A stochastic log-Laplace equation

arXiv:math/0410164v1 [math.PR] 6 Oct 2004

The Annals of Probability 2004, Vol. 32, No. 3B, 2362–2388 DOI: 10.1214/009117904000000540 c Institute of Mathematical Statistics, 2004

A STOCHASTIC LOG-LAPLACE EQUATION1 By Jie Xiong University of Tennessee We study a nonlinear stochastic partial differential equation whose solution is the conditional log-Laplace functional of a superprocess in a random environment. We establish its existence and uniqueness by smoothing out the nonlinear term and making use of the particle system representation developed by Kurtz and Xiong [Stochastic Process. Appl. 83 (1999) 103–126]. We also derive the Wong–Zakai type approximation for this equation. As an application, we give a direct proof of the moment formulas of Skoulakis and Adler [Ann. Appl. Probab. 11 (2001) 488–543].

1. Introduction and main results. 1.1. Introduction. We study the behavior of a branching interacting particle system in a random environment. For simplicity of notation, we assume that the particles move in the one-dimensional space R. The branching is critical binary; that is, at independent exponential times, each particle will die or split into two with equal probabilities. Between branchings, the motion of the ith particle is governed by an individual Brownian motion Bi (t) and a common Brownian motion W (t) which applies to all particles in the system: (1.1)

dηti = b(ηti ) dt + c(ηti ) dW (t) + e(ηti ) dBi (t),

i = 1, 2, . . . ,

where b, c, e are real functions on R (c, e ≥ 0), W , B1 , B2 , . . . are independent (standard) Brownian motions and ηti is the position of the ith particle at time t. Let MF (R) denote the set of all finite Borel measures on R. It is established by Skoulakis and Adler [18] that the high-density limit Xt of this system is the unique MF (R)-valued solution to the following martingale Received May 2002; revised June 2003. Supported in part by the NSA. AMS 2000 subject classifications. Primary 60G57, 60H15; secondary 60J80. Key words and phrases. Superprocess, random environment, Wong–Zakai approximation, particle system representation, stochastic partial differential equation. 1

This is an electronic reprint of the original article published by the Institute of Mathematical Statistics in The Annals of Probability, 2004, Vol. 32, No. 3B, 2362–2388. This reprint differs from the original in pagination and typographic detail. 1

2

J. XIONG

problem (MP): Xt is a continuous process with initial X0 = µ ∈ MF (R) such that, for any φ ∈ Cb2 (R), Mt (φ) ≡ hXt , φi − hµ, φi −

Z

t

0

hXs , bφ′ + aφ′′ i ds

is a continuous martingale with quadratic variation process hM (φ)it =

Z

0

t

(hXs , φ2 i + |hXs , cφ′ i|2 ) ds,

where a(x) = 21 (e(x)2 + c(x)2 ). Moment formulas are derived in [18]. A related model is studied by Wang [19] and Dawson, Li and Wang [4]. The log-Laplace equation has been used by many authors in deriving various properties for superprocesses (cf. [2, 5]). It is natural, as indicated in [18], to derive properties of Xt by making use of the corresponding backward stochastic log-Laplace equation (LLE): ys,t(x) = f (x) + (1.2) +

t

Z

s

t

Z

s

(b(x)∂x yr,t(x) + a(x)∂x2 yr,t (x) − yr,t(x)2 ) dr

ˆ r, c(x)∂x yr,t (x) dW

where f is the test function for the Laplace transform [cf. (1.8)], ∂x , ∂x2 are the first and second partial derivatives with respect to x and the last integral is the backward Itô integral. Since a solution to (1.2) is not established in [18], the moment formulas for Xt are derived based on other techniques. The establishment of a unique solution to (1.2) is posed by [18] as an interesting challenge. In this paper, we study the LLE (1.2). The main result is Theorem 1.2 in which we prove that the log-Laplace transform of Xt is indeed given by the solution to (1.2). For simplicity of notation, we consider the forward version of the LLE: yt (x) = f (x) + (1.3) +

Z

0

Z

0

t

(b(x)∂x yr (x) + a(x)∂x2 yr (x) − yr (x)2 ) dr

t

c(x)∂x yr (x) dWr .

The stochastic partial differential equation (SPDE) is an important field of current research. We refer the reader to [1, 11] and [17] for an introduction to this topic. Many authors studied linear SPDEs. Here we only mention two recent papers: [9] and [14]. Fine properties of the solutions are established. Nonlinear SPDEs have also been studied. Here we mention a sequence of papers by Kotelenez [12, 13] which are the closest to the present setting. In this case, the derivative of the solution is not involved in the noise term. To the best of our knowledge, the LLE (1.3) does not fit into the setups of existing theory of SPDE.

A STOCHASTIC LOG-LAPLACE EQUATION

3

1.2. Main results. First we study the existence and uniqueness for the solution to (1.3). We also establish its particle system representation in the spirit of Kurtz and Xiong [15]. To begin with, we introduce some notation needed in this paper. Let H0 = L2 (R) be the set of all square integrable functions on R, and let H0+ consist of all the nonnegative functions in H0 . Let Hm = {φ ∈ H0 : φ′ , . . . , φ(m) ∈ H0 }. Define the Sobolev norm on Hm by kφk2m

=

m Z X

j=1

|φ(j) (x)|2 dx.

Use h·, ·i to denote the inner product in H0 or the integral of a function with respect to a measure. Definition 1.1. An H0+ -valued (measurable) process yt is a solution to (1.3) if, for any φ ∈ C0∞ (R), hyt , φi = hf, φi + +

Z

0

t

Z

0

t

hyr , −(bφ)′ + (aφ)′′ − yr φi dr

hyr , −(cφ)′ i dWr ,

t ≥ 0.

Throughout this paper, we assume the following Boundedness condition (BC). f ≥ 0, b, c, e are bounded functions with bounded first and second derivatives. Denote a bound by K. Further, e is bounded away from 0, c has third continuous and bounded derivative and f is of compact support. Theorem 1.2.

Suppose that Condition (BC) holds. Then:

(i) The LLE (1.3) has a unique solution yt (x). (ii) yt is the unique solution of the following infinite particle system: i = 1, 2, . . . , (1.4) (1.5) (1.6)

dξti = e(ξti ) dBi (t) + (2a′ − b − cc′ )(ξti ) dt − c(ξti ) dWt ,

dmit = mit ((a′′ − b′ − yt )(ξti ) dt − c′ (ξti ) dWt ), n 1X mit δξ i t n→∞ n i=1

Yt = lim

a.s.

where, for any t ≥ 0, Yt is absolutely continuous with respect to Lebesgue measure and yt is the Radon–Nikodym derivative.

4

J. XIONG

Next, we consider the Wong–Zakai type approximation to LLE (1.3): ytε (x)

= f (x) +

(1.7) +

Z

t 0

Z

t 0

(¯b(x)∂x yrε (x) + a ¯(x)∂x2 yrε (x) − yrε (x)2 ) dr

˙ ε dr, c(x)∂x yrε (x)W r

¯(x) = 12 e(x)2 and, for kε ≤ r < (k + 1)ε, where ¯b(x) = b(x) − 12 c(x)c′ (x), a −1 ε ˙ = ε (W(k+1)ε − Wkε ). W r Theorem 1.3.

