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: