Table Of Content

Degenerate SDEs in Hilbert Spaces with Rough Drifts 5 ∗ 1 0 2 Feng-Yu Wanga),b) and Xicheng Zhangc) n a a) School of Mathematical Sciences, Beijing Normal University, Beijing 100875,China J 7 b) Department of Mathematics, Swansea University, Singleton Park,SA2 8PP, UK 1 c) School of Mathematics and Statistics, Wuhan University, Wuhan 430072,China ] R January 20, 2015 P . h t a m Abstract [ 1 The existence and uniqueness of mild solutions are proved for a class of degenerate v stochastic differential equations on Hilbert spaces where the drift is Dini continuous in 0 the component with noise and Hölder continuous of order larger than 2 in the other 5 3 1 component. Inthefinite-dimensionalcasetheDinicontinuity isfurtherweakened. The 4 main results are applied to solve second order stochastic systems driven by space-time 0 white noises. . 1 0 AMS subject Classification: 60H15, 35R60. 5 1 Keywords: : v Degenerate evolution equation, mild solution, regularization transform. i X r a 1 Introduction Let H (i = 1,2,3) be separable Hilbert spaces, and let L(H ;H ) be the class of all bounded i i j linear operators from H to H (1 i,j 3). We shall simply denote the norm and inner i j ≤ ≤ product by and , for Hilbert spaces, and let stand for the operator norm. |·| h· ·i k·k Let W be a cylindrical Brownian motion on H ; i.e. for an orthonormal basis h on t 3 i i≥1 H { } , we have 3 W = Bih , t t i i≥1 X where Bi is a family of independent one-dimensional Brownian motions. Let F { t}i≥1 { t}t≥0 be the natural filtration induced by W . t ∗ FW is supported in part by NNSFC (11131003, 11431014), the 985 project and the Laboratory of Mathematical and Complex Systems, XZ is supported partly by NNSFC (11271294,11325105). 1 H H H We consider the following degenerate stochastic evolution equation on := : 1 2 × dX = A X +BY dt, t 1 t t (1.1) (dYt = (cid:8)bt(Xt,Yt)+A(cid:9)2Yt dt+σtdWt, where B L(H ;H ), σ : [0, (cid:8)) L(H ;H ),(cid:9)b : [0, ) H H are measurable 2 1 3 2 2 ∈ ∞ → ∞ × → and locally bounded, and for every i = 1,2, (A ,D(A )) is a bounded above linear operator i i generating a strongly continuous semigroup etAi on H . We will let , (1) and (2) denote i ∇ ∇ ∇ the gradient operators on H,H and H respectively. 1 2 Definition 1.1. A continuous adapted process (X ,Y ) is called a mild solution to (1.1) t t t∈[0,ζ) with life time ζ, if ζ > 0 is an F -stopping time such that P-a.s. limsup ( X + Y ) = holds on ζ < and, P-a.s. fort all t [0,ζ), t↑ζ | t| | t| ∞ { ∞} ∈ t X = etA1X + e(t−s)A1BY ds, t 0 s  Z0 t t  Y = etA2Y + e(t−s)A2b (X ,Y )ds+ e(t−s)A2σ dW . t 0 s s s s s Z0 Z0   The purposeof this paper is to investigate the existence/uniqueness of the mild solution under some Dini’s type continuity conditions on the drift b. The main idea is to construct a map which transforms