Suppose that Condition (BC) holds. Then for any t ≥ 0, E

Z

|ytε (x) − yt (x)|2 dx → 0

as ε → 0. Now we consider the Wong–Zakai approximation to the measure-valued process X. Let PW be the conditional probability measure given W . Let X ε be the solution to the following conditional martingale problem (CMP): X ε is a continuous MF (R)-valued process such that, for any φ ∈ Cb2 (R), Mtε (φ) ≡ hXtε , φi − hX0ε , φi −

Z

t

0

˙ ε )φ′ + a ¯φ′′ i ds hXsε , (¯b + cW s

is a continuous PW -martingale with quadratic variation process hM ε (φ)it =

Z

0

t

hXsε , φ2 i ds.

¯ ≡ R∪{∂} be the one-point compactification of R. Denote by MF (R) ¯ Let R ¯ the space of all finite measures on R with the weak convergence topology. ¯ by extending the Note that MF (R) can be regarded as a subset of MF (R) measure at ∂ as 0. ¯ Theorem 1.4. As ε → 0, if X0ε → µ in MF (R), then X ε → X in C([0, ∞), MF (R)) W in conditional law P for almost all W . As a consequence, we have (1.8)

EW exp(−hXt , f i) = exp(−hµ, y0,t i)

a.s.

Finally, we derive the moment formulas of Xt . Note that these formulas have been obtained in [18] by a different method. Let p(t, x, y) and q(t, (x1 , x2 ), (y1 , y2 )) be the transition density functions of the Markov processes with generators L1 φ(x) = b(x)φ′ (x) + a(x)φ′′ (x)


5

and L2 F (x1 , x2 ) = b(x1 )∂x1 F + b(x2 )∂x2 F

+ a(x1 )∂x21 F + a(x2 )∂x22 F + c(x1 )c(x2 )∂x1 ∂x2 F,

respectively. Theorem 1.5. Suppose that Condition (BC) holds. For any bounded continuous function f , we have (1.9)

E(hXt , f i) =

ZZ

f (y)p(t, x, y) dyµ(dx)

and E(hXt , f i2 ) = (1.10)

Z

R4

f (y1 )f (y2 )q(t, (x1 , x2 ), (y1 , y2 )) dy1 dy2 µ(dx1 )µ(dx2 )

+2

Z Z tZ 0

p(t − s, x, y) ×

ZZ

f (z1 )f (z2 )q(s, (y, y), (z1 , z2 )) dz1 dz2 dy dsµ(dx).

We shall use K with a subscript to denote a constant. If it will be quoted, the subscript will be the equation where it is defined. Otherwise, we shall use K1 , K2 , . . . in the proof of a proposition and the sequence starts over again in the proof of a new proposition. For example, K1 may appear in the proofs of two different propositions to represent different constants. Note that the Wong–Zakai approximation is not really needed to obtain the results in Theorems 1.4 and 1.5. An easier approach in deriving (1.8) is available. We refer the reader to [16] for the treatment of a related model which adds immigration structure to a branching interacting system studied in [4] and [19]. In this paper, we use the Wong–Zakai approximation because this is part of the conjecture in [18] and the main purpose of the current paper is to solve that conjecture. Furthermore, the Wong–Zakai approximation is of interest on its own. 2. Stochastic log-Laplace equation. In this section, we prove Theorem 1.2. 2.1. Approximation. To establish the existence of a nonnegative solution to (1.3), we smooth and truncate its nonlinear term and consider ε

yt (x) = f (x)

6

J. XIONG

(2.1)

+

t

Z

0

+

Z

t

0

(b(x)∂x εyr (x) + a(x)∂x2 εyr (x) − (Tε εyrε (x))εyr (x)) dr c(x)∂x εyr (x) dWr ,

1 2 where Tε h(x) ≡ pε (x − z)h(z) dz, pε (x) = (2πε)−1/2 exp(− 2ε x ), εyrε (x) = ε εˆ y r yr (x) and

R

εˆy

with the convention that Lemma 2.1.

0 0

r=

= 0.

R

εy

Rr

(u) du ∧ ε−1 εy (u) du r

Equation (2.1) has a unique solution.

Proof. Consider the following infinite particle system: i = 1, 2, . . . , dξti = e(ξti ) dBi (t) + (2a′ − b − cc′ )(ξti ) dt − c(ξti ) dWt ,

(2.2)

ε,i ′′ ′ ε ε i ′ i dmε,i t = mt ((a − b − Tε Yt )(ξt ) dt − c (ξt ) dWt ), ε

n 1X mε,i t δξti n→∞ n i=1

Yt = lim

a.s., −1

where ∀ ν ∈ M+ (R)ν ε ∈ M+ (R) is defined by ν ε = ν(R)∧ε ν(R) ν. Now we show that the conditions of [15] are satisfied by the coefficients of the system (2.2). We only check those for dε (x, ν) ≡ −(Tε ν ε )(x). The verification for other √ coefficients is trivial. Note that pε (x) ≤ ( 2πε )−1 and 1 2 |x| 1 |∂x pε (x)| ≤ √ sup e−x /2ε √ = √ . ε 2πε x 2πeε Then

Let

Z √ ε |dε (x, ν)| = pε (x − y)ν (dy) ≤ ( 2πεε)−1 .

B1 = {g ∈ C(R) : |g(x)| ≤ 1, |g(x) − g(y)| ≤ |x − y| ∀ x, y ∈ R} and ρ(ν1 , ν2 ) = sup |hν1 − ν2 , gi|. g∈B1

7


For g ∈ B1 , we have |hν1ε − ν2ε , gi| ≤

ν1 (R) ∧ ε−1 |hν1 − ν2 , gi| ν1 (R) ν1 (R) ∧ ε−1 ν2 (R) ∧ ε−1 + |hν2 , gi| − ν1 (R) ν2 (R)

≤ ρ(ν1 , ν2 ) + |hν1 − ν2 , 1i| + |ν1 (R) ∧ ε−1 − ν2 (R) ∧ ε−1 | ≤ 3ρ(ν1 , ν2 ). Then |dε (x1 , ν1 ) − dε (x2 , ν2 )| Z

(pε (x1 − y) − pε (x2 − y))ν1ε (dy)

≤

Z

pε (x2 − y)ν1ε (dy) −

+

Z

pε (x2 − y)ν2ε (dy)

√ √ ≤ ( 2πeε2 )−1 |x1 − x2 | + ( 2πε )−1 (eε ∧ 1)−1/2 ρ(ν1ε , ν2ε ) p

≤ K1 |x1 − x2 |2 + ρ(ν1 , ν2 )2 .

By [15], εYt is the unique solution to ε

h Yt , φi = hf, φi +

Z

0

t

ε

′′

′

h Yr , (aφ) − (bφ)

− (Tε εYrε )φi dr

Further, εYt has density εyt which belongs to H0 .

−

Z

0

t

hεYr , (cφ)′ i dWr .

