A STOCHASTIC EVOLUTION EQUATION ARISING ... - Project Euclid

c 2004 International Press °

COMM. MATH. SCI. Vol. 2, No. 3, pp. 325–358

A STOCHASTIC EVOLUTION EQUATION ARISING FROM THE FLUCTUATIONS OF A CLASS OF INTERACTING PARTICLE SYSTEMS ∗ THOMAS G. KURTZ

† AND

JIE XIONG

‡

Abstract. In an earlier paper, we studied the approximation of solutions V (t) to a class of SPDEs by the empirical measure V n (t) of a system of n√interacting diffusions. In the present paper, we consider a central limit type problem, showing that n(V n − V ) converges weakly, in the dual of a nuclear space, to the unique solution of a stochastic evolution equation. Analogous results in which the diffusions that determine V n are replaced by their Euler approximations are also discussed. Key words. Stochastic partial differential equations, interacting infinite particle system, central limit theorem, Euler scheme MSC 2000 subject classifications: Primary 60H15, 60H35; Secondary 60B12, 60F17, 60F25, 60H10, 93E11.

1. Introduction In [26], we considered a class of nonlinear stochastic partial differential equations (SPDE) of the form Ã d d X 1 X ∂xi [bi (x,v(t,·))v(t,x)] ∂xi ∂xj [aij (x,v(t,·))v(t,x)] − dv(t,x) = 2 i,j=1 i=1 ! +d(x,v(t,·))v(t,x) dt ! Z Ã d X − β(x,v(t,·),u)v(t,x) + ∂xi [αi (x,v(t,·),u)] W (dudt) . U

(1.1)

i=1

The natural interpretation of v is as the density of a mass distribution V evolving in time, and in fact, since v will not have the regularity presumed in (1.1), to rigorously formulate the equation, we must use a weak form hφ,V (t)i − hφ,V (0)i Z t = hφd(·,V (s)) + L(V (s))φ,V (s)ids 0 Z ® + φβ(·,V (s),u) + ∇φT α(·,V (s),u),V (s) W (duds) . (1.2) U ×[0,t]

where L(v)φ(x) = ∗ Received:

X 1X aij (x,v)∂xi ∂xj φ(x) + bi (x,v)∂xi φ(x). 2 i,j i

February 4, 2003; accepted (in revised version): July 06, 2004. Communicated by Shi

Jin. † Departments of Mathematics and Statistics, University of Wisconsin-Madison, 480 Lincoln Drive, Madison, WI 53706-1388, USA ([email protected]). Research supported in part by NSF grant DMS 02-05034. ‡ Department of Mathematics, University of Tennessee, Knoxville, TN 37996-1300, USA, and Department of Mathematics, Hebei Normal University, Shijiazhuang 050016, PRC ([email protected]). Research supported partially by NSA and by Alexander von Humboldt Foundation.

325

326

STOCHASTIC EVOLUTION EQUATION FOR FLUCTUATIONS

Equations in this class arise in a variety of settings, including nonlinear filtering with both the Zakai and the Kushner-FKK equations being of this form. Other examples include McKean-Vlasov equations [30] and classes of SPDEs considered by Kotelenez [23] and Dawson and Vaillancourt [9]. In [26], we established a representation of the solution of (1.1) in terms of weighted empirical measures of the form n

1X Ai (t)δXi (t) , n→∞ n i=1

V (t) = lim

(1.3)

where δx is the Dirac measure at x and the limit exists in the weak* topology on M(Rd ). To be precise, let U be a Polish space and µ be a σ-finite measure on U . W will be a space-time Gaussian white noise on U × [0,∞) with covariance measure µ(du)dt, namely E(W (A × [0,t])W (B × [0,s])) = µ(A ∩ B)(t ∧ s). For 1 ≤ i, j ≤ d, aij , bi , d and αi , β will be real functions on Rd × M(Rd ) and on Rd × M(Rd ) × U respectively. Here M(Rd ) is the collection of all finite measures on Rd . Dot notation will represent a function in that variable alone. For example, d(·,v) is the real-valued function on Rd with v ∈ M(Rd ) fixed. b and α will denote the vectors with components bi and αi and a will denote the matrix ((aij )). The particle system {Xi ,Ai ,V } giving the solution is governed by the following equations: Z t Z t Xi (t) = Xi (0) + σ(Xi (s),V (s))dBi (s) + c(Xi (s),V (s))ds 0 0 Z + α(Xi (s),V (s),u)W (duds) (1.4) U ×[0,t]

and

Z t Z t Ai (t) = Ai (0) + Ai (s)γ T (Xi (s),V (s))dBi (s) + Ai (s)d(Xi (s),V (s))ds 0 0 Z + Ai (s)β(Xi (s),V (s),u)W (duds) , (1.5) U ×[0,t]

where the Bi are independent, standard Rd -valued Brownian motions, independent of W , and {(Xi (0),Ai (0))} is an exchangeable sequence of random variables in Rd × R that is independent of {Bi } and W . Here σ, c and γ are related to a, b, α, and β by Z T a(x,v) = σ(x,v)σ (x,v) + α(x,v,u)αT (x,v,u)µ(du) U

and

Z b(x,v) = c(x,v) + σ(x,v)γ(x,v) +

β(x,v,u)α(x,v,u)µ(du). U

The representation of the solution given by (1.3) suggests that the solution can be approximated by the weighted empirical measure n

V n (t) =

1X n A (t)δXin (t) , n i=1 i

(1.6)

THOMAS G. KURTZ AND JIE XIONG

327

of a finite particle system satisfying Z t Z t Xin (t) = Xi (0) + σ(Xin (s),V n (s))dBi (s) + c(Xin (s),V n (s))ds 0 0 Z n n + α(Xi (s),V (s),u)W (duds)

(1.7)

U ×[0,t]

Z t Z t Ani (t) = Ai (0) + Ani (s)γ T (Xin (s),V n (s))dBi (s) + Ani (s)d(Xin (s),V n (s))ds 0 0 Z + Ani (s)β(Xin (s),V n (s),u)W (duds), (1.8) U ×[0,t]

for i = 1,2,··· ,n. it was shown that for an appropriate metric ρe on M(Rd ), √ In [27], n { ne ρ(V (t),V (t))}n≥1 is stochastically bounded, that is, for each ² > 0, there is a constant K² such that ¡√ ¢ sup P ne ρ(V n (t),V (t)) > K² < ². (1.9) n

From (1.9), we see that the convergence rate has an upper bound of the order of √1n . A natural question to ask is whether √1n is the right order. To this end, we √ study the convergence of the process Sn (t) = n(V n (t) − V (t)) and show that for an appropriate space Φ−κ of distributions, Sn converges in distribution in CΦ−κ [0,∞). We characterize the limit S as the unique solution of a stochastic evolution equation of the form Z t hφ,S(t)i = hφ,S(0)i + hφ,M (t)i + hF1 (V (s))φ,S(s)ids 0 Z + hF2 (V (s),u)φ,S(s)iW (duds), (1.10) U ×[0,t]

where F1 and F2 are linear in φ and M is a distribution-valued martingale. F1 and F2 are defined in Assumption (S6) in Section 4 in terms of appropriate differentials of the coefficients of (1.2), reflecting the fact that we are rescaling the deviation of V n from V . This type of problem has been studied by various authors in the McKean-Vlasov setting, that is, α = 0 and Ani (t) ≡ 1 (cf. Hitsuda and Mitoma [19] and the references therein). Comparing the present results with those of [19], here the process V is not deterministic and the process S is not Gaussian. In [19], the limit S is characterized by its covariance structure which, because S is Gaussian, uniquely determines its distribution. A stochastic evolution equation is also derived in that paper. The uniqueness of the solution to that stochastic evolution equation is proved by Mitoma [33]. Another new feature in this paper is that the driving martingale M in the evolution equation (1.10) is not Gaussian and has to be defined by the particle system {Xi ,Ai ,V } itself. The main difficulty in establishing the uniqueness of the solution of (1.10) comes from the addition of the last term in (1.10) which does not appear in [19] and [33].

328


Limits of empirical measure processes for systems of interacting diffusions have been studied by various authors (see, for example, Chiang, Kallianpur and Sundar [4], Graham [18], Kallianpur and Xiong [22], Méléard [31], and Morien [34]) since the pioneering work by McKean [30]. Typically, the driving processes in the models are assumed to be independent, and the limit is then a deterministic, measure-valued function. Florchinger and Le Gland [14] consider particle approximations for stochastic partial differential equations in a setting that, in the notation above, corresponds to taking γ = σ = 0 and the other coefficients independent of V . Florchinger and Le Gland were motivated by approximations to the Zakai equation of nonlinear filtering. Del Moral [10] specifically studies this example. Kotelenez [23] introduces a model of n-particles with the same driving process for each particle and studies the empirical process as the solution of an SPDE. His model corresponds to taking γ = σ = d = β = 0, but the other coefficients are allowed to depend on V . In particular, the weights Ai are constants. Dawson and Vaillancourt [9] consider a model given as a solution of a martingale problem that corresponds to taking Ai (t) ≡ 1 in the current model. Bernard, Talay, and Tubaro [1] consider a system with time-varying weights and a deterministic limit. The paper is organized as follows: In the next section, we derive key estimates on the magnitude of the Ani and on the error in the approximation of (Xi ,Ai ) by (Xin ,Ani ). In Section 3, we prove that {Sn } is a tight sequence of Φ0 -valued processes (Φ0 being a conuclear space defined later). Then, in Section 4, we show that the limit S of {Sn } is the unique solution of (1.10). If one wants to use the finite system to simulate the solution of the SPDE, then the finite system must also be approximated. The simplest approach is to use an Euler approximation. In the last section of this paper, we analyze this approximation in the simplest setting, assuming that W is a one-dimensional Brownian motion (that is, U consists of a single point). Letting V n,1/n denote the weighted empirical measure for the Euler scheme approximating the finite system (cf. (5.1-5.3)), we consider the √ process Sen (t) = n(V n,1/n (t) − V n (t)). We prove tightness for {Sen } and characterize its limit as the unique solution of another stochastic evolution equation. Finally, we combine the two parts and derive a stochastic evolution equation for the limit of √ n(V n,1/n − V ). 2. Preliminaries In this section, we state the main results of [26] and [27] needed in the present paper for the convenience of the reader. The following assumptions were made in [26] for the existence and uniqueness of solutions of the SPDE (1.1). (S1) There exists a constant K such that for each x ∈ Rd , ν ∈ M(Rd ) Z |σ(x,ν)|2 + |c(x,ν)|2 +

|α(x,ν,u)|2 µ(du) U

Z

2

2

β(x,ν,u)2 µ(du) ≤ K 2 .

