4 1. THE DERIVATIVE OPERATOR

Proposition 1.6. Suppose that F is a smooth and cylindrical random variable

and h ∈ H. Then

(1.10) E( DF, h

H

) = E(F W (h)).

Proof. We can restrict the proof to the case where there exist orthonormal

elements of H, e1, . . . , en, such that h = e1 and

F = f(W (e1), . . . , W (en)),

where f ∈ Cp

∞(Rn).

Let φ(x) denote the density of the standard normal distribution

on Rn given in (1.2). Then we have

E( DF, h

H

) = E

∂f

∂x1

(W (e1), . . . , W (en)) =

Rn

∂f

∂x1

(x)φ(x)dx

=

Rn

f(x)φ(x)x1dx = E(F W (e1)),

which completes the proof.

Applying the previous result to a product F G, we obtain the following conse-

quence.

Proposition 1.7. Suppose that F and G are smooth and cylindrical random

variables, and h ∈ H. Then we have

(1.11) E(G DF, h

H

) = E(−F DG, h

H

+ F GW (h)).

Proof. Use the formula D(F G) = F DG + GDF .

The integration-by-parts formula (1.11) is a useful tool to show the closability

of the derivative operator. In this sense, we have the following result.

Proposition 1.8. The operator D is closable from

Lp(Ω)

to

Lp(Ω;

H) for any

p ≥ 1.

Proof. Let {FN , N ≥ 1} be a sequence of random variables in S such that

FN converges to 0 in Lp(Ω), and DFN converges to η in Lp(Ω; H), as N tends to

infinity. Then, we claim that η = 0. Indeed, for any h ∈ H and for any random

variable F = f(W (h1), . . . , W (hn))e−W

(h)2

such that f is infinitely differentiable

with compact support and 0, we have

E( η, h

H

F ) = lim

N→∞

E( DFN , h

H

F )

= lim

N→∞

E(−FN DF, h

H

+ FN F W (h))

= 0.

This implies that η = 0.

We can define the iteration of the operator D in such a way that for a random

variable F ∈ S, the iterated derivative

DkF

is a random variable with values in

H⊗k.

For every p ≥ 1 and any natural number k ≥ 1 we introduce the seminorm

on S defined by

F

k,p

=

⎡

⎣E(|F|p)

+

k

j=1

E

(

Dj

F

p

H⊗j

)

⎤1/p

⎦

.