the original equation into an equation with regular enough coefficients ensuring the pathwise uniqueness of thesolution. This idea goesback to [13, 15] where finite- dimensional SDEs with singular drift are investigated, see also [7, 14] for further develop- ments. Inrecent years, thisargument hasbeen developed in[1,2,3, 4, 11] fornon-degenerate SDEs in Hilbert spaces. The main difficulty of the study for the present degenerate equation is that the semigroup P0 associated to the linear equation (i.e. b = 0) has worse gradient t estimates with respect to x H . More precisely, unlike in the non-degenerate case one has 1 ∈ P0 t−1/2 for small t > 0 which is integrable over [0,1], for the present model one k∇ t k∞→∞ ≈ has (1)P0 t−3/2 which is not integrable, where P := sup Pf for k∇ t k∞→∞ ≈ k k∞→∞ kfk∞≤1k k∞ a linear operator P and the uniform norm . To reduce the singularity for small t > 0, ∞ k·k we will use some other norms to replace . This leads to different type continuity ∞→∞ k · k conditions on b. Indeed, we will need the Hölder continuity of b in the first component x H , and a Dini type continuity of b in the second component y H as in [11] where 1 2 ∈ ∈ the non-degenerate equation is concerned. To ensure the required gradient estimates on P0, we make the following assumptions on t the linear part. (H1) σ σ∗ is invertible in H with locally bounded (σ σ∗)−1 in t 0. t t 2 k t t k ≥ (H2) BB∗ is invertible in H , and BetA2 = etA1etA0B for some A L(H ,H ) and all t 0. 1 0 1 1 ∈ ≥ (H3) A is self-adjoint having discrete spectrum 0 < λ λ counting multiplicities 2 1 2 − ≤ ≤ ··· such that 1 < for some δ (0,1). i≥1 λ1−δ ∞ ∈ i P 2 Since σ is locally bounded in t 0, B is bounded, and A is bounded above so that t 1 ≥ eA1t ect holds for some constant c 0, it is well known from [5] that (H3) implies the k k ≤ ≥ existence, uniqueness and non-explosion of a continuous mild solution to the linear equation, i.e. (1.1) with b = 0. As Itô’s formula does not apply directly to the mild solution, in the study we will make finite-dimensional approximations. Throughout the paper, we let e i i≥1 { } be the eigenbasis of A , which is an orthonormal basis in H such that A e = λ e . For any 2 2 2 i i i n 1, let H(n) = span e , ,e , and let π(n) : H H(n) be the orthogon−al projection. ≥ 2 { 1 ··· n} 2 2 → 2 Next, let H(n) = BH(n) and π(n) : H H(n) be the orthogonal projection. Since BB∗ is 1 2 1 1 → 1 invertible, we have lim π(n)x = x for x H . Let n→∞ 1 ∈ 1 π(n) = (π(n),π(n)) : H H := H(n) H(n). 1 2 → n 1 × 2 In our study of finite-dimensional approximations, we will need the following assumption which is trivial in the finite-dimensional setting. (H4) There exists n 1 such that for any n n , π(n)B = Bπ(n) on H , and π(n)A = 0 ≥ ≥ 0 1 2 2 1 1 A π(n) on D(A ). 