+|γ(x,ν)| + |d(x,ν)| + U

(S2) For each x1 ,x2 ∈ Rd , ν1 ,ν2 ∈ M(Rd ) |σ(x1 ,ν1 ) − σ(x2 ,ν2 )|2 + |c(x1 ,ν1 ) − c(x2 ,ν2 )|2

329


Z +|γ(x1 ,ν1 ) − γ(x1 ,ν1 )|2 +

|α(x1 ,ν1 ,u) − α(x2 ,ν2 ,u)|2 µ(du) Z

U

2

|β(x1 ,ν1 ,u) − β(x2 ,ν2 ,u)|2 µ(du)

+|d(x1 ,ν1 ) − d(x2 ,ν2 )| + U

≤ K 2 (|x1 − x2 |2 + ρ(ν1 ,ν2 )2 ), where ρ(ν1 ,ν2 ) = sup{|hφ,ν1 i − hφ,ν2 i| : φ ∈ B1 } and © ª B1 = φ : |φ(x) − φ(y)| ≤ |x − y|,|φ(x)| ≤ 1,∀x,y ∈ Rd . By the same proof as in Proposition 2.1 of [26], we have the following result. Proposition 2.1. Suppose that Assumption (S1) holds and p is a positive number. i) If Eep|X1 (0)| < ∞,

(2.1)

then n

sup E sup ep|Xi (s)| < ∞. 1≤n≤∞

(2.2)

0≤s≤T

ii) If E|A1 (0)|p < ∞,

(2.3)

then sup E sup |Ani (s)|p < ∞.

1≤n≤∞

(2.4)

0≤s≤T

iii) If E|A1 (0)|er|X1 (0)| < ∞,

(2.5)

then n

sup E sup |Ani (s)|er|Xi (s)| < ∞.

1≤n≤∞

(2.6)

0≤s≤T

Remark 2.2. If (2.1) and (2.3) hold, then (2.5) holds with r = p − 1. A weaker form of the following assumption was used in [27]. (S3) There exist constants λ > 1 and K > 0 such that for any iid sequence (ξi ,ηi ), i = 1,2,··· and x ∈ Rd , ¯ Ã ¯2λ ! n ¯ ¯ 1X KEξ12λ ¯ ¯ E ¯σ x, , ξi δηi − σ(x,µ)¯ ≤ ¯ ¯ n nλ i=1

where µ(·) = E[ξ1 1η1 ∈· ], and a similar inequality holds for the other coefficients.

330


R Remark 2.3. If σ(x,µ) = σ1 (x,y)µ(dy) or σ(x,µ) = σ1 (x), then (S3) usually holds. For example, if |σ1 (x,y)| ≤ K, then (S3) holds. The following estimate is the key for the proof of the tightness of {Sn }. Theorem 2.4. Under the assumptions (S1)-(S3), there exists a constant c1 (T,m) such that   2  n X  n n 2λ  1 n n λ  Esup |Xin (t ∧ ηm ) − Xi (t ∧ ηm )| + |Anj (t ∧ ηm ) − Aj (t ∧ ηm )|  n j=1 t≤T ≤ where

c1 (T,m) , nλ (

) n k X X 1 1 n ηm = inf t : An (t)2 > m2 or lim Ai (t)2 > m2 . k→∞ k n i=1 i i=1

Proof: By Doob’s inequality and Holder’s inequality, we have 2λ

n n Esup |Xin (r ∧ ηm ) − Xi (r ∧ ηm )| r≤t

µ 2λ

≤3

2λ 2λ − 1

¶2λ Z t λ−1 n dst E |σ(Xin (s),V n (s)) − σ(Xi (s),V (s))|2λ 1s≤ηm Z

+32λ t2λ−1 E 0

0

t

n ds |c(Xin (s),V n (s)) − c(Xi (s),V (s))|2λ 1s≤ηm

¶2λ 2λ tλ−1 +3 2λ − 1 ¶λ Z t µZ n ds. ×E |α(Xin (s),V n (s),u) − α(Xi (s),V (s),u)|2 µ(du) 1s≤ηm µ

2λ

0

U

(2.7) Let n

1X Ve n (t) = Ai (t)δXi (t) n i=1

and

Vein (t) =

1 n−1

n X

Aj (t)δXj (t) .

j=1, j6=i

Then n E|σ(Xin (s),V n (s)) − σ(Xi (s),V (s))|2λ 1s≤ηm 2λ n n n e ≤ 3 E|σ(X (s),V (s)) − σ(Xi (s), V (s))|2λ 1s≤ηn

i

m

+32λ E|σ(Xi (s), Ve n (s)) − σ(Xi (s), Vein (s))|2λ n +32λ E|σ(Xi (s), Vein (s)) − σ(Xi (s),V (s))|2λ 1s≤ηm ´λ ³ n ≤ 32λ K 2λ E |Xin (s) − Xi (s)|2 + ρ(V n (s), Ve n (s))2 1s≤ηm n +32λ K 2λ Eρ(Ve n (s), Vein (s))2λ 1s≤ηm n X ¯ £ £ ¤¤ 1 +32λ E E |σ(Xi (s), Aj (s)δXj (s) ) − σ(Xi (s),V (s))2λ |¯W,Xi . n−1

j=1, j6=i

(2.8)

331


Note that, similar to the arguments in the proof of Theorem 2.1 in [26], we have ρ(V n (s), Ve n (s)) n n 1X n 1X n Aj (s)|Xjn (s) − Xj (s)| + |A (s) − Aj (s)| ≤ n j=1 n j=1 j ≤(

n n n 1 X n 2 1/2 1 X n 1X n Aj (s) ) ( |Xj (s) − Xj (s)|2 )1/2 + |A (s) − Aj (s)|. (2.9) n j=1 n j=1 n j=1 j

Similarly, n X 1 1 ρ(Ve n (s), Vein (s)) ≤ Ai (s) + Aj (s). n n(n − 1) j=1

Let 2λ

n n n fm (t) = Esup |Xin (r ∧ ηm ) − Xi (r ∧ ηm )|

,

r≤t

and 

2 n X 1 n n n λ gm (t) = Esup  |An (r ∧ ηm ) − Aj (r ∧ ηm )| . n j=1 j r≤t Then, for the right hand side of (2.8), ³ n n 2λ 1st term ≤ 32λ K 2λ 2λ E|Xin (s ∧ ηm ) − Xi (s ∧ ηm )| n ´ 1X n n n 2λ n |Xj (s ∧ ηm ) − Xj (s ∧ ηm )| + 22λ gm (s) n j=1 ¡ ¢ n n n ≤ 18λ K 2λ fm (s) + 4λ m2λ fm (s) + 4λ gm (s)

+22λ m2λ E

and

µ 2λ

2λ 2λ

2nd term ≤ 3 K 2

¶ 1 1 2λ 2λ Esup |Ai (r)| + m . n2λ r≤t (n − 1)2λ

Since, conditioning on (W,Xi ), (Aj ,Xj ), j 6= i, are iid, we have µ ¶ ¡ ¢ K 2λ 32λ K 2λ 2λ 2λ E A (s) |W,X = EA1 (s)2λ . 3rd term ≤ 3 E 1 i (n − 1)λ (n − 1)λ Hence, the first term on the right hand side of (2.7) is dominated by µ

¶2λ Z t ¡ ¢ 6λ n n 18λ K 2λ (1 + 2λ m2λ )fm (s) + 2λ gm (s) ds T λ−1 2λ − 1 0 ¶ µ ¶2λ · µ 1 1 6λ 2λ 2λ 2λ 2λ 2λ Esup |Ai (r)| + m + 3 K 2 2λ − 1 n2λ r≤t (n − 1)2λ ¸ 32λ K 2λ + EA1 (s)2λ T λ . (n − 1)λ

332


Similar estimates hold for other terms on the right hand side of (2.7). Therefore, there exist constants c2 (T,m) and c3 (T,m) such that Z t c3 (T,m) n n n fm (t) ≤ c2 (T,m) (fm (s) + gm (s))ds + . (2.10) nλ 0 By similar arguments as in (2.7) of [26] and (2.10) above, we have Z t c5 (T,m) n n n gm (t) ≤ c4 (T,m) (fm (s) + gm (s))ds + . nλ 0 Therefore Z n n fm (t) + gm (t) ≤ (c2 + c4 )

0

t

n n (fm (s) + gm (s))ds +

c3 + c5 . nλ

By Gronwall’s inequality, we have n n fm (t) + gm (t) ≤

by taking c1 = e

c1 nλ

(c2 +c4 )T

(c3 +c5 ) . c2 +c4

3. Tightness In this section, we prove tightness for {Sn } in an appropriate space. For simplicity of notation, we restrict our calculations to space dimension d = 1 in the rest of this paper. As in Hitsuda and Mitoma [19], we use the modified Schwartz ¡ ¢ R space Φ. Let ρ(x) = C exp −1/(1 − |x|2 ) 1|x| κ such that the embedding 0 0 Tκκ : Φκ0 → Φκ is a Hilbert-Schmidt operator. The adjoint Tκκ ∗ : Φ−κ → Φ−κ0 is also Hilbert-Schmidt. Φ0 = ∪∞ k=0 Φ−k gives a representation of the dual of Φ. (See [17], page 59.) We prove tightness for {Sn } in CΦ−κ [0,∞) for an appropriate κ. Theorem 3.1. Suppose that (S1)-(S3) hold and that (2.1) and (2.3) hold for p = max(4λ,λ/(λ − 1)). Then there exists κ such that {Sn } is tight in CΦ−κ [0,∞). Proof: Let ( ) n n ³ ´ X X n 1 1 n,p ηm = inf t : An (t)p ≥ mp or 1 + ep|Xi (t)| ∨ ep|Xi (t)| ≥ mp . n i=1 i n i=1 Then for T > 0, we have n,p sup P(ηm ≤T)≤ n

