1.2. DERIVATIVE OPERATOR: DEFINITION AND PROPERTIES 3
a Λ. This implies that E(F ) = 0 and F is orthogonal to all polynomials in the
random variables W (ei). As a consequence,
(1.7) E

⎝F
exp

⎝i
n
j=1
λj W (ej
)⎠⎠
⎞⎞
= 0
for all n 1, and λ Rn. Therefore
E(F |W (e1), . . . , W (en)) = 0,
and letting n tend to infinity we get F = 0.
For each integer n 1, we will denote by Hn the closed subspace of
L2(Ω,
F, P )
spanned by the random variables {Φa, a Λ, |a| = n}. The space Hn is the
nth Wiener chaos associated with the Gaussian family W . Notice that H1 is the
Gaussian subspace formed by the random variables {W (h), h H}. We denote
by H0 the space of constant random variables. In this way we have obtained the
orthogonal decomposition
L2(Ω,
F, P ) =
⊕n=0Hn.∞
For each n 1, we will denote by Jn the orthogonal projection on the nth Wiener
chaos.
The following proposition implies that the nth Wiener chaos Hn does not de-
pend on the basis {ei, i 1}.
Proposition 1.5. For each n 1, the set of random variables
{Hn(W (h)), h
H
= 1}
is a total subset of Hn.
Proof. The set {Hn(W (h)), h
H
= 1, n 0} is total in L2(Ω, F, P ), be-
cause a random variable F orthogonal to this set is orthogonal to all the powers
{W (h)n, n 0}. So, it satisfies (1.7), and it must be zero. Therefore, if we denote
by Hn the closed span of the family {Hn(W (h)), h
H
= 1} we also have an or-
thogonal decomposition L2(Ω, F, P ) = ⊕n=0Hn. Finally, to show that Hn = Hn
it suffices to remark that ⊕n=0Hn N coincides with the set of polynomials in the el-
ements of the Gaussian family {W (h), h H} of degree less than or equal to N,
and, as a consequence, ⊕n=0Hn N ⊕n=0Hn.N
1.2. Derivative operator: Definition and properties
Let S denote the class of smooth and cylindrical random variables of the form
(1.8) F = f(W (h1), . . . , W (hn)),
where f belongs to Cp
∞(Rn)
(f and all its partial derivatives have polynomial growth
order), h1, . . . , hn are in H, and n 1.
The derivative of F is the H-valued random variable given by
(1.9) DF =
n
i=1
∂f
∂xi
(W (h1), . . . , W (hn))hi.
For example, D(W (h)) = h, and D(W
(h)2)
= 2W (h)h.
The following result is an integration-by-parts formula.
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