1 1 1 We introduce the following classes of functions to characterize the continuity modulation of the drift b: D := φ : [0, ) [0, ) is increasing with φ(0) = 0 and φ(s) > 0 for s > 0 , 0 ∞ → ∞ n 1 φ(s) o D := φ D : φ2 is concave and ds < . 1 0 ∈ s ∞ (cid:26) Z0 (cid:27) We remark that the condition 1 φ(s)ds < is well known as Dini’s condition, due to 0 s ∞ the notion of Dini’s continuity. Obviously, the class D contains φ(s) := K for R 1 {log(c+s−1)}1+r constants K,r > 0 and large enough c e such that φ2 is concave. ≥ Theorem 1.1. Assume (H1)-(H4). (1) If for any n 1 there exist α (2,1), φ D and a constant K > 0 such that ≥ n ∈ 3 n ∈ 1 n b (x,y) b (x′,y′) K x x′ αn +φ ( y y′ ), t [0,n], (x,y) (x′,y′) n, t t n n | − | ≤ | − | | − | ∈ | |∨| | ≤ then for any (X ,Y ) H, the equation (1.1) has a unique mild solution (X ,Y ) 0 0 t t t∈[0,ζ) ∈ with life time ζ. (2) If moreover (1.2) b (x,y +y′),y ℓ ( x 2 + y 2)+h ( y′ ), x H ,y,y′ H ,t 0 t t t 1 2 h i ≤ | | | | | | ∈ ∈ ≥ holds for some increasing function ℓ,h : [0, ) [0, ) (0, ) such that ∞ ds = ∞ × ∞ → ∞ 1 ℓt(s) holds for all t 0, then the unique mild solution is non-explosive, i.e. the life time ζ∞= P-a.s. ≥ R ∞ To illustrate this result, we consider the following example of second order stochastic system driven by white noise. 3 Example 1.1. Let D Rd be a bounded open domain, and let ∆ be the Dirichlet Laplace ⊂ operator on D. Consider the equation ∂2u(t,x) =b u(t,x),∂ u(t,x)+( ∆)θu(t, )(x) ( ∆)2θu(t, )(x) t t t − · − − · (1.3) W(dt,dx) (cid:0)2( ∆)θ∂ u(t, )(x)+ , (cid:1)t 0,x D. t − − · dtdx ≥ ∈ Here, θ > 0 is a constant, W is a Brownian sheet (space-time white noise) on Rd, and b : [0, ) R2 R is measurable such that for any T > 0, ∞ × → (1.4) b (u,v) b (u′,v′) C u u′ α +φ( v v′ ) , t [0,T],u,v,u′,v′ R t t | − | ≤ | − | | − | ∈ ∈ holds for some constants C > 0,α (cid:0)(2,1), and some φ D(cid:1) . ∈ 3 ∈ 1 To solve this equation using Theorem 1.1, we take H = L2(D;dx) for i = 1,2,3, and i ∞ W = e e (x)W(ds,dx) t i i i=1 Z[0,t]×D X for e the unitary eigenbasis of ∆. Letting i i≥1 { } X = u(t, ), Y = ∂ u(t, )+( ∆)θu(t, ), t t t · · − · we reformulate (1.3) as dX = Y ( ∆)θX dt, t t t (1.5) − − (dYt = (cid:8)bt(Xt,Yt) ( (cid:9)∆)θYt dt+dWt. − − (cid:8) (cid:9) Obviously, assumptions (H1), (H2) and (H4) hold for B = σ = I (the identity operator) t and A = A = ( ∆)θ. Moreover, since the eigenvalues of ( ∆)θ satisfy λ ci2θ/d for 1 2 i − − − ≥ some constant c > 0 and all i 1, assumption (H3) holds for A = ( ∆)θ provided θ > d. ≥ 2 − − 2 Finally, by Jensen’s inequality it is easy to see that (1.4) implies b(f,g) b(f˜,g˜) C f f˜ +φ( g g˜ ) , f,g,f˜,g˜ L2(D;dx),t [0,T], L2(D) L2(D) L2(D) k − k ≤ k − k k − k ∈ ∈ (cid:0) (cid:1) where the constant C might be different if the volume of D is not equal to 1. Therefore, by Theorem 1.1, for any θ > d the equation (1.5) has a unique