µ ¶ 2 n p p|Xin (s)| sup 1 + E sup Ai (s) + E sup e , mp 1≤n≤∞ 0≤s≤T 0≤s≤T

and since by Proposition 2.1, for each T > 0, the right side goes to zero as m → ∞, it n,p )}. is enough to prove tightness for {Sn (· ∧ ηm By Itô’s formula, we have hφ,V n (t)i − hφ,V n (0)i µ n Z 1X t n = A (s) φ(Xin (s))γ(Xin (s),V n (s)) n i=1 0 i ¶ 0 n n n +φ (Xi (s))σ(Xi (s),V (s)) dBi (s) Z +

t

hφd(·,V n (s)) + L(V n (s))φ,V n (s)ids

0

Z

hφβ(·,V n (s),u) + φ0 α(·,V n (s),u),V n (s)iW (duds),

+ U ×[0,t]

where 1 L(v)φ(x) = a(x,v)φ00 (x) + b(x,v)φ0 (x). 2

(3.2)

334


Hence, by (1.2) and (3.2), we have hφ,Sn (t) − Sn (0)i µ n Z 1 X t n =√ A (s) φ(Xin (s))γ(Xin (s),V n (s)) n i=1 0 i ¶ 0 n n n +φ (Xi (s))σ(Xi (s),V (s)) dBi (s) Z + 0

Z +

t√

³ n hφd(·,V n (s)) + L(V n (s))φ,V n (s)i ´ −hφd(·,V (s)) + L(V (s))φ,V (s)i ds √ ³ n hφβ(·,V n (s),u) + φ0 α(·,V n (s),u),V n (s)i

U ×[0,t]

´ −hφβ(·,V (s),u) + φ0 α(·,V (s),u),V (s)i W (duds).

(3.3)

Note that hφ,Sn (t)i = Mφ1,n (t) + Anφ (t) + Mφ2,n (t) is a semimartingale with respect to the filtration {Ft } generated by W and the Bi . Setting √ ³ Gnφ (s,u) = n hφβ(·,V n (s),u) + φ0 α(·,V n (s),u),V n (s)i ´ −hφβ(·,V (s),u) + φ0 α(·,V (s),u),V (s)i , we have h i Mφ1,n

t n

1X = n i=1

Z 0

t

³ ´2 Ani (s)2 φ(Xin (s))γ(Xin (s),V n (s)) + φ0 (Xin (s))σ(Xin (s),V n (s)) ds

and £ 2,n ¤ M = t

Z tZ 0

U

Gnφ (s,u)2 µ(du)dt.

Let n ´2 1 X n 2³ Ai (s) φ(Xin (s))γ(Xin (s),V n (s)) + φ0 (Xin (s))σ(Xin (s),V n (s)) n Z i=1 Hφ2,n (s) = Gnφ (s,u)2 µ(du)

Hφ1,n (s) =

U 3,n Hφ (s) = n(hφd(·,V n (s)) + L(V n (s))φ,V n (s)i − hφd(·,V

2

(s)) + L(V (s))φ,V (s)i) .

It follows, for example, that for t,h > 0, we have Z t+h 3(Hφ1,n (s) + Hφ2,n (s) + hHφ3,n (s))ds|Ft ], (3.4) E[hφ,Sn (t + h) − Sn (t)i2 |Ft ] ≤ E[ t

and, applying Doob’s inequality, Z t E[suphφ,Sn (s) − Sn (0)i ] ≤ E[ 12(Hφ1,n (s) + Hφ2,n (s) + tHφ3,n (s))ds]. 2

s≤t

0

(3.5)


335

We need to estimate each of the Hφk,n . k k e e κ = sup e Let φe = φψ and |φ| x,0≤k≤κ |(d /dx )φ(x)|. Then |φ|κ ≤ constkφkκ+1 . It is easy to see that there exists a constant c7 such that e 0 e|x| |φ(x)| ≤ c7 |φ| and e 1 e|x| . |φ(x)| + |φ0 (x)| ≤ c7 |φ| Hence ¯ n ¯ ¯1 X ³ ´2 ¯ λ ¯ ¯ n 2 n n n 0 n n n n,p E¯ Ai (s) φ(Xi (s))γ(Xi (s),V (s)) + φ (Xi (s))σ(Xi (s),V (s)) ¯ 1{ηm ≥s} ¯n ¯ i=1 ¯λ ¯ n ¯1 X ¯ n ¯ e 2 ¯¯ 1{ηn,p ≥s} ≤ E¯ Ani (s)2 K 2 c7 e2|Xi (s)| |φ| 1 m ¯n ¯ i=1 ¯ n ¯ λ2 n ¯ ¯ X X 2λ λ e 2λ ¯ 1 n 41 4|Xin (s)| ¯ n,p Ai (s) e ≤ K c7 |φ|1 E ¯ ¯ 1{ηm ≥s} ¯ ¯n n i=1 i=1 ¯ ¯λ e 2λ ¯m4 23 m4 ¯ 2 ≤ K 2λ cλ7 |φ| 1 e 2λ , ≡ c8 (m,λ,s)|φ| 1 and we have 2λ e 2λ n,p E[Hφ1,n (s)λ 1{ηm ≥s} ] ≤ c8 (m,λ,s)|φ|1 ≤ c8 (m,λ,s)kφk2 .

Observing that

√

(3.6)

n(hφd(·,V n (s))),V n (s)i − hφd(·,V (s))),V (s)i) E √ D = n φd(·,V n (s))),V n (s) − Ve n (s) E √ D + n φ(d(·,V n (s)) − d(·, Ve n (s)), Ve n (s) E √ D + n φ(d(·, Ve n (s)) − d(·,V (s))), Ve n (s) E √ D + n φd(·,V (s)), Ve n (s) − V (s) ,

we have ¯√ D E¯ ¯ ¯ ¯ n φd(·,V n (s))),V n (s) − Ve n (s) ¯ ¯ ¯ n ¯ 1 X ¯ ¯ ¯ n n n n n =¯√ (Ai (s)φ(Xi (s))d(Xi (s),V (s)) − Ai (s)φ(Xi (s))d(Xi (s),V (s)))¯ ¯ n ¯ i=1 n

1 X ≤√ |φ(Xin (s)) − φ(Xi (s))||d(Xin (s),V n (s))|Ani (s) n i=1 n

1 X +√ |φ(Xi (s))||d(Xin (s),V n (s)) − d(Xi (s),V n (s))|Ani (s) n i=1 n

1 X n |A (s) − Ai (s)||φ(Xi (s))d(Xi (s),V n (s))| +√ n i=1 i

336

STOCHASTIC EVOLUTION EQUATION FOR FLUCTUATIONS n

K X 0 ≤√ |φ (θXin (s) + (1 − θ)Xi (s))||Xin (s) − Xi (s)|Ani (s) n i=1 n

K X |φ(Xi (s))||Xin (s) − Xi (s)|Ani (s) +√ n i=1 n

K X |φ(Xi (s))||Ani (s) − Ai (s)| +√ n i=1 2K X ³ |Xin (s)| |Xi (s)| ´ e ≤√ c7 e ∨e |φ|1 |Xin (s) − Xi (s)|Ani (s) n i=1 n

n

K X |Xi (s)| e c7 e |φ|0 |Ani (s) − Ai (s)|, +√ n i=1 and hence

¯√ D E¯2λ ¯ ¯ n,p E ¯ n φd(·,V n (s))),V n (s) − Ve n (s) ¯ 1{ηm ≥s} Ã n X √ e 2λ (2 nc7 K)2λ E 1 ≤ 22λ−1 |φ| |X n (s) − Xi (s)|2λ 1 n i=1 i Ã n !2λ−1 ! 2λ 1 X ¯¯³ |Xin (s)| |Xi (s)| ´ n ¯¯ 2λ−1 n,p × 1{ηm ∨e Ai (s)¯ ¯ e ≥s} n i=1 !2 ÃÃ n 1X n λ 2λ−1 e 2λ λ 2λ 2λ |A (s) − Ai (s)| +2 |φ|0 n K c7 E n i=1 i Ã n !λ−1 ! λ 1 X λ−1 |Xi (s)| n,p × e 1{ηm ≥s} n i=1 e 2λ . ≤ c9 (m,λ,s)|φ| 1

As

¯√ D E¯ ¯ ¯ ¯ n φ(d(·,V n (s)) − d(·, Ve n (s)), Ve n (s) ¯ ¯ ¯ n ¯ 1 X ¯ ¯ ¯ Ai (s)φ(Xi (s))(d(Xi (s),V n (s)) − d(Xi (s), Ve n (s)))¯ =¯√ ¯ n ¯ i=1 n

1 X ≤√ Ai (s)|φ(Xi (s))|Kρ(V n (s), Ve n (s)), n i=1 we have

¯√ D E¯2λ ¯ ¯ n,p E ¯ n φ(d(·,V n (s)) − d(·, Ve n (s)), Ve n (s) ¯ 1{ηm ≥s} ¯ ¯2λ n ¯1 X ¯ ¯ e 0 ρ(V n (s), Ve n (s))¯¯ 1{ηn,p ≥s} ≤ K 2λ nλ E ¯ Ai (s)c7 e|Xi (s)| |φ| m ¯n ¯ i=1 "Ã n !λ n X 1X 2λ λ 2λ e 2λ 21 2|Xi (s)| Ai (s) e ≤ K n c7 |φ|0 E n i=1 n i=1

