1.2. DERIVATIVE OPERATOR: DEFINITION AND PROPERTIES 7
converges in
L2(Ω;
H) as N tends to infinity if and only if (1.12) holds. Hence,
using again that the operator D is closed it follows that F
D1,2
if and only if
(1.12) holds. Finally, (1.13) is also easy to check.
By iteration we obtain
Dk(JnF
) =
Jn−k(DkF
) for all k 2 and n k.
Furthermore,
(1.15) E(
DkF
2
H⊗k
) =

n=k
n(n 1) ··· (n k + 1) JnF
2
2
,
and F
Dk,2
if and only if
∑∞
n=1
nk
JnF
2
2
∞.
An immediate consequence of Proposition 1.12 is the fact that if F
D1,2,
and
DF = 0, then F = E(F ).
The following technical result is very useful to show that a given random vari-
able belongs to the space
D1,2.
Lemma 1.13. Let {Fn, n 1} be a sequence of random variables in D1,2 that
converges to F in L2(Ω) and such that
sup
n
E
(
DFn
2
H
)
∞.
Then F belongs to
D1,2,
and the sequence of derivatives {DFn, n 1} converges to
DF in the weak topology of
L2(Ω;
H).
Proof. By Proposition 1.12, to show that F belongs to D1,2 it suffices to check
that (1.12) holds true, and this is an immediate consequence of Fatou’s lemma:

m=1
m JmF
2
2
=

m=1
m lim
n→∞
JmFn
2
2
liminfn→∞

m=1
m JmFn
2
2
sup
n
E( DFn
2
2
) ∞.
There exists a subsequence {Fn(k), k 1} such that the sequence of derivatives
DFn(k) converges in the weak topology of L2(Ω; H) to some element α L2(Ω; H).
We claim that α = DF . In fact, for any random variable G in the Nth Wiener
chaos, N 0, and for any h H we have
lim
k→∞
E( DFn(k), h
H
G) = E( α, h
H
G).
On the other hand, Proposition 1.12 implies that
E( DFn(k), h
H
G) = E(JN ( DFn(k), h
H
)G)
converges to E(JN ( DF, h
H
)G) = E( DF, h
H
G) as k tends to infinity. Hence,
E( α, h
H
G) = E( DF, h
H
G),
which implies α = DF . Finally, for any weakly convergent subsequence of {DFn, n
1} the limit must be equal to DF by the preceding argument, and this implies the
weak convergence of the whole sequence to DF .
The chain rule can be extended to the case of a Lipschitz function:
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