mild solution on L2(D;dx) 2 × L2(D;dx) which is non-explosive. Next, we consider the finite-dimensional case, i.e. consider the following degenerate SDE on Rm Rd: × dX = AX +BY dt, t t t (1.6) (dYt = b(cid:8)t(Xt,Yt)dt+(cid:9)σtdWt, 4 where A is an m m-matrix, B is an m d-matrix, σ : [0, ) L(H ;Rd) is measurable 3 and locally bound×ed, and W is a cylindric×al Brownian motio∞n o→n H . In this case, Theorem t 3 1.1 can be improved by using the following larger class D to replace D : 2 1 1 dt D := φ D : φ2 is concave, = , 2 ( ∈ 0 Z0 t 1+ t1 φ(ss)ds 2 ∞) which includes φ(s) := K for some constant(cid:0)K >R0 and lar(cid:1)ge enough constant c > 0. √log(c+s−1) Theorem 1.2. Let H = Rm and H = Rd be finite-dimensional. Assume that BB∗ and 1 2 σ σ∗ are invertible with (σ σ∗)−1 locally bounded in t 0. t t t t ≥ (1) If for any n 1 there exist α (2,1), φ D and a constant K > 0 such that ≥ n ∈ 3 n ∈ 2 n b (x,y) b (x′,y′) K x x′ αn +φ ( y y′ ), t [0,n], (x,y) (x′,y′) n, t t n n | − | ≤ | − | | − | ∈ | |∨| | ≤ then for any (X ,Y ) Rm+d, the equation (1.6) has a unique solution (X ,Y ) 0 0 t t t∈[0,ζ) ∈ with life time ζ. (2) If moreover (1.7) b (x,y),y ℓ ( x 2 + y 2), x Rm,y Rd,t 0 t t h i ≤ | | | | ∈ ∈ ≥ holds for some increasing function ℓ,h : [0, ) [0, ) (0, ) such that ∞ ds = ∞ × ∞ → ∞ 1 ℓt(s) holds for all t 0, then the unique mild solution is non-explosive, i.e. the life time ζ∞= P-a.s. ≥ R ∞ Example 1.2. Consider the following second order stochastic differential equation on Rd: d2X dX t = b X , t +σW˙ , dt2 t t dt t (cid:16) (cid:17) where W isthed-dimensional Brownian, σ Rd Rd isinvertible, andb : [0, ) Rd Rd t Rd is measurable such that for any T > 0 th∈e con⊗dition (1.4) holds for some c∞onst×ants×C >→0, α (2,1) and some function φ D . By letting m = d and Y = dXt, we reformulate this ∈ 3 ∈ 2 t dt equation as dX = Y dt, t t (1.8) (dYt = bt(Xt,Yt)dt+σdWt. According to Theorem 1.2, for any initial point this equation has a unique solution which is non-explosive. We would like to point out that in the finite-dimensional setting, the pathwise uniqueness for equation (1.6) with Hölder continuous drifts has been investigated in a preprint by 5 Chaudru de Raynal (http://hal.archives-ouvertes.fr/hal-00702532/document). However, we found some gaps in the proof, for instance, the probabilistic representation of the solution is wrongly used and this is crucial in related calculations. Obviously, in Theorem 1.2 the condition on b along the second component y is much weaker than the Hölder continuity. The remainder of the paper is organized as follows. In Section 2, we investigate gradient estimates on the semigroup P0 associated to the linear equation (i.e. b = 0). These gradient s,t estimates are then used in Section 3 to construct and study the regularization transform. In Section 4 we use the regularization transform to represent the mild solution to (1.1), which enables us to prove Theorems 1.1 and Theorem 1.2 in Section 5. 