337


Ã

! # n n 1X n 1X n 2 λ 2λ n,p ×2 m ( |X (s) − Xi (s)| ) + ( |A (s) − Ai (s)|) 1{ηm ≥s} n i=1 i n i=1 i ¡ ¢ e 2λ m2 c(m,2) λ 22λ−1 ≤ K 2λ nλ c2λ 7 |φ|0   Ã n !2 n X X 1 1 2λ λ  n,p ×E m2λ |X n (s) − Xi (s)| + |An (s) − Ai (s)| 1{ηm ≥s} n i=1 i n i=1 i 2λ−1

2λ

e 2λ . ≤ c10 (m,λ,s)|φ| 0 As ¯√ D E¯ ¯ ¯ ¯ n φ(d(·, Ve n (s)) − d(·,V (s))), Ve n (s) ¯ ¯ ¯ n ¯ ¯ 1 X ¯ ¯ n e Ai (s)φ(Xi (s))(d(Xi (s), V (s))) − d(Xi (s),V (s)))¯ , =¯√ ¯ ¯ n i=1 we have ¯√ D E¯2λ ¯ ¯ n,p E ¯ n φ(d(·, Ve n (s)) − d(·,V (s))), Ve n (s) ¯ 1{ηm ≥s} !2λ−1 "Ã n 2λ 2λ 1X |X (s)| 2λ 2λ λ e 2λ Ai (s) 2λ−1 e 2λ−1 i ≤ c7 K n |φ|0 E n i=1 n ¯2λ 1 X ¯¯ ¯ n,p × ¯d(Xi (s), Ve n (s)) − d(Xi (s),V (s))¯ 1{ηm ≥s} n i=1

#

e 2λ , ≤ c11 (m,λ,s)|φ| 0 where the last inequality follows by arguments similar to the estimate for the third term on the right side of (2.8). Let Nn =

n X

(Ai (s)φ(Xi (s))d(Xi (s),V (s)) − hφd(·,V (s)),V (s)i),

n ≥ 1.

i=1

Then {Nn : n = 1,2,···} is a discrete-time P(·|W )-martingale with, using the notation of Burkholder [3], Sn (N )2 =

n X

2

(Ai (s)φ(Xi (s))d(Xi (s),V (s)) − hφd(·,V (s)),V (s)i) .

i=1

(Do not confuse Sn (N ) here with our process Sn .) By Theorem 3.2 in Burkholder [3], there exists a constant Cλ such that E(Nn2λ |W ) ≤ Cλ E(Sn (N )2λ |W ).

338


Therefore ¯√ D E¯2λ ¯ ¯ E ¯ n φd(·,V (s)), Ve n (s) − V (s) ¯ 1 E(E(Nn2λ |W )) nλ n £ £¡ X ¤¤ 1 2 ¢λ ¯¯ ≤ λ Cλ E E (Ai (s)φ(Xi (s))d(Xi (s),V (s)) − hφd(·,V (s)),V (s)i) W n i=1 ´ ³ 2λ = Cλ E |A1 (s)φ(X1 (s))d(X1 (s),V (s)) − hφd(·,V (s)),V (s)i| =

e 2λ , ≤ c12 (m,λ,s)|φ| 0 and we have ¯√ ¯2λ e 2λ n,p E[¯ n(hφd(·,V n (s))),V n (s)i − hφd(·,V (s))),V (s)i)¯ 1{ηm ≥s} ] ≤ c13 (m,λ,s)|φ|1 . Similar arguments give ¡√ ¢2λ e 2λ n,p E[ n|hL(V n (s))φ,V n (s)i − hL(V (s))φ,V (s)i| 1{ηm ≥s} ] ≤ c14 (m,λ,s)|φ|3 , where the estimate in terms of the higher derivatives is required because of the differential operator, and we have 2λ e 2λ n,p E[Hφ3,n (s)λ 1{ηm ≥s} ] ≤ c15 (m,λ,s)|φ|3 ≤ c15 (m,λ,s)kφk4 .

(3.7)

Finally, again applying similar arguments, we can show 2λ e 2λ n,p E[Hφ2,n (s)λ 1{ηm ≥s} ] ≤ c16 (m,λ,s)|φ|2 ≤ c16 (m,λ,s)kφk3 .

(3.8)

Without loss of generality, we can assume that all of the cl (m,λ,s) are nondecreasing in s. Applying (3.6), (3.7), and (3.8), (3.5) gives n,p E[suphφ,Sn (s ∧ ηm ) − Sn (0)i2 ] ≤ c17 (m,p,t)kφk24 . s≤t

4 For κ sufficiently large, the embedding Tκ−1 : Φκ−1 → Φ4 is Hilbert-Schmidt and hence, P ∞ if {φk } is an orthonormal basis for Φκ−1 , k=1 kφk k24 < ∞. (See [17], Lemma 1 and Theorem 2, pages 33-34.) Consequently,

n,p E[sup kSn (s ∧ ηm ) − Sn (0)k2−(κ−1) ] ≤ E[ s≤t

∞ X

k=1

n,p suphφk ,Sn (s ∧ ηm ) − Sn (0)i2 ] s≤t

≤ c17 (m,p,t)

∞ X

kφk k24 < ∞.

k=1

It follows that for each t ≥ 0 and ² > 0, there exists kt,² > 0 such that n,p sup P{sup kSn (s ∧ ηm )k−(κ−1) > kt,² } ≤ ². n

s≤t

n,p But {ψ ∈ Φ−κ : kψk−(κ−1) ≤ kt,² } is a compact subset of Φ−κ , so {Sn (· ∧ ηm )} satisfies the compact containment condition in Φ−κ .

339


Similarly, by (3.4), for t < t + h ≤ T , h < 1, n,p n,p 2 E[kSn ((t + h) ∧ ηm ) − Sn (t ∧ ηm )k−(κ−1) |Ft ] Z ∞ t+h X n,p ≤ E[ (Hφ1,n (s) + Hφ2,n (s) + hHφ3,n (s))1{ηm ≥s} ds|Ft ] k k k k=1

t

≤ h(λ−1)/λ E[

∞ Z X (

T

0

k=1

1/λ n,p (Hφ1,n (s) + Hφ2,n (s) + hHφ3,n (s))λ 1{ηm |Ft ]. ≥s} ds) k k k

Then E[

∞ Z X (

T

0

k=1

= E[

1/λ n,p ] (s))λ 1{ηm (s) + hHφ3,n (s) + Hφ2,n (Hφ1,n ≥s} ds) k k k

∞ X

k=1 ∞ X

≤ E[

Z kφk k24 (

≤(

k=1

×

³

T

³

0

Z 2 kφk k4 (

0

k=1 ∞ X

T

(Hφ1,n (s) + Hφ2,n (s) + hHφ3,n (s))/kφk k24 k k k (s) + Hφ2,n (s) + hHφ3,n (s))/kφk k24 (Hφ1,n k k k

´λ ´λ

1/λ n,p 1{ηm ] ≥s} ds)

1/λ n,p ] 1{ηm ≥s} ds)

kφk k24 )(λ−1)/λ Ã

∞ X

Z kφk k24

k=1

T

0

³ ´λ n,p E[ (Hφ1,n (s) + Hφ2,n (s) + hHφ3,n (s))/kφk k24 1{ηm ≥s} ]ds k k k

!1/λ

< ∞. Since n,p n,p 2 n,p n,p 2 kSn ((t + h) ∧ ηm ) − Sn (t ∧ ηm )k−κ ≤ kSn ((t + h) ∧ ηm ) − Sn (t ∧ ηm )k−(κ−1) ,

we have verified the conditions of Theorem 4.20 of [24] (Theorem 3.8.6 of [13]) with γn (δ) = δ (λ−1)/λ

∞ Z X ( k=1

T

0

1/λ n,p (Hφ1,n (s) + Hφ2,n (s) + δHφ3,n (s))λ 1{ηm . ≥s} ds) k k k

Note that since the Sn are continuous, tightness of {Sn } in DΦ−k [0,∞) implies tightness in CΦ−k [0,∞). The same argument gives tightness for {M 1,n }, and we have the following additional result. Lemma 3.1. Under the conditions of Theorem 3.1, {M 1,n } is tight in CΦ−κ [0,∞). 4. Characterization of the limit We need the following additional assumptions. (S4) There exists δ > 0 such that Z |σ(x,ν)T z|2 − δ |z · α(x,ν,u)|2 µ(du) ≥ 0 U

∀x,z ∈ Rd , ν ∈ M(Rd ).

340


(S5) The coefficients σ, c, d, a, b, γ, α, and β are differentiable with respect to the measure in the sense that, for example, there exists a bounded, continuous function ∂d on Rd × M(Rd ) × Rd such that for ν1 ,ν2 ∈ M(Rd ), Z 1Z d(x,ν2 ) − d(x,ν1 ) = 0

Rd

∂d(x,(1 − r)ν1 + rν2 ,y)(ν2 (dy) − ν1 (dy))dr.