2.2. Boundedness. In this section, we establish a comparison result for SPDEs of the form (2.1). As a consequence, we obtain the boundedness of εy . t Lemma 2.2.

For all r, x, we have ε

yr (x) ≤ kf k∞

a.s.,

where kf k∞ is the supremum of f . Proof. Let m ˜ it be given by dm ˜ it = m ˜ it ((a′′ − b′ )(ξti ) dt − c′ (ξti ) dWt ) and let n 1X Y˜t = lim m ˜ it δξti n→∞ n i=1

a.s.

8

J. XIONG

˜ it and hence, for φ ≥ 0, Then mε,i t ≤m

hεYt , φi ≤ hY˜t , φi.

(2.3)

Similarly to Lemma 2.1, it is easy to show that (2.4)

hY˜t , φi = hf, φi +

Let φt be given by (2.5)

t

Z

0

hf, φt i = hf, φi +

hY˜r , (aφ)′′ − (bφ)′ i dr −

Z

0

t

haf ′′ + bf ′ , φr i dr +

Z

t 0

t

Z

0

hY˜r , (cφ)′ i dWr .

˜ r, hcf ′ , φr i dW

˜ is an independent copy of W . The existence of a solution to (2.5) where W follows from [15]. By Itô’s formula, we see that −αhY˜t ,φi

e

t

−

Z

t

−

Z

−αhY˜s ,φi

e

0

2

α αhaY˜s′′ + bY˜s′ , φi + hcY˜s′ , φi2 ds 2

and −αhf,φt i

e

−αhf,φs i

e 0

α2 αhaf + bf , φs i + hcf ′ , φs i2 ds 2

′′

′

are martingales. By a duality argument (cf. [6], page 188), we have ˜

Ee−αhYt ,φi = Ee−αhf,φt i . This implies that hY˜t , φi and hf, φt i have the same distribution. Taking f ≡ 1 in (2.5), it is clear that

Then

Z

φt (x) dx =

Z

φ(x) dx

a.s.

hf, φt i ≤ kf k∞

Z

φ(x) dx

a.s.

hY˜t , φi ≤ kf k∞

Z

φ(x) dx

a.s.

and hence

This implies the conclusion of the lemma. From the proof of Lemma 2.2 , we have the following: Corollary 2.3. sup |εˆy t − 1| → 0

0≤t≤T

as ε → 0.

a.s.


9

Proof. From (2.4) and Condition (BC), it is easy to see that sup hY˜t , 1i < ∞

a.s.

0≤t≤T

The conclusion then follows from (2.3). 2.3. Estimates on Sobolev norm. Now we give an estimate for the Sobolev norm of εyt . Lemma 2.4. E sup kεyt k41 ≤ K2.6 .

(2.6)

0≤t≤T

Proof. We freeze the nonlinear term and consider εyt (x) as the unique solution to the following linear equation: ztε (x)

t

Z

= f (x) +

0

(2.7) +

Z

t

0

(b(x)∂x zrε (x) + a(x)∂x2 zrε (x) − (Tε εyrε (x))zrε (x)) dr

c(x)∂x zrε (x) dWr .

By [17], the solution has derivatives and their estimates depend on the bounds of b, a, Tε εyrε , c and their derivatives. Since the bound of the derivative of Tε εyrε may depend on ε, we cannot apply Rozovskii’s estimate directly. Instead, we derive our estimate here. Note that hztε , φi =

hf, φi + +

t

Z

0

Z

t

0

hb∂x zrε + a∂x2 zrε − (Tε εyrε )zrε , φi dr

hc∂x zrε , φi dWr .

By Itô’s formula, we have hztε , φi2

2

= hf, φi + +

Z

t

0

Z

t

0

2hzrε , φihb∂x zrε + a∂x2 zrε − (Tε εyrε )zrε , φi dr

2hzrε , φihc∂x zrε , φi dWr +

Z

0

t

hc∂x zrε , φi2 dr.

Adding over φ in a complete orthonormal system (CONS) of H0 , we have kztε k20

=

kf k20 +

Z

0

+

Z

0

t

t

2hzrε , b∂x zrε + a∂x2 zrε − (Tε εyrε )zrε i dr

2hzrε , c∂x zrε i dWr +

Z

0

t

kc∂x zrε k20 dr.

10

J. XIONG

Applying Itô’s formula, we have kztε k40 (2.8)

= kf k40 +

Z

+

Z

+

Z

0

t 0 t 0

t

4kzrε k20 hzrε , b∂x zrε + a∂x2 zrε − (Tε εyrε )zrε i dr

4kzrε k20 hzrε , c∂x zrε i dWr

+

Z

t

2kzrε k20 kc∂x zrε k20 dr

0

4hzrε , c∂x zrε i2 dr.

Note that the only coefficient in (2.8) which depends on ε is −(Tε εyrε ). Since this term is negative, it can be discarded. The other terms in (2.8) can be estimated as follows: By (3.4) in [15], we have |hzrε , b∂x zrε i| ≤ K1 kzrε k20

(2.9)

and |hzrε , c∂x zrε i| ≤ K2 kzrε k20 .

By (3.8) in [15] (with δ = 0 there), we have 2hzrε , a∂x2 zrε i + kc∂x zrε k20 ≤ K3 kzrε k20 . Therefore, Z

kztε k40 ≤ kf k40 + K4

t

0

kzrε k40 dr +

Z

t

0

4kzrε k20 hzrε , c∂x zrε i dWr .

By the Burkholder–Davis–Gundy inequality and (2.9), we then have E sup kzsε k40 ≤ kf k40 + K4

Z

t

≤ kf k40 + K4

Z

t

≤ kf k40 + K7

Z

t

s≤t

0

0

0

kzrε k40 dr + K5 E

t

Z

0

kzrε k40 hzrε , c∂x zrε i2 dr

kzrε k40 dr + K6 E sup kzsε k20 s≤t

Z

0

t

kzrε k40 dr

kzrε k40 dr + 12 E sup kzsε k40 . s≤t

Therefore E sup kzsε k40 ≤ 2kf k40 + K2.10

(2.10)

s≤t

Z

0

t

Ekzrε k40 dr,

where K2.10 is a constant. Gronwall’s inequality implies that E sup kztε k40 ≤ K2.11 .

(2.11)

0≤t≤T

Let uεr = ∂x zrε . Note that ε

yr (x)∂x (Tε (ˆ yrε yrε )(x)) = εyr (x)ˆ yrε Tε uεr = εyrε (x)Tε uεr .

1/2

1/2

11


Then uεt (x) = f ′ (x) + Z

+

t

0

Z

0

t

(c(x)∂x uεr (x) + c′ (x)uεr (x)) dWr

(b1 (x)∂x uεr (x) + a(x)∂x2 uεr (x) + cε1 (x)uεr (x) − εyrε (x)Tε uεr (x)) dr,

where b1 = b + a′ , cε1 = b′ − Tε εyrε . So kuεt k20 = kf ′ k20 + + +

Z

Z

t

0 t

0

t

Z

0

kc∂x uεr + c′ uεr k20 dr

2huεr , b1 ∂x uεr + a∂x2 uεr + cε1 uεr − εyrε Tε uεr i dr

2huεr , c∂x uεr + c′ uεr i dWr .