2 Gradient estimates on P0 s,t For any s 0, consider the linear equation ≥ dX0 = A X0 +BY0 dt, (2.1) s,t 1 s,t s,t (dYs0,t = A(cid:8)2Ys0,tdt+σtdW(cid:9)t, t ≥ s. By (H1)-(H3) and Duhamel’s formula, the unique solution of this equation starting at (x,y) H at time s is given by: ∈ t X0 = e(t−s)A1x+ e(t−r)A1BY0 dr, s,t s,r (2.2)  Zs t  Y0 = e(t−s)A2y + e(t−r)A2σ dW . s,t r r Zs   Toindicatethedependenceontheinitialpoint,wealsodenotethesolutionby(X0 ,Y0)(x,y). s,t s,t Let P0 be the Markov operator associated to (X0 ,Y0) , i.e. s,t s,t s,t P0 f(x,y) = Ef((X0 ,Y0)(x,y)), t s 0,(x,y) H,f B (H). s,t s,t s,t ≥ ≥ ∈ ∈ b By the Markov property, we have P0 P0 = P0 for 0 s r t. s,r r,t s,t ≤ ≤ ≤ We first present a Bismut type derivative formula for P0 . Let s,t t Qt = s(t s)esA0BB∗esA∗0ds, t > 0, − Z0 where A is in (H2). Since BB∗ is invertible and A is bounded, Q−1 is invertible for every 0 0 t t > 0, and for any T > 0 there exists a constant c > 0 such that c (2.3) Q−1 , t (0,T]. k t k ≤ t3 ∈ 6 Next, for any s [0,T) and v = (v ,v ) H, let 1 2 ∈ ∈ T T r Vsv,T := Q−T−1s v1 + T −se(r−s)A∗0Bv2dr , (cid:20) Zs − (cid:21) v d Φv (r) := e(r−s)A2 2 + (r s)(T r)B∗e(r−s)A∗0 Vv . s,T T s dr − − s,T (cid:20) − (cid:21) (cid:8) (cid:9) Theorem 2.1. For any s [0,T),v = (v ,v ) H, f B (H), and (x,y) H, there holds 1 2 b ∈ ∈ ∈ ∈ T (2.4) ( P0 f)(x,y) = E f (X0 ,Y0 )(x,y) σ∗(σ σ∗)−1Φv (r),dW . ∇v s,T s,T s,T r r r s,T r (cid:20) Zs (cid:21) (cid:0) (cid:1) (cid:10) (cid:11) Proof. We use the argument of coupling by change of measures as in [6] where the finite- dimensional case is considered. For any ε [0,1), let (Xε ,Yε ) solve the equation ∈ s,t s,t t≥s dXε = A Xε +BYε dt, Xε = x+εv , (2.5) s,t 1 s,t s,t s,s 1 (dYsε,t = (cid:8)A2Ysε,t −εΦvs,T(t(cid:9)) dt+σtdWt, Ysε,s = y +εv2. (cid:8) (cid:9) Noticing that d(Xε X0 ) = A (Xε X0 )+B(Yε Y0) dt, s,t − s,t 1 s,t − s,t s,t − s,t (d(Ysε,t−Ys0,t) = (cid:8)A2(Ysε,t −Ys0,t)−εΦvs,T(t) dt, (cid:9) by Duhamel’s formula and the defi(cid:8)nition of Φv , we have (cid:9) s,T t Yε Y0 = εe(t−s)A2v ε e(t−r)A2Φv (r)dr s,t − s,t 2 − s,T (2.6) Zs T t = εe(t−s)A2 T −sv2 −(t−s)(T −t)B∗e(t−s)A∗0Vsv,T . (cid:20) − (cid:21) On the other hand, by (H2), we also have t Xε X0 = εe(t−s)A1v + e(t−r)A1B(Yε Y0 )dr s,t− s,t 1 s,r − s,r (2.7) Zs t T r = εe(t−s)A1 v1 + e(r−s)A0 T −sBv2 −(r −s)(T −r)BB∗e(r−s)A∗0Vsv,T dr . (cid:20) Zs (cid:16) − (cid:17) (cid:21) In particular, by the definition of Vv , (2.6) and (2.7) imply s,T (2.8) (Xε ,Yε ) = (X0 ,Y0 ), ε (0,1). s,T s,T s,T s,T ∈ Now, since sup Φv (t) < , by Girsanov’s theorem, t∈[s,T]| s,T | ∞ t Wε := W ε σ∗(σ σ∗)−1Φv (r)dr, t [s,T] t t − r r r s,T ∈ Zs 7 is a cylindrical Brownian motion on H