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

.
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