Note that cε1 is bounded by a constant which does not depend on ε. Similar to arguments leading to (2.11), we have E sup kuεt k40 ≤ K2.12 .

(2.12)

0≤t≤T

The conclusion then follows from (2.11) and (2.12). 2.4. Existence and uniqueness. In this section, we prove the first part of Theorem 1.2. Let zt (x) ≡ ztε,η (x) ≡ εyt (x) − ηyt (x).

Then zt (x) =

Z

t 0

+

(b(x)∂x zr (x) + a(x)∂x2 zr (x) − (Tε εyrε (x)εyr (x) − Tη ηyrη (x)ηyr (x))) dr

Z

t

0

Note that

c(x)∂x zr (x) dWr .

Tε εyrε εyr − Tη ηyrη ηyr = εˆy r (Tε εyr )zr + εˆy r (Tε zr )ηyr

+ (εˆy r − ηˆy r )(Tε ηyr )ηyr + ηˆy r (Tε ηyr − Tη ηyr )ηyr .

Similarly to (2.10), we have E sup

0≤s≤t

(2.13)

kzs k40

≤ K2.13

Z

t 0

Ekzr k40 dr

+ 3kf k4∞ E + K2.13 E

Z t Z 0

Z

0

t

|Tε ηyr (x) − Tη ηyr (x)|2 dx

|εˆyr − ηˆyr |4 dr.

2

dr

12

J. XIONG

As Tε ηyr (x) − Tη ηyr (x) =

ZZ

1

0

√ √ √ √ ∂x ηyr (x + (θ ε + (1 − θ) η )a)( ε − η )a dθp(a) da,

we have, when ε, η → 0, (2.14)

Z

√ √ |Tε ηyr (x) − Tη ηyr (x)|2 dx ≤ k∂x ηyr k20 ( ε − η )2 → 0,

where p(a) is the standard normal density. By Corollary 2.3 and the dominated convergence theorem, we have (2.15)

E

t

Z

0

|εˆy r − ηˆy r |4 dr → 0.

It follows from Gronwall’s inequality, (2.13)–(2.15) that E sup kεyt − ηyt k40 → 0

as ε, η → 0.

0≤t≤T

Hence, there exists yt s.t. εyt → yt in H0 . Note that hεyt , φi = hf, φi + +

Z

0

t

Z

0

t

hεyr , −(bφ)′ + (aφ)′′ − (Tε εyrε )φi dr

hεyr , − (cφ)′ i dWr .

We consider the limit of the nonlinear term only, since the other terms clearly converge to the counterpart with εy replaced by y: Z t Z

E

ε

yr (x)(Tε εyrε )(x)φ(x) dx dr −

0

≤E

Z tZ 0

+E +E → 0.

0

yr (x)2 φ(x) dx dr

|Tε (εyrε − yr )|(x)εyr (x)|φ(x)| dx dr

Z tZ 0

Z tZ 0

Z tZ

|Tε yr − yr |(x)εyr (x)|φ(x)| dx dr |εyr − yr |(x)yr (x)|φ(x)| dx dr

It is then easy to show that yt solves (1.2). To prove the uniqueness, we assume that yt and y˜t are two solution to (1.3). Similar to (2.13), we have (2.16)

E sup kyt − y˜t k40 s≤t

≤ K2.16

Z

0

t

Ekyr − y˜r k40 dr.

13


The uniqueness then follows from Gronwall’s inequality. Lemma 2.5. E sup k∂x yt k40 ≤ K2.12 . 0≤t≤T

Proof. Note that E sup

0≤t≤T

k∂x yt k40

=E

sup 0≤t≤T

=E

sup 0≤t≤T

=E

sup 0≤t≤T

≤ lim inf E ε→0

2

X

h∂x yt , φi i

X

hyt , φ′i i2

i

i

X i

!2

!2

ε

lim h

ε→0

sup 0≤t≤T

yt , φ′i i2

X i

ε

h

!2

yt , φ′i i2

!2

= lim inf E sup k∂x εyt k40 ε→0

0≤t≤T

≤ K2.12 , where {φi } is a CONS of H0 . 2.5. Particle representation. In this section, we verify Theorem 1.2(ii). Let yt be the solution to (1.3) and let Yt (dx) = yt (x) dx. Let (ξti , mit ) be given by (1.4) and (1.5). Denote the process given by the right-hand side of (1.6) by Y˜t . Now we only need to verify that Y˜t coincides with Yt . Applying Itô’s formula to mit φ(ξti ), it is easy to show that

(2.17)

hY˜t , φi = hf, φi + +

Z

0

t

Z

0

t

hY˜r , (aφ)′′ − (bφ)′ − yr φi dr

hY˜r , − (cφ)′ i dWr .

By (1.3), we see that (2.17) holds with Y˜t replaced by Yt . Similar to last section, we have uniqueness for the solution of (2.17). This proves Yt = Y˜t and hence, Yt has the particle representation given in Theorem 1.2. 3. Wong–Zakai approximation. In this section, we prove Theorem 1.3.

14

J. XIONG

3.1. Some estimates on ytε . For the convenience of the reader, we state a definition and a theorem which are simplified versions of a definition on page 141 and Theorem 4.6 on page 142 in [8]. Let Lu = a ˜∂x2 u + ˜b∂x u + c˜u. Definition 3.1. A fundamental solution of the parabolic operator L − ∂/∂t in R × [0, T ] is a function Γ(x, t; ξ, τ ) defined for all (x, t) and (ξ, τ ) in R × [0, T ], t > τ , satisfying the following condition: For any continuous function φ(x) with compact support, the function u(x, t) =

Z

Γ(x, t; ξ, τ )φ(ξ) dξ

R

satisfies

∂u =0 ∂t u(x, t) → φ(x)

Lu −

if x ∈ R, τ < t ≤ T, if t → τ + .

To state the next theorem, we need the following conditions: (A1) There is a positive constant K such that a ˜(x, t) ≥ K

for all x ∈ R and t ∈ [0, T ].

(A2) The coefficients of L are bounded continuous functions in R × [0, T ]. (A3) The coefficients of L are Hölder continuous in x, uniformly with respect to (x, t) in compact subsets of R × [0, T ]. Theorem 3.2. Let (A1)–(A3) hold. Let g(x, t) be a bounded continuous function in R × [0, T ], H¨ older continuous in x uniformly with respect to (x, t) in compact subsets, and let φ(x) be a bounded continuous function in R. Then there exists a solution of the Cauchy problem (3.1)

M u ≡ Lu(x, t) −

∂u(x, t) = g(x, t) ∂t

in R × [0, T ]

with the initial condition (3.2)

u(x, 0) = φ(x)

on R.

The solution is given by u(x, t) =

Z

Rn

Γ(x, t; ξ, 0)φ(ξ) dξ −

Z tZ 0

Rn

Γ(x, t; ξ, τ )g(ξ, τ ) dξ dτ.

