Stein's method, Malliavin calculus and infinite-dimensional Gaussian

Proposition 6.1 1. For every q 1, the qth Wiener chaos C q (X) is a Hilbert subspace of
L 2 (P), and the application
h 7! I q (h) , h 2 H q ,
de…nes a Hilbert space isomorphism between H q , endowed with the scalar product q! h; i H q,
and C q (X).
2. For every q; q 0 0 such that q 6= q 0 , the spaces C q (X) and C q 0 (X) are orthogonal in
L 2 (P) :
3. Let F be a functional of the isonormal Gaussian process X satisfying E[F (X) 2 ] < 1:
then, there exists a unique sequence ff q : q 1g such that f q 2 H q , and
F = E (F ) +
1X
I q (f q ) =
q=1
1X
I q (f q ) , (6.6)
q=0
where we have used the notation I 0 (f 0 ) = E (F ), and the series converges in L 2 (P).
4. Suppose that H = L 2 (Z; Z; ), where is -…nite and non-atomic. Then, for q 2,
the symmetric power H q can be identi…ed with L 2 s (Z q ; Z q ; q ) and, for every f 2 H q ,
the random variable I q (f) coincides with the Wiener-Itô integral of f with respect to the
Gaussian measure given by A 7! X (1 A ), A 2 Z .
Remark. The combination of Point 1 and Point 2 in the statement of Proposition 6.1
implies that, for every q; q 0 1,
E I q (f) I q 0 f 0 = 1 q=q 0q! f; f 0 H q :
From the previous statement, one also deduces the following Hilbert space isomorphism:
L 2 ( (X)) '
1M
H q , (6.7)
q=0
where ' stands for a Hilbert space isomorphism, and each symmetric power H q is endowed
with the modi…ed scalar product q! h; i H q. The direct sum on the RHS of (6.7) is called the
symmetric Fock space associated with H.
6.3 Contractions and products
We start by introducing the notion of contraction in the context of powers of Hilbert spaces.
De…nition 6.3 Consider a real separable Hilbert space H, and let fe i : i 1g be an orthonormal
basis of H. For every n; m 1, every r = 0; :::; n ^ m and every f 2 H n and g 2 H m , we
de…ne the contraction of order r, of f and g, as the element of H n+m 2r given by
f r g =
1X
i 1 ;:::;i r=1
and we denote by f e r g its symmetrization.
hf; e i1 e ir i H r hg; e i1 e ir i H r ; (6.8)
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