(S6) For κ given by Theorem 3.1, φ ∈ Φκ+l+2 , ν1 ,ν2 ∈ M(Rd ), and u ∈ U , F1 (ν1 ,ν2 )φ ≡ d(·,ν2 )φ + L(ν2 )φ Z Z 1³ + φ(x)∂d(x,rν2 + (1 − r)ν1 ,·) Rd

0

´ +∂L(rν2 + (1 − r)ν1 ,·)φ(x) drν1 (dx)

(4.1)

and F2 (ν1 ,ν2 ,u)φ ≡ φβ(·,ν2 ,u) + ∇φT α(·,ν2 ,u) Z Z 1³ − φ(x)∂β(x,rν2 + (1 − r)ν1 ,u,·) Rd

0

´ +∇φT (x)∂α(x,rν2 + (1 − r)ν1 ,u,·) drν1 (dx)(4.2)

are in Φκ+l for 0 ≤ l ≤ 2. For φ ∈ Φκ+2 , the mappings (ν1 ,ν2 ,v) ∈ M(Rd ) × M(Rd ) × Φ−κ → hF1 (ν1 ,ν2 )φ,vi ∈ R

(4.3)

and (ν1 ,ν2 ,v) ∈ M(Rd ) × M(Rd ) × Φ−κ → hF2 (ν1 ,ν2 ,·)φ,vi ∈ L2 (U,µ)

(4.4)

are continuous. (S7) For each ν ∈ M(Rd ), the mappings from x ∈ Rd to aij (x,ν), bi (x,ν), d(x,ν) ∈ R and αi (x,ν,·), β(x,ν,·) ∈ L2 (U,µ) have bounded derivatives with respect to x up to order q ≡ κ + 2. For each x ∈ Rd , u ∈ U and ν ∈ M(Rd ), ∂aij (x,ν,·), ∂bi (x,ν,·), ∂d(x,ν,·), ∂αi (x,ν,u,·), ∂β(x,ν,u,·) are in Φq , and there exists a constant K such that X X k∂aij (x,ν,·)k2q + k∂bi (x,ν,·)k2q + k∂d(x,ν,·)k2q i,j

+

Z ÃX U

i

!

k∂αi (x,ν,u,·)k2q + k∂β(x,ν,u,·)k2q

µ(du) ≤ K.

i

Remark 4.1. If ν1 = ν2 = ν, we write Fi (ν) rather than Fi (ν,ν). Condition (S6) implies smoothness and growth conditions on the coefficients of the differential operators. Continuity of the mapping (ν1 ,ν2 ) ∈ M(Rd ) × M(Rd ) → F1 (ν1 ,ν2 )φ ∈ Φk


341

would imply (4.3). Continuing to restrict the calculations to dimension d = 1, by (3.3), Z t ® hφ,Sn (t)i = hφ,Sn (0)i + φ,M 1,n (t) + hF1 (V (s),V n (s))φ,Sn (s)ids 0 Z + hF2 (V (s),V n (s),u)φ,Sn (s)iW (duds),

(4.5)

U ×[0,t]

where ® Mφ1,n (t) = φ,M 1,n (t) µ n Z t X −1/2 n =n Ai (s) φ(Xin (s))γ(Xin (s),V n (s)) i=1

√

0

¶ +φ0 (Xin (s))σ(Xin (s),V n (s)) dBi (s),

n(hφd(·,V n (s)) + L(V n (s))φ,V n (s)i − hφd(·,V (s)) + L(V (s))φ,V (s)i) = hφd(·,V n (s)) + L(V n (s))φ,Sn (s)i √ + nhφd(·,V n (s)) + L(V n (s))φ − φd(·,V (s)) + L(V (s))φ,V (s)i = hφd(·,V n (s)) + L(V n (s))φ,Sn (s)i Z Z 1 + φ(x)∂d(x,rV n (s) + (1 − r)V (s),·) R 0 ® +∂L(rV n (s) + (1 − r)V (s),·)φ(x),Sn (s) drV (s,dx) = hF1 (V (s),V n (s))φ,Sn (s)i,

and ´ √ ³ n hφβ(·,V n (s),u) + φ0 α(·,V n (s),u),V n (s)i − hφβ(·,V (s),u) + φ0 α(·,V (s),u),V (s)i = hφβ(·,V n (s),u) + φ0 α(·,V n (s),u),Sn (s)i Z Z 1 − φ(x)∂β(x,rV n (s) + (1 − r)V (s),u,·) R 0

® +φ0 (x)∂α(x,V (s),u,·),Sn (s) drV (s,dx) = hF2 (V (s),V n (s),u)φ,Sn (s)i. Let H = L2 (U,µ). In the terminology of Kurtz and Protter [25], we define a R × H# -semimartingale Y by setting Z Y (a,h,t) = at + B h (t) = at + h(u)W (duds), U ×[0,t]

for a ∈ R and h ∈ H. Let Un (t) = Sn (0) + M 1,n (t), and for φ ∈ Φκ+2 , let hF (V (s),V n (s))φ,Sn (s)i denote the R × H-valued process given by hF (V (s),V n (s))φ,Sn (s)i = (hF1 (V (s),V n (s))φ,Sn (s)i,hF2 (V (s),V n (s),u)φ,Sn (s)i).

342


Then (4.5) can be rewritten in the notation of Kurtz and Protter [25] as hφ,Sn (t)i = hφ,Un (t)i + hF (V (·),V n (·))φ,Sn i · Y (t). h i Note that for each φ ∈ C 1 (R), h ∈ H, and 1 ≤ i ≤ n, Mφ1,n ,B h = 0, h

t

i

Mφ1,n ,Bi t Z ¡ ¢2 1 t n = (Ai (s))2 φ(Xin (s))γ(Xin (s),V n (s)) + φ0 (Xin (s))σ(Xin (s),V n (s)) ds, n 0 and h

Mφ1,n

i t

=

Z tD ¡ 0

E ¢2 φγ(·,V n (s)) + φ0 σ(·,V n (s)) ,V2n (s) ds,

Pn 1

where V2n (s) = n j=1 Anj (s)2 δXjn (s) . We should emphasize that we are proving convergence in distribution for {Sn }. The limit will not “live” on the original probability space. To be precise, for a countable dense subset {hj } ⊂ H, the sequence {(V n ,M 1,n ,Sn ,{Y (hj )},{Bi },{Xi },{Ai })} is relatively compact in CM(R)×(Φ−κ )2 ×(R∞ )4 [0,∞). Denoting a limit point by (V,M,S,{Y (hi )},{Bi },{Xi },{Ai }) (even though these are not the V , Y , {Bi }, {Xi }, {Ai } on the original probability space, they will have the same distribution), M (and hence S) will not be adapted Y,{Bi } to the filtration {Ft } generated by Y and {Bi }. Note that {Y (hi )} determines Y (h) (and hence B h ) for all h ∈ H and the Y (h) determine W . £ ¤ For any limit point, M will be a Φ−κ -valued local martingale with Mφ ,B h t = 0 for every φ ∈ Φ and h ∈ H and Z tD E ¡ ¢2 [Mφ ]t = φγ(·,V (s)) + φ0 σ(·,V (s)) ,V2 (s) ds, 0

where n

1X Ai (t)2 δXi (t) . n→∞ n i=1

V2 (t) = lim

Lemma 4.1. For φ ∈ Φκ , E[eihφ,S(0)+M (t)i |W ] Z t ¢ 1¡ = exp{− hφ2 ,V2 (0)i − hφ,V (0)i2 + h(φγ(·,V (s)) + φ0 σ(·,V (s)))2 ,V2 (s)ids } 2 0 and E[eihφ,M (t+r)−M (t)i |σ(W ) ∨ FtM ] Z ¢ 1 t+r h(φγ(·,V (s)) + φ0 σ(·,V (s)))2 ,V2 (s)ids }, = exp{− 2 t


343

which determine the joint distribution of W and M . Proof: Define D E fφn (t) = φ, M fn (t) M µ n Z t X −1/2 =n Ai (s) φ(Xi (s))γ(Xi (s),V (s)) i=1

0

¶ +φ (Xi (s))σ(Xi (s),V (s)) dBi (s), 0

and observe that h i fφn Mφ1,n − M t n Z 1 X t³ n = Ai (s)φ(Xin (s))γ(Xin (s),V n (s)) + φ0 (Xin (s))σ(Xin (s),V n (s)) n i=1 0 ¶2 −Ai (s)φ(Xi (s))γ(Xi (s),V (s)) − φ0 (Xi (s))σ(Xi (s),V (s)) ds. fn must have the same limit. Again, to converges to zero. It follows that M 1,n and M be precise, one should say that any limit point of fn ,Sn ,{Y (hj )},{Bi },{Xi },{Ai })} {(V n ,M 1,n , M will be of the form (V,M,M,S,{Y (hi )},{Bi },{Xi },{Ai }). For a σ(W ) measurable random variable Z, exchangeability implies E[eihφ,S(0)+M (t)i Z] fn (t)i

= lim E[eihφ,Sn (0)+M n→∞

Z]

1 fφ,1 }|W ]n Z] = lim E[E[exp{i √ (A1 (0)φ(X1 (0)) − hφ,V (0)i + M n→∞ n Z t ¢ 1¡ 2 2 = E[exp{− hφ ,V2 (0)i − hφ,V (0)i + h(φγ(·,V (s)) + φ0 σ(·,V (s)))2 ,V2 (s)ids }Z] 2 0 where Z fφ,1 (t) = M

t

µ ¶ A1 (s) φ(X1 (s))γ(X1 (s),V (s)) + φ0 (X1 (s))σ(X1 (s),V (s)) dB1 (s).

0

The proof of the second identity is similar. Let U (t) = S(0) + M (t) and hF (V (s))φ,S(s)i = (hF1 (V (s))φ,S(s)i,hF2 (V (s),u)φ,S(s)i). Then (Sn ,Un ) is relatively compact in CΦ−κ ×Φ−κ [0,∞), and by the continuity assumptions on F1 and F2 and Theorem 5.5 in [25], for any limit point (S,U ), we have hφ,S(t)i = hφ,U (t)i + hF (V (·))φ,Si · Y (t).

344


Specifically, any limit point of {Sn } satisfies (1.10). To prove uniqueness for the solution to (1.10), suppose that S1 and S2 are solutions and set ξ = S1 − S2 . Then ξ satisfies Z

Z

t

hF1 (V (s))φ,ξ(s)ids +

hφ,ξ(t)i = 0

hF2 (V (s),u)φ,ξ(s)iW (duds).

(4.6)

U ×[0,t]

We adapt arguments of Rozovskii [36] to establish that ξ ≡ 0 is the unique solution to (4.6) and hence establish uniqueness for (1.10). Lemma 4.2. Suppose that the assumptions (S1)-(S7) hold. Then ξ = 0 a.s. R Proof: Let q = κ + 2, and for ν ∈ M(R), define ρν ≡ R ψ(x)−1 ν(dx) < ∞. LemmaA.6,

By

Z 2hv,F1∗ (ν)vi−q +

U

kF2∗ (ν,u)vk2−q µ(du) ≤ c18 ρ2ν kvk2−q ,

(4.7)

for all v ∈ Φ−κ . Note that ξ(t) takes values in Φ−κ ⊂ Φ−q . Let {φqj } be an orthonormal basis for Φq . Applying Itô’s formula, we have

®2 φqj ,ξ(t) =

Z

t

0

® ® 2 φqj ,ξ(t) F1 (V (s))φqj ,ξ(s) ds Z ® ® + 2 φqj ,ξ(t) F2 (V (s),u)φqj ,ξ(s) W (duds) U ×[0,t]

Z tZ + 0

U

®2 F2 (V (s),u)φqj ,ξ(s) µ(du)ds.