under the probability measure dP := R dP, where 2 ε ε T ε2 T (2.9) R := exp ε σ∗(σ σ∗)−1Φv (r),dW σ∗(σ σ∗)−1Φv (r) 2dr . ε r r r s,T r − 2 r r r s,T (cid:20) Zs Zs (cid:21) (cid:10) (cid:11) (cid:12) (cid:12) Hence, if we write (2.5) as (cid:12) (cid:12) dXε = A Xε +BYε dt, Xε = x+εv , s,t 1 s,t s,t s,s 1 (dYsε,t = A(cid:8)2Ysε,tdt+σtdW(cid:9)tε, Ysε,s = y +εv2, then by the weak uniqueness of the solution, we obtain Ps0,Tf(x+εv1,y +εv2) = EPε f(Xsε,T,Ysε,T) = E Rεf(Xsε,T,Ysε,T) . Combining this with (2.8) and (2.9), we ar(cid:2)rive at (cid:3) (cid:2) (cid:3) R 1 ( P0 f)(x,y) = limE ε − f (X0 ,Y0 )(x,y) ∇v s,T ε↓0 ε s,T s,T (cid:20) (cid:21) (cid:0) T (cid:1) = E f (X0 ,Y0 )(x,y) σ∗(σ σ∗)−1Φv (r),dW . s,T s,T r r r s,T r (cid:20) Zs (cid:21) (cid:0) (cid:1) (cid:10) (cid:11) The proof is finished. Remark 2.1. In formula (2.4), although the operator A does not appear explicitly, it is 1 used in (2.7) implicitly though assumption (H2). Of course, in the finite-dimensional case this assumption is not needed, see [6, 14]. We note that the derivative formula in Theorem 2.1 also applies to Hilbert-valued map f B (H;H˜) by expanding f along an orthonormal basis of H˜, where H˜ is a separable b ∈ Hilbert space. Moreover, by the semigroup property, formula (2.4) also implies high order derivative formulas. For instance, for t (s,T) and v,v˜ H, (2.2) implies ∈ ∈ t v := (X0 ,Y0) = e(t−s)A1v + e(t−r)A1Be(r−s)A2v dr, e(t−s)A2v . t ∇v s,t s,t 1 2 2 (cid:18) Zs (cid:19) Then by P0 f = P0 P0 f and (2.2), (2.4), we have s,T s,t t,T t P0 f = E (P0 f)(X0 ,Y0) σ∗(σ σ∗)−1Φv˜ (r),dW ∇v∇v˜ s,T ∇v t,T s,t s,t r r r s,t r (cid:20) Zs (cid:21) t(cid:10) (cid:11) = E P0 f(X0 ,Y0 ) σ∗(σ σ∗)−1Φv˜ (r),dW ∇v t,T s,t s,t r r r s,t r (cid:20) Zs (cid:21) (cid:0) (cid:1) t(cid:10) (cid:11) (2.10) = E P0 f (X0 ,Y0) σ∗(σ σ∗)−1Φv˜ (r),dW ∇vt t,T s,t s,t r r r s,t r (cid:20) Zs (cid:21) (cid:0) (cid:1) T (cid:10) (cid:11) = E f(X0 ,Y0 ) σ∗(σ σ∗)−1Φvt (r),dW s,T s,T r r r t,T r (cid:20) (cid:18)Zt (cid:19) (cid:10)t (cid:11) σ∗(σ σ∗)−1Φv˜ (r),dW . × r r r s,t r (cid:18)Zs (cid:19)(cid:21) (cid:10) (cid:11) 8 We will use (2.4) and (2.10) to estimate derivatives of P0 f for f B (H;H˜) in terms s,T ∈ b of the norm f(x,y) f(x′,y′) f := f + sup | − | , k kφ,ψ k k∞ (x,y)6=(x′,y′)∈H φ( x x′ )+ψ( y y′ ) | − | | − | where φ,ψ D and is the uniform norm. Let 0 ∞ ∈ k·k C (H;H˜) := f B (H;H˜) : f < . φ,ψ b φ,ψ ∈ k k ∞ n o Then (C (H;H˜), ) is a Banach space. In particular, for any α [0,1], if we let φ,ψ φ,ψ γ (s) = sα1 (s)k, t·hken for α,β [0,1], C (H;H˜) is the usual Hölder∈space and α (0,∞) ∈ γα,γβ f(x,y) f(x′,y′) f = f + sup | − | . k kγα,γβ k k∞ (x,y)6=(x′,y′)∈H x x′ α + y y′ β | − | | − | Note that f f . k kγ0,γ0 ≈ k k∞ Corollary 2.2. Assume (H1)-(H3) and let T > 0 be fixed. Let (i) denote the gradient ∇ operator on H ,i = 