cm+n

√τ3 − τ2

p2(ǫ2+τ2−τ1)(xm−2 − yn+2).

Note that τ3 − τ2 is exponential with parameter no greater than (m − 1).Note that, for the next step, the worst case will involve the second case of thepresent step. Therefore, the τ4 − τ3 there will be exponential with parameterno greater than 2(m− 2). The patten will be k(m− k). In summary, the worstcase estimate will be

ft ≤ Πj(τj − τj−1)−1/2Πk

1

δ

∫ δ

0dǫkp2ǫ+ak

(xik − yjk).

Note thatE

(

(τj − τj−1)−1/2

)

≤ cmm!.

HenceE(〈Xt, φ〉m 〈ρt, ψ〉n) ≤ cm+nm!.

Then

E ((〈Xt, φ〉 + 〈ρt, ψ〉)m) ≤m∑

k=0

(

m

k

)

cmk! ≤ cmm!.

Note that

∑

m≥1

(

E

(

(〈Xt, φ〉 + 〈ρt, ψ〉)2m))−1/2m

≥∑

m≥1

(

c2m(2m)!)−1/2m

≥ c∑

m≥1

(2m)−1 = ∞.

Namely, Carleman’s condition is satisfied and the moment problem determinethe distribution (cf. Chung [2]).

4. The Existence of Density

In this section, we study the absolute continuity of the MF (R)-valued randomelement Xt. Namely, we prove the existence of the density by a standard secondmoment calculation. We continue to assume X0 and ρ0 to be finite.

Proof of Theorem 4. By (25), we have

E 〈Xt, g〉〈Xt, h〉


=

∫ ∫

X0(dx1)X0(dx2)

∫

Prr(t, (x1, x2), d(u1, u2))g(u1)h(u2)

+ limδ→0

∫ t

0ds

∫

X0(dx1)

∫

ρ0(dx2)

∫

Prc(t− s, (x1, x2), d(y1, y2))

× 1

δ

∫ δ

0dǫ

∫

dzpǫ(z − y1)pǫ(z − y2)

∫

Prr(s, (z, z), d(u1, u2))g(u1)h(u2)

=

∫ ∫

X0(dx1)X0(dx2)

∫

Prr(t, (x1, x2), d(u1, u2))g(u1)h(u2)

+

∫ t

0ds

∫

X0(dx1)

∫

ρ0(dx2)

∫

dzPrc(t− s, (x1, x2), (z, z))

×∫

Prr(s, (z, z), d(u1 , u2))g(u1)h(u2). (33)

Takeg = p(ǫ, x, ·) and h = p(ǫ′, x, ·) .

Note that∫ ∫

Prr(t, (x1, x2), d(u1, u2))p(ǫ, x, u1)p(ǫ

′, x, u2)

→ Prr(t, (x1, x2), (x, x)) (34)

and∫ ∫


′, x, u2)

≤ cp(t+ ǫ, x1 − x)p(t+ ǫ′, x2 − x)

→ cp(t, x1 − x)p(t, x2 − x). (35)

As∫ T

0dt

∫

dx

∫ ∫

X0(dx1)X0(dx2)p(t+ ǫ, x1 − x)p(t+ ǫ′, x2 − x)

=

∫ T

0dt

∫ ∫

X0(dx1)X0(dx2)p(2t+ ǫ+ ǫ′, x1 − x2),

andp(2t+ ǫ+ ǫ′, x1 − x2) ≤

c√t,

by the dominated convergence theorem, we have

∫ T

0dt

∫

dx

∫ ∫

X0(dx1)X0(dx2)p(t+ ǫ, x1 − x)p(t+ ǫ′, x2 − x)


→∫ T

0dt

∫

dx

∫ ∫

X0(dx1)X0(dx2)p(t, x1 − x)p(t, x2 − x). (36)

Making use of the extended dominated convergence theorem (cf. Kallenberg[9], p. 12), by (34), (35) and (36), we have

∫ T

0dt

∫

dx

∫ ∫

X0(dx1)X0(dx2)

×∫ ∫


′, x, u2)

→∫ T

0dt

∫

dx

∫ ∫

X0(dx1)X0(dx2)Prr(t, (x1, x2), (x, x)). (37)

Note that∫ ∫

Prr(s, (z, z), d(u1, u2))p(ǫ, x, u1)p(ǫ

′, x, u2)

≤ cp(s + ǫ, z − x)p(s+ ǫ′, z − x)

≤ c√sp(s+ ǫ, z − x).