By Proposition 2.1, if (2.1) and (2.3) hold, then (2.6) holds giving E[supt≤T ρV (t) ] < ∞. Let τk = inf{t : ρV (t) ≥ k}. Stopping the processes at τk , taking expectations, and summing over j, (4.7) gives t∧τk µ

Z Ekξ(t ∧ τk )k2−q

=E Z ≤ 0

0 t

¶

Z 2hξ(s),F1∗ (V

(s))ξ(s)i−q +

U

kF2∗ (V

(s),u)ξ(s)k2−q µ(du)

ds

c18 k 2 Ekξ(s ∧ τk )k2−q ds.

Then uniqueness follows from Gronwall’s inequality and the fact that τk → ∞ as k → ∞. Finally, we have our main result. Theorem 4.2. Under assumptions (S1)-(S7), we have Sn ⇒ S and S is the unique solution to the stochastic evolution equation (1.10). 5. CLT for Euler scheme £ ¤ Now we consider the CLT for the Euler scheme used in [27]. Let ηδ (s) = δs δ, and for some partition {Uk } of U and uk ∈ Uk , define ξδ (u) = uk , u ∈ Uk , k = 1,2,....


345

Let {(Xin,δ ,An,δ i ),i = 1,...,n} satisfy Z t σ(Xin,δ (ηδ (s)),V n,δ (ηδ (s)))dBi (s) Xin,δ (t) = Xi (0) + 0 Z t + c(Xin,δ (ηδ (s)),V n,δ (ηδ (s)))ds Z0 + α(Xin,δ (ηδ (s)),V n,δ (ηδ (s)),ξδ (u))W (duds),

(5.1)

Z t n,δ n,δ An,δ (t) = A (0) + An,δ (ηδ (s)))dBi (s) i i i (s)γ(Xi (ηδ (s)),V 0 Z t n,δ n,δ + An,δ (ηδ (s)))ds i (s)d(Xi (ηδ (s)),V 0 Z n,δ n,δ + An,δ (ηδ (s)),ξδ (u))W (duds), i (s)β(Xi (ηδ (s)),V

(5.2)

U ×[0,t]

U ×[0,t]

where n

V n,δ (t) =

1 X n,δ A (t)δX n,δ (t) . i n i=1 i

(5.3)

In this paper, we only analyze the simplest case in which W is a one-dimensional Brownian motion, that is, U consists of a single point. Modifying Theorem 3.3 in [27] in a way similar to the proof of Theorem 2.4 of the current paper, we have the following result. Theorem 5.1. Under the assumptions (S1)-(S5), we have   2  n ¯ ¯2λ X 1 ¯ ¯ n λ  λ E sup ¯Xin,δ (t) − Xin (t)¯ +  |An,δ n,δ ≤ c(T,m)δ ,  1t m or A (t) > m . n i=1 i n i=1 i

Applying the same arguments as those in Section 3, we can prove that the sequence √ Sen ≡ n(V n,1/n − V n ) is tight. Now we characterize its limit points.

346


Note that Sen (0) = 0. As in (3.3), we have D E φ, Sen (t) ( · n Z 1 X t n,1/n n,1/n n,1/n =√ Ai (s) φ(Xi (s))γ(Xi (η n1 (s)),V n,1/n (η n1 (s))) n i=1 0 ¸ n,1/n n,1/n 0 n,1/n +φ (Xi (s))σ(Xi (η n1 (s)),V (η n1 (s))) ) −Ani (s)[φ(Xin (s))γ(Xin (s),V n (s)) + φ0 (Xin (s))σ(Xin (s),V n (s))] dBi (s) n

1 X +√ n i=1

· Z t( n,1/n n,1/n n,1/n Ai (s) φ(Xi (s))d(Xi (η n1 (s)),V n,1/n (η n1 (s))) 0

¸

+L(V

n,1/n

n,1/n (η n1 (s)) (η n1 (s)))φ(Xi

)

−Ani (s)[φ(Xin (s))d(Xin (s),V n (s)) + L(V n (s))φ(Xin (s))] n

1 X +√ n i=1

ds

· Z t( n,1/n n,1/n n,1/n Ai (s) φ(Xi (s))β(Xi (η n1 (s)),V n,1/n (η n1 (s))) 0

¸ 0

+φ

n,1/n n,1/n (Xi (s))α(Xi (η n1 (s)),V n,1/n (η n1 (s)))

−Ani (s)[φ(Xin (s))β(Xin (s),V n (s)) + φ0 (Xin (s))α(Xin (s),V n (s))] ≡ I1 + I2 + I3 .

) dW (s) (5.4)

Lemma 5.1. Let ξin (t) be processes satisfying n

1X n 2 ξ (t) ≤ K, n i=1 i

∀t, ∀ω.

Then n

1 X √ n i=1

Z

t

0

n

ξin (s)(Bi (s) − Bi (η n1 (s)))2 ds → 0,

1 X √ n i=1 n

1 X √ n i=1 n

1 X √ n i=1

(5.6)

ξin (s)(W (s) − W (η n1 (s)))2 ds → 0,

(5.7)

0

ξin (η n1 (s))(Bi (s) − Bi (η n1 (s)))ds → 0,

(5.8)

t

0

t

t

ξin (s)(s − η n1 (s))2 ds → 0,

Z

Z

Z

(5.5)

0

347

THOMAS G. KURTZ AND JIE XIONG n

1 X √ n i=1

n

1 X √ n i=1

n

1 X √ n i=1

(5.9)

ξin (η n1 (s))(W (s) − W (η n1 (s)))ds → 0,

(5.10)

ξin (η n1 (s))(Bi (s) − Bi (η n1 (s)))dW (s) → 0,

(5.11)

t

0

t

0

t

ξin (η n1 (s))(s − η n1 (s))ds → 0,

Z

Z

Z

n

1 X √ n i=1

Z

0

t

0

ξin (η n1 (s))(s − η n1 (s))dW (s) → 0.

(5.12)

Proof: Note that ¯ ¯2 n Z t ¯ 1 X ¯ ¯ 2 ¯ n E¯ √ ξi (s)(Bi (s) − Bi (η n1 (s))) ds¯ ¯ n ¯ i=1 0 Z t X n n 1 1X ≤ nE ξin (s)2 (Bi (s) − Bi (η n1 (s)))4 ds n i=1 0 n i=1 n Z t X ≤K 3(s − η n1 (s))2 ds i=1

≤ 3Kn

0

[nt] Z X j n

j=0

= Kn[nt]

j+1 n

j (s − )2 ds n

1 → 0. n3

This proves (5.5). (5.6), (5.7) and (5.9) can be proved similarly. For k = 1, 2, ···, let n

1 X Mk = √ n i=1

Z 0

k n

ξin (η n1 (s))(Bi (s) − Bi (η n1 (s)))ds.

Then Mk is a discrete-time, square integrable martingale with quadratic variation process

[M ]k =

k X j=1

Ã

n

1 X n √ ξ (η 1 (s)) n i=1 i n

Z

j n j−1 n

!2 (Bi (s) − Bi (η n1 (s)))ds

.

348


Hence 2 EM[nt] =

[nt] X

ÃZ E

=

j=1

≤K

n

j−1 n

j=1 [nt] Z X

j n

Z

j n j−1 n

1 X n j −1 j −1 √ ξ ( )(Bi (s) − Bi ( ))ds n n n i=1 i

j n j−1 n

[nt] Z X

1 n

Z

j=1 0

1 n

!2

µ ¶ n 1X j −1 n j −1 2 E ξi ( ) (s1 ∧ s2 − )ds1 ds2 n i=1 n n s1 ∧ s2 ds1 ds2

0

Z

= 2K[nt]

1 n

1 s1 ( − s1 )ds1 n

0

K[nt] = → 0. 3n3 This proves (5.8). (5.10) can be proved similarly. Finally, ¯ ¯2 n Z t ¯ ¯ 1 X ¯ ¯ E¯ √ ξin (η n1 (s))(Bi (s) − Bi (η n1 (s)))dW (s)¯ ¯ ¯ n i=1 0 !2 Z tÃ n 1 X n √ =E ξ (η 1 (s))(Bi (s) − Bi (η n1 (s))) ds n i=1 i n 0 Z t X n 1 = Eξin (η n1 (s))2 (s − η n1 (s))ds 0 n i=1 ≤K

[nt] Z X j=0

j+1 n

j (s − )ds n

j n

= K([nt] + 1)

1 → 0, 2n2

which proves (5.11). (5.12) can be proved similarly. Lemma 5.2. Let f n (t) = W

Z t√ 0

2n(W (s) − W (η n1 (s)))dW (s).

fn ⇒ W f and W f is a one-dimensional Brownian motion independent of W . Then W f n is a sequence of martingales and Proof: It is clear that W Z t h i n f W = 2n(W (s) − W (η n1 (s)))2 ds t 0 " # Z nj nt j −1 2 1 X 2 (W (s) − W ( 2n t )) ds = j−1 nt j=1 n n Z n1 → 2n2 t sds = t. 0

349


By (5.10), we have h

f n ,W W

Z

i t

=

t√

0

n(W (s) − W (η n1 (s)))ds → 0,

and the lemma follows by the martingale central limit theorem Note that n,1/n

Xi

n,1/n

(s) − Xi

n,1/n

= σ(Xi

(η n1 (s)),V n,1/n (η n1 (s)))(Bi (s) − Bi (η n1 (s)))

n,1/n

+c(Xi

(η n1 (s)),V n,1/n (η n1 (s)))(s − η n1 (s))

n,1/n

+α(Xi

(η n1 (s))

