68.4. THE SKOROKHOD INTEGRAL 2345

Consider the terms involved. The first term is just E(∥∥F̂h

∥∥2H

)= E

(∥Fh∥2

H

). Now con-

sider the third term. It equals

−E(D(Dp+1

(F̂))

, F̂h)=−E

(D2

p+1(F̂)

F̂)

=−1(√2π)p+1

∫Rp

∫R

D2p+1(F̂ (x)

)F̂ (x)e−

12 |x|

2dxp+1dx̂p+1

=1(√

2π)p+1

∫Rp

∫R

Dp+1(F̂ (x)

)Dp+1

(F̂ (x)

)e−

12 |x|

2dxp+1dx̂p+1

− 1(√2π)p+1

∫Rp

∫R

Dp+1(F̂ (x)

)(xp+1F̂ (x)

)e−

12 |x|

2dxp+1dx̂p+1

= E((

Dp+1F̂)2)−E

(W (h)Dp+1

(F̂)

F̂)

= E((

Dp+1F̂)2)−E

(W (h)D

(F̂), F̂h

)Hence 68.4.23 reduces to

E(∥∥F̂h

∥∥2)+E

((Dp+1F̂

)2)

= E(∥∥F̂h

∥∥2)+E

(⟨D(F̂),h⟩2)

= E(∥Fh∥2

)+E

(⟨D(F) ,h⟩2

)This assumed that ∥h∥= 1. For arbitrary nonzero h,

E(

δ (Fh)2)

= ∥h∥2 E

(δ

(F

h∥h∥

)2)

= ∥h∥2

(E

(∥∥∥∥Fh∥h∥

∥∥∥∥2)+E

(⟨D(F) ,

h∥h∥

⟩2))

= E(∥Fh∥2

)+E

(⟨D(F) ,h⟩2

)Next consider a generalization, u = ∑

mj=1 Fjh j where the

{h j}

is an orthonormal set ofvectors. Say Fj = Fj

(W (k1) , · · · ,W

(kn j

)). Let

{h1, · · · ,hm,e1, · · · ,ep

}= {gi}m+p

i=1 be anorthonormal basis for the span of all the h j and ki. Thus gi = hi for i≤ m. Then let

Fj(W (k1) , · · · ,W

(kn j

))= F̂j (W (h1) , · · · ,W (hm) ,W (e1) , · · · ,W (ep))

The computations will be done with respect to this orthonormal set because it will be sim-pler. Also, the above argument using the density function for the normal distribution willbe used without explicitly repeating it.

It is desired to consider E(

δ (u)2)

. Recall that

D(δ (Fh)) = Fh+W (h)D(F)−D⟨DF,h⟩ .