Now we come back to our equation (1.7). We shall take ˙ ε )∂x . L=a ¯∂x2 + (¯b + cW

15


Lemma 3.3. Ekytε k40 ≤ K3.3 .

(3.3)

Proof. Given W , let q W (y, t; x, s) be the fundamental solution of the parabolic operator L − ∂t . Then, by Theorem 3.2 and (1.7), ytε (x) = ≤ so

Z

Z

kytε k40

W

q (x, t; y, 0)f (y) dy −

Z tZ

q W (x, t; u, s)ysε (u)2 du ds

0

q W (x, t; y, 0)f (y) dy,

≤

Z Z

=

Z Z Z

W

2

q (x, t; y, 0) dx f (y) dy W

q (x1 , t; y1 , 0) dx1

Z

2

W

q (x2 , t; y2 , 0) dx2

× f (y1 )2 f (y2 )2 dy1 dy2 . Note that q W (x, t; y, 0) = q ∗W (y, 0; x, t), q ∗W is the fundamental solution of L∗ + ∂t where ˙ ε )φ)′ + (¯ L∗ φ = −((¯b + cW aφ)′′ ˙ ε )φ − (¯b − 2¯ ˙ ε )φ′ + a = −(¯b′ − a ¯′′ + c′ W a + cW ¯φ′′ . Let ˙ ε ) dt. dξtε = e(ξtε ) dBt − (¯b − 2¯ a + cW t By the Feymann–Kac formula, Z

W exp − q ∗W (y, 0; x, t) dx = Ey,0

Z

t

0

˙ ε )(ξ ε ) dr (¯b′ − a ¯′′ + c′ W r r

W ≤ e2Kt Ey,0 exp −

Z

0

t

˙ rε dr , c′ (ξrε )W

W denotes the conditional distribution of ξ ε given W and ξ ε = y. where Ey,0 t 0 Hence [assume t = (k + 1)ε],

e−4Kt E

Z

q ∗W (y, 0; x, t) dx

≤ E exp −2

k X i=0

′

c

2

ε (ξiε )(W(i+1)ε

!

− Wiε )

16

J. XIONG

× exp −2

k Z X

(i+1)ε Z r

× exp −2

k Z X

(i+1)ε Z r

i=0 iε

iε

i=0 iε

˙ rε dr ((2¯ a − ¯b)c′′ + a ¯c′′′ )(ξsε ) dsW ′′

c

iε

(ξsε )(e(ξsε ) dBs

!

˙ sε ds)W ˙ rε dr − c(ξsε )W

!!

≤ (I1 I2 I3 I4 )1/4 , where I1 = E exp −8

k X

I2 = E exp −8

k Z X

(i+1)ε Z r

I3 = E exp −8

k Z X

(i+1)ε Z r

′

c

ε (ξiε )(W(i+1)ε

i=0

i=0 iε

iε

i=0 iε

!

− Wiε ) ,

((2¯ a − ¯b)c + a ¯c ′′

′′

c

iε

′′′

˙ rε dr )(ξsε ) dsW

˙ rε dr (ξsε )e(ξsε ) dBs W

!

,

!

and I4 = E exp 8

k Z X

(i+1)ε Z r

i=0 iε

iε

′′

c

˙ ε dsW ˙ ε dr (ξsε )c(ξsε )W s r

!

.

ε ) for iε ≤ s < (i + 1)ε. Let P ˜ be the probability meaDefine cε (s) = −8c′ (ξiε sure given by

dP˜ = exp dP

Z

t

0

1 cε (s) dWs − 2

Z

t

0

2

|cε (s)| ds .

Then, by the Girsanov formula, ˜ exp I1 = E

Z 1 2

0

t

|cε (s)|2 ds ≤ exp(32kc′ k2∞ t),

˜ denotes the expectation under the measure P˜ . Note that for ε where E small, more precisely, for ε < min((4k(2¯ a − ¯b)c′′ + a ¯c′′′ k∞ )−1/2 , (8kec′′ k∞ )−1 ), we have I2 ≤ E exp 4k(2¯ a − ¯b)c + a ¯c k∞ ′′

′′′

k X i=0

ε|W(i+1)ε − Wiε |

≤ E exp 2k(2¯ a − ¯b)c + a ¯c k∞ t + ε ′′

′′′

k X i=0

! 2

|W(i+1)ε − Wiε |

!!

17


≤ exp(2k(2¯ a − ¯b)c′′ + a ¯c′′′ k∞ t)(1 − 4k(2¯ a − ¯b)c′′ + a ¯c′′′ k∞ ε2 )−k/2 ≤ exp(10k(2¯ a − ¯b)c′′ + a ¯c′′′ k∞ t), I3 = EEW exp −8

k Z X

(i+1)ε

i=0 iε

c′′ (ξsε )e(ξsε )ε−1 × ((i + 1)ε − s)(W(i+1)ε − Wiε ) dBs

≤ E exp 32 ≤ E exp ≤

k Y

k Z X

(i+1)ε

i=0 iε

32kec′′ k2∞

k X i=0

c′′ (ξsε )2 e(ξsε )2 (W(i+1)ε − Wiε )2 ds 2

!

!

(W(i+1)ε − Wiε ) ε

(1 − 64kec′′ k2∞ ε2 )−1/2

i=0

≤ exp(32kec′′ k2∞ εt) and I4 ≤ E exp 8

k X i=0

2

kcc′′ k∞ (W(i+1)ε − Wiε )

!

≤ exp(32kcc′′ k∞ t). The conclusion then follows easily. We now turn to the estimation on the norm of ∂x ytε . Lemma 3.4. ∃ α > 0, p > 1,

Suppose that {N (x) : x ∈ R} is a random field such that E(|N (x) − N (y)|p ) ≤ K|x − y|1+α .

Then for any λ > 0, E sup(|N (x)|p e−λ|x| ) < ∞. x∈R

Proof. It follows from Theorem 4 in [10] that, for any In = [n, n + 1],

E sup |N (x) − N (y)|p x,y∈In

1/p

≤C ≤

Z

0

1

Ku(1+α)/p du u1+1/p

CKp ≡ K1 . α

18

J. XIONG

Note that |N (x) − N (0)|p e−λ|x| ≤

X

−λ|n|/p

sup |N (y) − N (z)|e

n y,z∈In

≤ (2(1 − e−λ/p ))(1−p)/p Hence

X

!p

sup |N (y) − N (z)|p e−λ|n|/p .

n y,z∈In

E sup(|N (x) − N (0)|p e−λ|x| ) x∈R

≤ (2(1 − e−λ/p ))(1−p)/p ≤ (2(1 − e−λ/p ))

X n

X (1−p)/p

E sup |N (y) − N (z)|p e−λ|n|/p y,z∈In

K1p e−λ|n|/p

n

≤ K1p (2(1 − e−λ/p ))(1−2p)/p

< ∞.

The conclusion of the lemma then follows easily. Lemma 3.5. Ek∂x ytε k40 ≤ K3.4 .