(η n1 (s)),V n,1/n (η n1 (s)))(W (s) − W (η n1 (s)))

(5.13)

and µ n,1/n n,1/n Ai (s) = Ai (η n1 (s))exp n,1/n

+D(Xi

n,1/n

+β(Xi

n,1/n

γ(Xi

(η n1 (s)),V n,1/n (η n1 (s)))(Bi (s) − Bi (η n1 (s)))

(η n1 (s)),V n,1/n (η n1 (s)))(s − η n1 (s))

¶ (η n1 (s)),V n,1/n (η n1 (s)))(W (s) − W (η n1 (s))) , n,1/n

where D = d − 12 (γ 2 + β 2 ). (5.5)-(5.7) then justify the replacement of Ai (s) and n,1/n n,1/n n,1/n 1 1 Xi (s) by Ai (η n (s)) and Xi (η n (s)) in the calculations below, where the notation ≈ means that the difference converges to zero in probability. Lemma 5.3. Let ψ : R2 → R be bounded and continuous and have bounded, continuous first derivative ∂2 ψ with respect to the second variable. Then Z

n D E 1 X n,1/n n,1/n √ (η n1 (s)),·),V n,1/n (s) − V n,1/n (η n1 (s)) dW (s) Ai (η n1 (s)) ψ(Xi n i=1

t

0

Z

t

f n (s). hα(·,V (s))∂2 ψ(∗,·) + β(·,V (s))ψ(∗,·),V (s) ⊗ V (s)idW

≈ 0

Proof: Note that Z

t

n D E 1 X n,1/n n,1/n √ Ai (η n1 (s)) ψ(Xi (η n1 (s)),·),V n,1/n (s) − V n,1/n (η n1 (s)) dW (s) n i=1

t

n n 1 X n,1/n 1 X n,1/n √ Ai (η n1 (s)) A (s) n j=1 j n i=1

0

Z = 0

n,1/n

Z + 0

(ψ(Xi t

n X

n,1/n

(η n1 (s)),Xj

n,1/n

(s)) − ψ(Xi

n X

n,1/n

(η n1 (s)),Xj

1 1 n,1/n n,1/n n,1/n √ A (η n1 (s)) (A (s) − Aj (η n1 (s))) n j=1 j n i=1 i

(η n1 (s)))dW (s)

350

STOCHASTIC EVOLUTION EQUATION FOR FLUCTUATIONS n,1/n

ψ(Xi

Z

n,1/n

(η n1 (s)),Xj

n X

t

(η n1 (s))dW (s)

n X

1 1 n,1/n n,1/n n,1/n n,1/n A (η n1 (s)) A (s)∂2 ψ(Xi (η n1 (s)),Xj (η n1 (s)) n i=1 i n j=1 j

≈ 0

n,1/n

Z + 0

α(Xj t

f n (s) (η n1 (s)),V n,1/n (η n1 (s)))dW

n n 1 X n,1/n 1 X n,1/n n,1/n Ai (η n1 (s)) A (η n1 (s))β(Xj (η n1 (s)),V n,1/n (η n1 (s))) n i=1 n j=1 i n,1/n

ψ(Xi

n,1/n

(η n1 (s)),Xj

f n (s). (η n1 (s))dW

The conclusion of the lemma then follows. Theorem 5.2. For φ ∈ Φκ , define D E fφ (t) ≡ φ, M f(t) M Z t f (s) h(α∂1 β)(·,V (s))φ,V (s)idW = 0 Z t f (s) + hα(·,V (s))∂2 ∂3 β(∗,V (s),·) + β(·,V (s))∂2 β(∗,V (s),·))φ(∗),V (s) ⊗ V (s)idW 0 Z t f (s) + h(α∂1 α)(·,V (s))φ,V (s)idW 0 Z t f (s), + hα(·,V (s))∂2 ∂3 α(∗,V (s),·) + β(·,V (s))∂2 α(∗,V (s),·))φ0 (∗),V (s) ⊗ V (s)idW 0

where ∂1 and ∂3 are derivatives with respect to the corresponding variables and ∂2 fφ is a martingale satisfying [W, M fφ ]t = 0. refers to the operator defined in (S5). Then M e e e Let S be a limit point of {Sn }. Then S is the unique solution of D

Z tD E D E Z tD E E e f(t) + e e φ, S(t) = φ, M F1 (V (s))φ, S(s) ds + F2 (V (s))φ, S(s) dW (s). 0

0

(5.14)

Proof: Recall that I1 , I2 , I3 are defined by (5.4). It is easy to see that I1 → 0. Note that n Z 1 X t n,1/n n,1/n I2 = √ A (s)φ(Xi (s)) n i=1 0 i · ¸ n,1/n n,1/n d(Xi (η n1 (s)),V n,1/n (η n1 (s))) − d(Xi (s),V n,1/n (s)) ds n

1 X +√ n i=1

Z 0

t

· n,1/n L(V n,1/n (η n1 (s)))φ(Xi (η n1 (s)))

n,1/n Ai (s)

n,1/n

−L(V n,1/n (s))φ(Xi

¸ (s)) ds


√ + n

Z t ³D

351

E d(·,V n,1/n (s))φ + L(V n,1/n (s)φ,V n,1/n (s)

0

´ −hd(·,V n (s))φ + L(V n (s)φ,V n (s)i ds

≡ I21 + I22 + I23 , By Lemma 5.1, it is easy to show that I21 → 0 and I22 → 0, and by the definition of F1 in (4.1), Z tD E I23 = F1 (V n (s),V n,1/n (s))φ, Sen (s) ds 0

which converges to the second term on the right of (5.14). Similarly Z t n 1 X n,1/n n,1/n √ I3 = Ai (s)φ(Xi (s)) n 0 i=1 · ¸ n,1/n β(Xi (η n1 (s)),V n,1/n (η n1 (s))) − β(Xin (s),V n (s)) dW (s) Z

n

1 X n,1/n n,1/n √ Ai (s)φ0 (Xi (s)) n 0 i=1 ¸ · n,1/n n,1/n n n 1 1 (η n (s))) − α(Xi (s),V (s)) dW (s) α(Xi (η n (s)),V

+

t

Z tD +

E F2 (V n (s),V n,1/n (s)φ, Sen (s) dW (s)

0

≡ I31 + I32 + I33 . By (5.13) and Lemma 5.3, we have µ Z t n 1 X n,1/n n,1/n n,1/n n,1/n n,1/n √ Ai (s)φ(Xi (s) ∂1 β(Xi (s),V n,1/n (s))(Xi (η n1 (s)) − Xi I31 ≈ (s)) n 0 i=1 D E¶ n,1/n + ∂2 β(Xi (s),V n,1/n (s),·),V n,1/n (η n1 (s))) − V n,1/n (s) dW (s) Z

t

f n (s) h(α∂1 β)(·,V (s))φ,V (s)idW

≈ 0

Z

t

+

f n (s) hα(·,V (s))∂2 ∂3 β(∗,V (s),·) + β(·,V (s))∂2 β(∗,V (s),·))φ(∗),V (s) ⊗ V (s)idW

0

and Z

t

f n (s) h(α∂1 α)(·,V (s))φ,V (s)idW

I32 ≈ 0

Z

+

t

hα(·,V (s))∂2 ∂3 α(∗,V (s),·) 0

Then I31 + I32 of (5.14).

f n (s). +β(·,V (s))∂2 α(∗,V (s),·))φ0 (∗),V (s) ⊗ V (s)idW D E f and I33 converges to the third term on the right converges to φ, M

352


Uniqueness of the solution follows from Lemma 4.2 giving the desired result. Finally, we combine the results of Sections 4 and 5. √ Theorem 5.3. Let Sn∗ (t) = n(V n,1/n (t) − V (t)). Then Sn∗ ⇒ S ∗ where S ∗ is the unique solution of the stochastic evolution equation D E Z t f(t) + hφ,S ∗ (t)i = hφ,S(0)i + φ,M (t) + M hF1 (V (s))φ,S ∗ (s)ids 0 Z t + hF2 (V (s)φ,S ∗ (s)idW (s), (5.15) 0

φ ∈ Φκ . Remark 5.4. Comparing (5.15) to (1.10), we see that the difference lies in the f. M f arises directly from the discrete-time apadditional martingale driving term M proximation of the driving Brownian motion W through the limit in Lemma 5.2. Appendix A. Proof of monotonicity. We can represent Φ−q as the space of equivalence classes of (q + 1)-tuples v = {(v0 ,v1 ,··· ,vq )}, where vj ∈ Lq ≡ L2 (R,(1 + x2 )2q dx), such that hv,f i =

q Z X

(1 + x2 )2q vk (x)

k=0 R

dk (f (x)ψ(x))dx. dxk

(A.1)

The (q + 1)-tuples u and v are equivalent if the right side of (A.1) does not change when v is replaced by u. Then ( q ) X 2 2 kuk kLq : u = (u0 ,...,uq ) ∼ v . (A.2) kvk−q = inf k=0

By the Riesz representation theorem, for each v ∈ Φ−q , there exists a unique φ ≡ θq v ∈ Φq such that hv,f i = hφ,f iq =

q X

h(φψ)(k) ,(f ψ)(k) iLq .

k=0

It follows that v ∼ {(φψ,(φψ)0 ,··· ,(φψ)(q) )} and kvk2−q =

q X

k(φψ)(k) k2Lq .

k=0

In particular, the infimum in (A.2) is achieved. For each u = {(u0 ,u1 ,··· ,uq )} ∈ Φ−q , hu,vi−q =

q D X k=0

E uk ,(φψ)(k) )

Lq

does not depend on the choice of the (q + 1)-tuple in the class of u. Note also, that since Φq+2 is dense in Φq , {v ∈ Φ−q : θq v ∈ Φq+2 } is dense in Φ−q .