1,2. i (1) There exists a constant C > 0 such that for any α [0,1], ∈ C f (1)P0 f k kγα,γ0 , 0 s < t T,f C (H;H˜). k∇ s,t k∞ ≤ 3(1−α) ≤ ≤ ∈ γα,γ0 (t s) 2 − (2) There exists a constant C > 0 such that for any α [0,1] and φ D with φ2 concave, 0 ∈ ∈ C f k∇(2)Ps0,tfk∞ ≤ √ktkγαs,φ (t−s)α(22+δ) +φ C(t−s)δ2 − h i (cid:0) (cid:1) holds for all 0 s < t T and f C (H;H˜), where δ (0,1) is in (H3). In ≤ ≤ ∈ γα,φ ∈ particular, C f (2)P0 f k k∞, 0 s < t T,f B (H;H˜). k∇ s,t k∞ ≤ √t s ≤ ≤ ∈ b − Proof. (1) By the interpolation theorem (cf. [9, Theorem 1.2.1]), it suffices to prove it for α = 0,1. (1a) Let α = 1. For any v H , (2.2) implies 1 1 ∈ (2.11) (1)Y0 = 0, (1)X0 = e(t−s)A1v . ∇v1 s,t ∇v1 s,t 1 So, for any f C1(H;H˜), ∈ b (1)P0 f = E (1) f (X0 ,Y0) f e(t−s)A1v f v . |∇v1 s,t | (cid:12) (cid:20)(cid:18)∇∇(v11)Xs0,t (cid:19) s,t s,t (cid:21)(cid:12) ≤ k k1,0| 1| ≤ k k1,0| 1| (cid:12) (cid:12) (cid:12) (cid:12) (cid:12) 9 (cid:12) Thus, assertion (1) is proved for α = 1. (1b) Let α = 0 and 0 s < t T,v H. By (H1) and the definitions of Φv and Vv , ≤ ≤ ∈ s,t s,t there exists a constant C > 0 such that 1 t v 2 v 2 (2.12) σ∗(σ σ∗)−1Φv (r) 2dr C | 1| + | 2| . | r r r s,t | ≤ 1 (t s)3 t s Zs (cid:20) − − (cid:21) Combining this with (2.4) we obtain t P0 f 2 (P0 f 2) σ∗(σ σ∗)−1Φv (r) 2dr |∇v s,t | ≤ s,t| | | r r r s,t | (2.13) Zs v 2 v 2 C (P0 f 2) | 1| + | 2| . ≤ 1 s,t| | (t s)3 t s (cid:20) − − (cid:21) In particular, with v = 0 this implies assertion (1) for α = 0. 2 (2) For v H and (x,y) H, let (X0 ,Y0 ) = (X0 ,Y0 )(x,y) and 2 ∈ 2 ∈ s,t s,t s,t s,t t x˜ = e(t−s)A1x+ e(t−r)A1Be(r−s)A2ydr, y˜= e(t−s)A2y. Zs Moreover, let t r t ξ = e(t−r)A1Bdr e(r−r′)A2σr′dWr′, η = e(t−r)A2σrdWr. Zs Zs Zs Since E t σ∗(σ σ∗)−1Φv (r),dW = 0, applying (2.4) with v = (0,v ) and using (2.12), s r r r s,t r 2 we obtain R (cid:10) (cid:11) t (2)P0 f (x,y) E f(X0 ,Y0) f(x˜,y˜) σ∗(σ σ∗)−1Φv (r),dW |∇v2 s,t | ≤ s,t s,t − r r r s,t r (cid:12) (cid:20)n oZs (cid:21)(cid:12) (cid:12) t (cid:10) (cid:11) (cid:12) (2.14) (cid:12)f E ξ α+φ( η ) σ∗(σ σ∗)−1Φv (r),dW (cid:12) ≤ k(cid:12) kγα,φ | | | | r r r s,t r (cid:12) (cid:20)(cid:16) (cid:17)(cid:12)Zs (cid:12)(cid:21) C f v (cid:12) (cid:10) (cid:11)(cid:12) k kγα,φ| 2| E( ξ 2α +(cid:12)φ( η ))2. (cid:12) (cid:12) (cid:12) ≤ √t s | | | | − p Noting that (H3) implies t t e(t−r)A2σ 2 dr c e−2λi(t−r)dr k rkHS ≤ 1 Zs i≥1 Zs (2.15) X 1 e−2λi(t−s) (2λ (t s))δ c − c i − = c (t s)δ, 1 1 2 ≤ 2λ ≤ 2λ − i i i≥1 i≥1 X X for c := sup σ 2,c := 2δ−1c 1 < , by Jensen’s inequality we have 1 t∈[0,T]k tk 2 1 i≥1 λ1−δ ∞ i E ξ 2α c (t s)(2+δ)α, PEφ( η )2 (φ(E η ))2 φ [c (t s)]δ/2 2 3 2 | | ≤ − | | ≤ | | ≤ − for some constant c > 0. Combining this with (2.14) we pr(cid:0)ov(cid:0)e the first as(cid:1)s(cid:1)ertion in (2), 3 which implies the second assertion by taking α = 0 and φ = γ = 1. 0 10