(3.4) Proof. Note that ∂x ytε

′

=f +

Z

t

0

˙ rε )∂x yrε ((¯b′ − 2yrε + c′ W ˙ rε )∂x2 yrε + a + (¯b + a ¯′ + cW ¯∂x3 yrε ) dr.

Let q1W be the fundamental solution of L1 − ∂t , where ˙ ε )φ′ + (¯b′ − 2y ε + c′ W ˙ ε )φ. L1 φ = a ¯φ′′ + (¯b + a ¯′ + cW r

r

r

Then

∂x ytε Note that

=

Z

q1W (x, t; y, 0)f ′ (y) dy.

˙ ε )φ)′ + (¯b′ − 2y ε + c′ W ˙ ε )φ L∗1 φ = (¯ aφ)′′ − ((¯b + a ¯′ + cW r r r ′′ ′ ε ′ ε ¯ ˙ =a ¯φ + (¯ a − b − cW )φ − 2y φ. r

r

Similarly to Lemma 3.3, we have, for any λ and p > 1, (3.5)

E

Z

eλ|x| q1W (x, t; y, 0) dx

p

≤ K1

19


and Z

q1W (x, t; y, 0) dy

=

EW 0,x exp k¯b′ k∞

≤e

Z

t

0

˙ rε )(ηrε,x ) dr (¯b′ − 2yrε + c′ W k Z X

EW 0,x exp

(i+1)ε

′

c

i=0 iε

˙ iεε (ηrε,x ) dr W

!

,

where ηtε,x , with initial x, solves ˙ ε dt + e(η ε,x ) dBt . ¯)(η ε,x ) dt + c(η ε,x )W dη ε,x = (¯b + a t

t

t

Note that for iε ≤ r ≤ (i + 1)ε, ε,x )+ c′ (ηrε,x ) = c′ (ηiε

+

Z r

Z

r

iε

t

t

c′′ (ηsε,x )e(ηsε,x ) dBs

(¯ a + ¯b)c′′ +

iε

e2 ′′′ c ds + 2

Z

r iε

˙ iεε . cc′′ dsW

As Z r k Z (i+1)ε Z r X 2 e ˙ iεε dr W ˙ iεε (¯ a − ¯b)c′′ + c′′′ ds + cc′′ dsW 2 iε iε iε i=0

k X

e2 ′′′ ′′ ¯

a − b)c + c ≤ ε(W(i+1)ε − Wiε )

(¯ 2 ∞

i=0

+

k X i=0

kcc′′ k∞ (W(i+1)ε − Wiε )2

1 ≡ log M (W ) − k¯b′ k∞ , 4 we have Z

q1W (x, t; y, 0) dy

4

≤ M (W )N1 (x, W )N2 (x, W ),

where N1 (x, W ) = EW 0,x exp

−4

k X

′

c

ε,x (ηiε )(W(i+1)ε

i=0

!

− Wiε )

and N2 (x, W ) = EW 0,x exp

−4

k Z X

(i+1)ε Z r

i=0 iε

iε

′′

c

˙ rε dr (ηsε,x )e(ηsε,x ) dBs W

!

.

By arguments similar to Lemma 3.3, it is easy to see that M (W ) has finite moments.

20

J. XIONG

First take EW and then take expectation with respect to W , for t ∈ [iε, (i + 1)ε] and even integer p; we have E|ηtε,x − ηtε,y |p ε,x ≤ E|ηiε

+ pE

ε,y p − ηiε | t

Z

iε

+

Z

t

iε

K1 E|ηsε,x − ηsε,y |p ds

˙ sε ds (c(ηsε,x ) − c(ηsε,y ))(ηsε,x − ηsε,y )p−1 W

ε,x ε,y p ≤ E|ηiε − ηiε | +

Z

Z tZ

+ p(p − 1)E

iε s

iε

iε

t

K2 E|ηsε,x − ηsε,y |p ds

(c(ηrε,x ) − c(ηrε,y ))2 × (ηrε,x − ηrε,y )p−2 dr dsε−2 (W(i+1)ε − Wiε )2

ε,x ε,y p ≤ (1 + K3 ε)E|ηiε − ηiε | .

By induction we have

E|ηtε,x − ηtε,y |p ≤ K2 |x − y|p .

Therefore

E|Ni (x, W ) − Ni (y, W )|p ≤ K3 |x − y|p/2 ,

i = 1, 2.

By Lemma 3.4 we have

E sup |Ni (x, W )|p e−λ|x| ≤ K4 . x

Therefore Z

E sup x

q1W (x, t; y, 0) dye−λ|x|

4 −2λ|x|

≤ E M (W ) sup N1 (x, W )e

sup N2 (x, W )e x

≤ K5 .

Note that Z

x

−2λ|x|

(∂x ytε )(x)2 dx ≤

Z Z

q1W (x, t; y, 0)|f ′ (y)| dy

≤

Z Z


× sup x

Z

Z

q1W (x, t; y, 0) dy dxkf ′ k∞

|f ′ (y)| dy

q1W (x, t; y, 0) dye−λ|x| kf ′ k∞ .

21


Hence (Ek∂x ytε k40 )2

≤

kf ′ k4∞ E

Z Z

× E sup x

≤ kf ′ k4∞ E

Z

q1W (x, t; y, 0) dye−λ|x|

Z Z

× |f ′ (y)|2 dy ≤


kf ′ k4∞ K1 kf ′ k20

|f ′ (y)|2/3 dy

Z

′

2/3

|f (y)|

3

dy

′

|f (y)| dy

4

4


Z

4

K5

3

K5 < ∞.

This proves the conclusion of the lemma. Corollary 3.6.

(i) For any α ≥ 0 and p ≥ 0, we have Z p ε 1+α E |∂x yt (x)| dx ≤ K3.6 .

(3.6)

R

(ii)

Ek∂x2 ytε k40 ≤ K3.7 .

(3.7)

Proof. The proof of Lemma 3.5 can be modified to verify (i). Part (ii) follows from the same proof as well; note that (i) implies Ek(∂x ytε )2 k40 ≤ K3.6 . 3.2. Proof of Theorem 1.3. Now we prove Theorem 1.3. In this proof, the quantity h∂x2 zrε , f i for f smooth is understood as hzrε , ∂x2 f i. To make use of Itô’s formula, we need that ytε is adapted. We shall use ε yt−ε to replace ytε . However, for simplicity of notation, we still use ytε . Let ztε = ytε − yt . Then hztε , φi =

Z

t

0

hb∂x zrε + a∂x2 zrε − (yrε + yr )zrε , φi dr

+

Z

−

Z

t

0

0

ε ˙ r−ε hc∂x yrε , φiW dr

t

hc∂x yr , φi dWr −

By Itô’s formula, we have hztε , φi2 =

Z

0

t

Z

0

t

h 12 cc′ ∂x yrε + 21 c2 ∂x2 yrε , φi dr.

2hzrε , φihb∂x zrε + a∂x2 zrε − (yrε + yr )zrε , φi dr

22

J. XIONG t

Z

+

˙ ε dr − 2hzrε , φihc∂x yrε , φiW r−ε

0 t

−

Z

t

+

Z

kztε k20 =

Z

0

0

t

Z

0

2hzrε , φihc∂x yr , φi dWr

hzrε , φihcc′ ∂x yrε + c2 ∂x2 yrε , φi dr hc∂x yr , φi2 dr.