353


Lemma A.1. Suppose that f and its derivatives up to order q are bounded. Then for γ,φ ∈ Φq+1 , q Z X

(1 + x2 )2q (f γ 0 ψ)(k) (φψ)(k) dx

k=0 R

Z

(1 + x2 )2q f (γψ)(q+1) (φψ)(q) dx + O(kγkq kφkq ) R Z = − (1 + x2 )2q f (γψ)(q) (φψ)(q+1) dx + O(kγkq kφkq ), =

R

and hence for γ = φ, q Z X k=0 R

(1 + x2 )2q (f φ0 ψ)(k) (φψ)(k) dx = O(kφk2q ).

In addition R

Pq

2 2q 0 (k) (φψ)(k) dx k=0 R (1 + x ) (f γ ψ)

sup γ∈Φq

kγkq

sZ

≤ R

sZ

X

(1 + x2 )2q |f (φψ)(q+1) |2 dx + c19

(1 + x2 )2q |f (i) (φψ)(j) |2 dx

1≤i,j≤q

R

(A.3)

0

Proof: Let f1 = fψψ and note that if f and its derivatives are bounded, then f1 and its derivatives are bounded. Then q Z X

= =

k=0 R q Z X k=0 R q Z X

(1 + x2 )2q (f γ 0 ψ)(k) (φψ)(k) dx (1 + x2 )2q (f (γψ)0 − f1 γψ)(k) (φψ)(k) dx (1 + x2 )2q (f (γψ)0 )(k) (φψ)(k) dx −

k=0 R

(1 + x2 )2q f (γψ)(q+1) (φψ)(q) dx + R

+ − Z

(1 + x2 )2q (f1 γψ)(k) (φψ)(k) dx

k=0 R

Z =

q Z X

q−1 Z X k=0 R q Z X

q−1 µ X l=0

q l

¶Z

(1 + x2 )2q f (q−l) (γψ)(l+1) (φψ)(q) dx R

(1 + x2 )2q (f (γψ)0 )(k) (φψ)(k) dx (1 + x2 )2q (f1 γψ)(k) (φψ)(k) dx

k=0 R

(1 + x2 )2q f (γψ)(q+1) (φψ)(q) dx + O(kγkq kφkq ).

= R

354


Integrating the first term in the fourth expression above by parts gives µ ¶ Z 4qx (q) − (1 + x2 )2q f (γψ)(q) (φψ)(q+1) + (φψ) dx 1 + x2 R Z − (1 + x2 )2q f 0 (γψ)(q) (φψ)(q) dx R

+

q−1 µ ¶Z X q l=0

+ −

l

q−1 Z X k=0 R q Z X

(1 + x2 )2q f (q−l) (γψ)(l+1) (φψ)(q) dx

R

(1 + x2 )2q (f (γψ)0 )(k) (φψ)(k) dx (1 + x2 )2q (f1 γψ)(k) (φψ)(k) dx

k=0 R

Z

(1 + x2 )2q f (γψ)(q) (φψ)(q+1) dx + O(kγkq kφkq ).

=− R

Since kγkq ≥ k(γψ)(j) kLq , (A.3) follows by the Cauchy-Schwartz inequality. Finally, if γ = φ, we can add the two identities to obtain q Z X k=0 R

(1 + x2 )2q (f φ0 ψ)(k) (φψ)(k) dx = O(kφk2q ).

Write F1 = F11 + F12 and F2 = F21 + F22 , where 1 F11 φ = aφ00 + bφ0 , 2 For ν ∈ M(R), let ρν =

R R

F21 φ = αφ0 .

(A.4)

ψ(x)−1 ν(dx).

Lemma A.2. For v ∈ Φ−q such that φ = θq v ∈ Φq+2 , Z ∗ 2hv,F11 vi−q = − a(x,ν)(1 + x2 )2q |(φψ)(q+1) (x)|2 dx + O(kvk2−q ), R

where |O(kvk2−q )| ≤ c20 kvk2−q with c20 independent of ν. Proof: By definition, ∗ ∗ hv,F11 vi−q = hφ,F11 vi = hF11 φ,vi = hF11 φ,φiq Z q X 1 = (1 + x2 )2q (φψ)(k) ( aφ00 ψ)(k) dx 2 k=0 R q Z X + (1 + x2 )2q (φψ)(k) (bφ0 ψ)(k) dx,

(A.5)

k=0 R

where the third equality follows from the fact that F11 φ ∈ Φq . By Lemma A.1, the term involving b is bounded by a constant times kφk2q ≡ kvk2−q .

THOMAS G. KURTZ AND JIE XIONG 00

355

0

2aψ For the term involving a, let a1 = aψ ψ and a2 = ψ . Then q Z X

=

(1 + x2 )2q (φψ)(k) (aφ00 ψ)(k) dx

k=0 R q Z X

(1 + x2 )2q (φψ)(k) (a(φψ)(2) − a1 (ψφ) − a2 (ψφ0 ))(k) dx

k=0 R q Z X

=−

k=0 R

(1 + x2 )2q (φψ)(k+1) a(φψ)(k+1) dx + O(kφk2q )

Z =−

R

a(1 + x2 )2q |(φψ)(q+1) |2 dx + O(kφk2q ),

where the second equality follows by integrating by parts and applying Lemma A.1.

Lemma A.3. There exists a constant c21 such that for ² > 0, v ∈ Φ−κ such that φ = θq v ∈ Φq+2 , and ν ∈ M(R), Z ∗ kF21 (ν,u)vk2−q µ(du) U Z Z 2 2q ≤ (1 + ²) (1 + x ) |α(·,ν,u)|2 µ(du)|(φψ)(q+1) |2 dx + c21 (1 + ²−1 )kvk2−q . R

U

Proof: Noting that ∗ hF21 v,γi hφ,F21 γiq = sup , kγkq kγkq γ∈Φq γ∈Φq

∗ kF21 (ν,u)vk−q = sup

by (A.3), ∗ kF21 (ν,u)vk2−q

³

sZ (1 + x2 )2q |α(x,ν,u)(φψ)(q+1) |2 dx

≤ R

+c19

sZ

X 1≤i,j≤q

´2 (1 + x2 )2q |α(i) (x,ν,u)(φψ)(j) |2 dx .

R

Consequently, there exists a constant c22 such that Z ∗ kF21 (ν,u)vk2−q ≤ (1 + ²) (1 + x2 )2q |α(x,ν,u)(φψ)(q+1) |2 dx R X Z −1 +c22 (1 + ² ) (1 + x2 )2q |α(i) (x,ν,u)(φψ)(j) |2 dx2 , 1≤i,j≤q R

and integrating with respect to µ, the boundedness of α(i) in L2 (U,µ) implies the existence of c21 . Lemma A.4. For v ∈ Φ−κ , ∗ 2hv,F12 vi−q ≤ c23 ρν kvk2−q .

356


Proof: Assume that φ = θq v ∈ Φq+2 . As in Lemma A.2, ∗ hv,F12 vi−q = hF12 φ,φiq .

Write hF12 (ν)φ,φiq = hd(·,ν)φ,φiq + hG1 (ν)φ,φiq

(A.6)

where Z G1 (ν)φ =

1 (∂d(x,ν,·)φ(x) + ∂a(x,ν,·)φ00 (x) + ∂b(x,ν,·)φ0 (x))ν(dx). 2 R

The boundedness of the derivatives of d implies that the first term on the right of (A.6) is O(kφkq ) uniformly in ν. Note that Z Z |φ(x)ψ(x)|2 ≤ |(φ(y)ψ(y))0 |2 (1 + y 2 )2 dy (1 + y 2 )−2 dy, R

R

so |φ(x)ψ(x)| ≤ O(kφk1 ). 0

00

(z)| (z)| Similarly, letting K1 = supz |ψψ(z) and K2 = supz |ψψ(z) ,

|φ0 (x)ψ(x)| ≤ |(φ(x)ψ(x))0 | + K1 |φ(x)ψ(x)| ≤ O(kφk2 ) and |φ00 (x)ψ(x)| ≤ |(φ(x)ψ(x))00 | + 2K1 |φ0 (x)ψ(x)| + K2 |φ(x)ψ(x)| ≤ O(kφk3 ). Consequently, Z

1 (k∂d(x,ν,·)kq |φ(x)ψ(x)| + k∂a(x,ν,·)kq |φ00 (x)ψ(x)| 2 R +k∂b(x,ν,·)kq |φ0 (x)ψ(x)|)ψ −1 (x)ν(dx) ≤ c31 ρν kφk3 ≤ c31 ρν kφkq = c31 ρν kvk−q .

kG1 (ν)φkq =

Combining this inequality with the estimate on the first term gives the result. Lemma A.5. There exists c24 such that for v ∈ Φ−κ , Z ∗ kF22 (ν,u)vk2−q µ(du) ≤ c24 ρ2ν kvk2−q .

(A.7)

U

∗ Proof: To bound kF22 vk−q , note that ∗ hγ,F22 vi hF22 γ,φiq kF22 γkq = sup ≤ sup kφkq . kγk kγk q q γ∈Φq+2 γ∈Φq+2 γ∈Φq+2 kγkq

∗ kF22 vk−q = sup

Then kF22 γkq ≤ c25 (u)ρν kγkq by the same argument used to estimate G1 in Lemma A.4, and c25 can be selected to be integrable with respect to µ.


357

Combining the previous lemmas we have Lemma A.6. For v ∈ Φ−κ , Z 2hv,F1∗ (ν)vi−q +

U

kF2∗ (ν,u)vk2−q µ(du) ≤ c18 ρ2ν kvk2−q .

(A.8)

Proof: Selecting ² > 0 so that (1 + ²)2 ≤ (1 + δ) for δ in (S4 ), Z ∗ 2hv,F1 (ν)vi−q + kF2∗ (ν,u)vk2−q µ(du) U Z ∗ ∗ ≤ 2hv,F11 (ν)vi−q + (1 + ²) kF21 (ν,u)vk2−q µ(du) U Z ∗ −1 ∗ +2hv,F12 (ν)vi−q + (1 + ² ) kF22 (ν,u)vk2−q µ(du) U ¶ Z µ Z 2 2 ≤− a(x,ν) − (1 + ²) |α(x,ν,u)| µ(du) (1 + x2 )2q |(φψ)(q+1) (x)|2 dx R

U

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