Similar to the arguments leading to (37), we have that

∫ T

0dt

∫

dx

∫ ∫

X0(dx1)ρ0(dx2)

∫

dzPrc(t− s, (x1, x2), (z, z))

×∫ ∫

Prr(s, (z, z), d(u1 , u2))p(ǫ, x, u1)p(ǫ

′, x, u2)

→∫ T

0dt

∫

dx

∫ ∫

X0(dx1)ρ0(dx2)

×∫

dzPrc(t− s, (x1, x2), (z, z))Prr(s, (z, z), (x, x)). (38)

(33), (37) and (38) show that

∫ T

0

∫

E(〈Xt, p(ǫ, x, ·)〉⟨

Xt, p(ǫ′, x, ·)

⟩

)dxdt (39)

→∫ T

0dt

∫

dx

∫ ∫

X0(dx1)X0(dx2)Prr(t, (x1, x2), (x, x))

+

∫ T

0dt

∫

dx

∫ t

0ds

∫

X0(dx1)

∫

ρ0(dx2)


×∫

dzPrc(t− s, (x1, x2), (z, z))Prr(s, (z, z), (x, x)).

Therefore, as ǫ → 0, 〈Xt, p(ǫ, x, ·)〉 converges weakly as elements in L2(Ω ×[0, T ]×L2(R)). This implies the existence of the density Xt(x) of Xt. Further,∫ T0 dt

∫

dxEXt(x)2 <∞.

5. Acknoledgements

The first author’s research is supported partially by Alexander von HumboldtFoundation and by NSA. The second author’s research is supported by NSERCGrant N00761.

References

[1] M.T. Barlow, S.N. Evans, E.A. Perkins, Collision local times and measure-valued processes, Canadian Journal of Mathematics, 43 (1991), 897-938.

[2] K.L. Chung, A Course in Probability Theory, Volume 21 of Probability and

Mathematical Statistics, Academic Press, Second Edition (1974).

[3] D.A. Dawson, A.M. Etheridge, K. Fleischmann, L. Mytnik, E.A. Perkins,J. Xiong, Mutually catalytic branching in the plane: finite measure states,Ann. Probab., 30, No. 4 (2002), 1681–1762.

[4] D.A. Dawson, K. Fleischmann, Catalytic and mutually catalytic branch-ing, Infinite Dimensional Stochastic Analysis, Amsterdam (1999), 145–170;Verh. Afd. Natuurkd. 1. Reeks. K. Ned. Akad. Wet., 52, R. Neth. Acad.Arts Sci., Amsterdam (2000).

[5] D.A. Dawson, K. Fleischmann, A continuous super-Brownian motion in asuper-Brownian medium, J. Theoret. Probab., 10 (1997), 213-276.

[6] D.A. Dawson, Z. Li, H. Wang, Superprocesses with dependent spatial mo-tion and general branching densities, Electr. J. Probab., 6, No. 25 (2001),1-33.

[7] D.A. Dawson, J. Vaillancourt, H. Wang, Stochastic partial differentialequations for a class of interacting measure-valued diffusions, Ann. Inst.

H. Poincar Probab. Statist., 36, No. 2 (2000), 167–180.


[8] A. Friedman, Stochastic Differential Equations and Applications, Volume1, Academic Press (1975).

[9] O. Kallenberg, Foundations of Mordern Probability, Second Edition,Springer (2001).

[10] G. Kallianpur, J. Xiong, Stochastic differential equations in infinite di-mensional spaces, IMS Lecture Notes-Monograph Series, 26, Institute ofMathematical Statistics (1995).

[11] A. Klenke, A review on spatial catalytic branching, Stochastic Models, Ot-tawa, (1998), 245-263; In: MS Conf. Proc., 26, Amer. Math. Soc., Provi-dence, RI (2000).

[12] Z. Li, H. Wang, J. Xiong, A degenerate stochastic partial differential equa-tion for superprocesses with singular interaction, Probab. Th. Relat. Fields,130 (2004), 1-17.

[13] Z. Li, H. Wang, J. Xiong, Conditional Log-Laplace functionals of immigra-tion superprocesses with dependent spatial motion, WIAS Preprint, No.900 (2004).

[14] Z. Li, H. Wang, J. Xiong, Conditional excursion representation for a classof interacting superprocesses, WIAS Preprint, No. 935 (2004).

[15] G. Skoulakis, R.J. Adler, Superprocesses over a stochastic flow, Ann. Appl.

Probab., 11, No. 2 (2001), 488-543.

[16] H. Wang, State classification for a class of measure-valued branching dif-fusions in a Brownian medium, Probab. Theory Relat. Fields, 39 (1997),39-55.

[17] H. Wang, A class of measure-valued branching diffusions in a randommedium, Stochastic Anal. Appl., 16 (1998), 753-786.

[18] H. Wang, State classification for a class of interacting superprocesses withlocation dependent branching, Electr. Comm. Probab., 7, No. 16 (2002),157-167.

[19] J. Xiong, A stochastic Log-Laplace equation, Ann. Probab., 32 (2004),2362-2388.

[20] J. Xiong, Long-term behavior for superprocesses over a stochastic flow,Electronic Communications in Probability, 9 (2004), 36-52.


[21] J. Xiong, X. Zhou, On the duality between coalescing Brownian motions,Canada J. Math. (2004), To Appear.

380

Documents

ijpam.eu · International Journal of Pure and Applied Mathematics ————————————————————————– Volume 17 No. 3 2004, 351-379