Adding over φ in a CONS of H0 , we have t

0

2hzrε , b∂x zrε + a∂x2 zrε − (yrε + yr )zrε i dr

Z

t

Z

t

− +

Z

+

ε ˙ r−ε dr 2hzrε , c∂x yrε iW

0

(3.8)

0

−

Z

0

t

2hzrε , c∂x yr i dWr

hzrε , cc′ ∂x yrε + c2 ∂x2 yrε i dr

t

0

kc∂x yr k20 dr.

We now estimate the second term on the right-hand side of (3.8). For (i − 1)ε ≤ r < iε, note that ε hzrε , φi = hz(i−1)ε , φi +

+

Z

−

Z

r

(i−1)ε

Z

r

(i−1)ε

hb∂x zsε + a∂x2 zsε − (ysε + ys )zsε , φi ds

ε ˙ s−ε hc∂x ysε , φiW ds

r

(i−1)ε

hc∂x ys , φi dWs −

Z

r

(i−1)ε

h 12 cc′ ∂x ysε + 21 c2 ∂x2 ysε , φi ds

and ε hc∂x yrε , φi = hc∂x y(i−1)ε , φi +

+

Z

r (i−1)ε

Z

r

(i−1)ε

hc∂x (¯b∂x ysε + a ¯∂x2 ysε − (ysε )2 ), φi ds

ε ˙ s−ε hc∂x (c∂x ysε ), φiW ds.

Similarly to (3.8), we have ε ε hzrε , c∂x yrε i − hz(i−1)ε , c∂x y(i−1)ε i

=

Z

r

(i−1)ε

+ (3.9)

Z

hc∂x ysε , b∂x zsε + a∂x2 zsε − (ysε + ys )zsε i ds

r

(i−1)ε

ε ˙ s−ε kc∂x ysε k20 W ds −

Z

r

(i−1)ε

hc∂x ysε , c∂x ys i dWs


−

Z

+

Z

r

(i−1)ε r

(i−1)ε

Z

+

r

(i−1)ε

23

hc∂x ysε , 12 cc′ ∂x ysε + 21 c2 ∂x2 ysε i ds hzsε , c∂x (¯b∂x ysε + a ¯∂x2 ysε − (ysε )2 )i ds ˙ ε ds. hzsε , c∂x (c∂x ysε )iW s−ε

Let t = kε. Then E

t

Z

0

(3.10)

ε ˙ r−ε 2hzrε , c∂x yrε iW dr

=E

k−1 X Z (i+1)ε

=E

k−1 X

i=0 iε

(i+1)ε

Z

2

iε

i=0

ε ˙ r−ε 2hzrε , c∂x yrε iW dr ε ε ε ˙ r−ε (hzrε , c∂x yrε i − hz(i−1)ε , c∂x y(i−1)ε i)W dr.

Apply (3.9) to (3.10). We only consider the second, third and sixth terms in (3.9) since it√ is easy to verify that the other terms result in quantities bounded by K ε. Note that E

k−1 X

2

i=0

Z

(i−1)ε

iε

≈E (3.11)

(i+1)ε Z r

k−1 X

2

+E

(i+1)ε Z iε

Z

k−1 X

2

k−1 X i=0

≈E

Z

Z

(i+1)ε Z r iε

iε

i=0

=

(i−1)ε

iε

i=0

˙ ε dsW ˙ ε dr kc∂x ysε k20 W s−ε r−ε ε ˙ ε dsW ˙ ε dr kc∂x y(i−2)ε k20 W s−ε r−ε

ε kc∂x y(i−2)ε k20 ds drε−2 (Wiε − W(i−1)ε )2

ε ε2 Ekc∂x y(i−2)ε k20 ε−2 ε t

0

kc∂x yrε k20 dr,

√ where x ≈ y means that |x − y| ≤ K ε. Similarly, E (3.12)

k−1 X i=0

2

Z

(i+1)ε Z r

(i−1)ε

iε

≈E

Z

0

t

˙ ε dsW ˙ ε dr hzsε , c∂x (c∂x ysε )iW s−ε r−ε

hzrε , c∂x (c∂x yrε )i dr.

24

J. XIONG

Note that E

k−1 X i=0

(3.13)

2

Z

(i+1)ε Z r

(i−1)ε

iε

≈E

k−1 X

2

i=0

≈ 2E

Z

0

t

Z

˙ ε dr hc∂x ysε , c∂x ys i dWs W r−ε

(i+1)ε Z r

iε

(i−1)ε

ε ˙ ε dr hc∂x y(i−2)ε , c∂x y(i−2)ε i dWs W r−ε

hc∂x yrε , c∂x yr i dr.

By (3.8) and (3.10)–(3.13), we have Ekztε k20 ≤ K1

Z

t

0

√ Ekzsε k20 ds + K2 ε.

Gronwall’s inequality then implies the conclusion of the theorem. 4. Log-Laplace transform of Xt. In this section we prove Theorem 1.4. Since X ε solves the CMP defined in Section 1.2, it is easy to show that E suphXsε , 1i4 ≤ K1 . s≤t

For any φ ∈ Cc (R), it is then easy to show that

EhXtε − Xsε , φi4 ≤ K2 |t − s|2 .

¯ This implies the tightness of {X ε } in C([0, ∞), MF (R)). It is then easy to verify that any one of the limit points solves the MP. The uniqueness for the solution to the MP implies the weak convergence of X ε . Now we prove (1.8). First we assume that µ ∈ H0 and fix X0ε = µ. Let ψ be a bounded continuous function on C([0, t], R). Then E(exp(−hXt , f i)ψ(W )) = lim E(exp(−hXtε , f i)ψ(W )) ε→0

ε = lim E(exp(−hµ, y0,t i)ψ(W )) ε→0

= E(exp(−hµ, y0,t i)ψ(W )).

For general µ, we take µε ∈ H0 converging to µ in MF (R). Denote the solution of the MP with µ replaced by µε by X (ε) . Then (ε)

E(exp(−hXt , f i)ψ(W )) = lim E(exp(−hXt , f i)ψ(W )) ε→0

= lim E(exp(−hµε , y0,t i)ψ(W )) ε→0

= E(exp(−hµ, y0,t i)ψ(W )), where the last equation follows since y0,t is bounded and continuous.


25

5. Moments of Xt. In this section, we prove Theorem 1.5. Let ytα be the solution of ytα (x) = αf (x) + (5.1) +

Z

t 0

t

Z

0

(b(x)∂x yrα (x) + a(x)∂x2 yrα (x) − yrα (x)2 ) dr

c(x)∂x yrα (x) dWr .

Let zt and ht be solutions to zt (x) = f (x) + (5.2)

0

t

(b(x)∂x zr (x) + a(x)∂x2 zr (x)) dr

t

Z

+

Z

c(x)∂x zr (x) dWr

0

and ht (x) = (5.3)

t

Z

0

(b(x)∂x hr (x) + a(x)∂x2 hr (x) − 2zr (x)2 ) dr t

Z

+

c(x)∂x hr (x) dWr .

0

Define ztα = α−1 ytα − zt . Then ztα (x) =

t

Z

0

+

(b(x)∂x zrα (x) + a(x)∂x2 zrα (x)) dr t

Z

0

c(x)∂x zrα (x) dWr −

Z

t 0

α−1 yrα (x)2 dr.

Similarly to arguments in previous sections, we have Ekztα k20 → 0

as α → 0.

Define hαt = α−2 (yt2α − 2ytα ) − ht . Then hαt (x)

=

Z

t

0

+

(b(x)∂x hαr (x) + a(x)∂x2 hαr (x)) dr

Z

t

0

− Note that E

Z

0

t

c(x)∂x hαr (x) dWr (α−2 (yr2α (x)2 − 2yrα (x)2 ) − 2zr (x)2 ) dr.

|yrα (x)| ≤ αkf k∞ Z

and |zr (x)| ≤ kf k∞ . Hence

(α−2 (yr2α (x)2 − 2yrα (x)2 )−2zr (x)2 )2 dx

26

J. XIONG

=E

2

y 2α (x) − zr (x) 4 r 2α

Z

2

y α (x) −2 r − zr (x) α

y 2α (x) − yrα (x) − αzr (x) + 4zr (x) r α

→ 0.

2

dx

Similarly to above, we have Ekhαt k20 → 0

as α → 0.

Therefore zt = ∂α ytα |α=0 and ht = ∂α2 ytα |α=0 . Note that

E(hXt , f i|W ) = hµ, zt i and E(hXt , f i2 |W ) = hµ, zt i2 − hµ, ht i. Taking expectations on both sides of (5.2), we have Ezt (x) = f (x) + E

Z

0

t

(b(x)∂x zr (x) + a(x)∂x2 zr (x)) dr,

and hence, (1.9) holds. Applying Itô’s formula to (5.2), we have Ezt (x1 )zt (x2 ) = f (x1 )f (x2 ) + E

Z

(b(x1 )∂x1 zr (x1 )zr (x2 ) + b(x2 )∂x2 z(x1 )zr (x2 ) + a(x1 )∂x21 z(x1 )zr (x2 ) + a(x2 )∂x22 z(x1 )zr (x2 ) + c(x1 )c(x2 )∂x1 ∂x2 z(x1 )zr (x2 )) dr.

Hence (5.4)

Ezt (x1 )zt (x2 ) =

ZZ

f (y1 )f (y2 )q(t, (x1 , x2 ), (y1 , y2 )) dy1 dy2 .

Taking the expectation on both sides of (5.3), we have (5.5)

Eht (x) = E

Z

0

t

(b(x)∂x hr (x) + a(x)∂x2 hr (x) − 2zr (x)2 ) dr.

Hence, making use of (5.4) and solving (5.5), we obtain Eht (x) = −2

Z tZ 0

p(t − s, x, y)Ezs (y)2 dy ds


= −2

Z tZ 0

p(t − s, x, y)

ZZ

27

f (z1 )f (z2 ) × q(s, (y, y), (z1 , z2 )) dz1 dz2 dy ds.

This proves (1.10). Acknowledgments. I am grateful to Robert Adler for his suggestion which led to Theorem 1.4. I would like to thank the referee for pointing out two gaps overlooked by me in the original manuscript. I would like to thank both the Associate Editor and the referee for helpful suggestions. I thank Klaus Fleischmann for going over the referee’s report with me and for a careful reading of the manuscript. REFERENCES [1] Da Prato, G. and Tubaro, L., eds. (2002). Stochastic Partial Differential Equations and Applications. Lecture Notes in Pure and Appl. Math. 227. Springer, Berlin. MR1919498 ´ ´ e de Proba[2] Dawson, D. A. (1993). Measure-valued Markov processes. Ecole d ’Et´ bilités de Saint-Flour XXI. Lecture Notes in Math. 1541 1–260. Springer, Berlin. MR1242575 [3] Dawson, D. A. and Fleischmann, K. (1997). A continuous super-Brownian motion in a super-Brownian medium. J. Theoret. Probab. 10 213–276. MR1432624 [4] Dawson, D. A., Li, Z. H. and Wang, H. (2001). Superprocesses with dependent spatial motion and general branching densities. Electron. J. Probab. 6 1–33. MR1873302 [5] Dynkin, E. (1993). Superprocesses and partial differential equations. Ann. Probab. 21 1185–1262. MR1235414 [6] Ethier, S. N. and Kurtz, T. G. (1985). Markov Processes: Characterization and Convergence. Wiley, New York. MR838085 [7] Fleischmann, K. and Xiong, J. (2001). A cyclically catalytic super-Brownian motion. Ann. Probab. 2 820–861. MR1849179 [8] Friedman, A. (1975). Stochastic Differential Equations and Applications 1. Academic Press, New York. MR494490 ¨ ngy, I. (2002). Approximations of stochastic partial differential equations. [9] Gyo Stochastic Partial Differential Equations and Applications. Lecture Notes in Pure and Appl. Math. 227 287–307. Springer, Berlin. MR1919514 [10] Ibragimov, I. A. (1983). On smoothness conditions for trajectories of random functions. Theory Probab. Appl. 28 240–262. MR700208 [11] Kallianpur, G. and Xiong, J. (1995). Stochastic Differential Equations in Infinite Dimensional Spaces. IMS, Hayward, CA. MR1465436 [12] Kotelenez, P. (1992). Existence, uniqueness and smoothness for a class of function valued stochastic partial differential equations. Stochastics Stochastics Rep. 41 177–199. MR1275582 [13] Kotelenez, P. (1992). Comparison methods for a class of function valued stochastic partial differential equations. Probab. Theory Related Fields 93 1–19. MR1172936

28

J. XIONG

[14] Krylov, N. V. (2002). Some new results in the theory of SPDEs in Sobolev spaces. Stochastic Partial Differential Equations and Applications. Lecture Notes in Pure and Appl. Math. 227 325–336. Springer, Berlin. MR1919516 [15] Kurtz, T. G. and Xiong, J. (1999). Particle representations for a class of nonlinear SPDEs. Stochastic Process. Appl. 83 103–126. MR1705602 [16] Li, Z. H., Wang, H. and Xiong, J. (2004). Conditional log-Laplace functionals of immigration superprocesses with dependent spatial motion. WIAS Preprint No. 900. [17] Rozovskii, B. L. (1990). Stochastic Evolution Systems. Linear Theory and Applications to Nonlinear Filtering. Kluwer, Dordrecht. MR1135324 [18] Skoulakis, G. and Adler, R. J. (2001). Superprocesses over a stochastic flow. Ann. Appl. Probab. 11 488–543. MR1843056 [19] Wang, H. (1998). A class of measure-valued branching diffusions in a random medium. Stochastic Anal. Appl. 16 753–786. MR1632574 Department of Mathematics University of Tennessee Knoxville, Tennesee 37996-1300 USA e-mail: [email protected]