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

denote by F t the …ltration generated by W and by the P-null sets of (G), and we say that
a stochastic process u (t; !) is adapted, if u (t) 2 F t for every t 2 [0; 1]. If u is adapted and
R
1
E
0 u (t)2 dt < 1, then the Itô stochastic integral R 1
0 u (t) dW t is a well-de…ned element of
L 2 ( (X)). Moreover, in this case one has that
(u) =
Z 1
0
u (t) dW t
(see [65, Proposition 1.3.11]).
7.2.3 A formula on products
We conclude with a general (useful) formula involving products of Malliavin di¤erentiable random
variables and elements of dom (). The framework is that of a general isonormal process
X = fX (h) : h 2 Hg.
Proposition 7.5 Let F 2 D 1;2 and u 2 dom () be such that: (i) F u 2 L 2 ( (X) ; H), (ii)
F (u) 2 L 2 ( (X)), and (iii) hDF; ui H
2 L 2 ( (X)). Then, F u 2 dom (), and also
(F u) = F (u) hDF; ui H
: (7.20)
Proof. Consider a random variable G equal to the RHS (7.2), with f 2 C0
1
E
hDG; F ui H
= E
hF DG; ui H = E hD (F G) GDF; uiH
= E
F (u) hDF; ui H G .
(Rm ). Then,
Since random variables such as G generate (X), the conclusion is obtained.
7.3 The Ornstein-Uhlenbeck Semigroup and Mehler’s formula
7.3.1 De…nition, Mehler’s formula and vector-valued Markov processes
Let X = fX (h) : h 2 Hg be an isonormal Gaussian process over some real separable Hilbert
space H.
De…nition 7.4 The Ornstein-Uhlenbeck semigroup fT t : t 0g is the set of contraction
operators de…ned as
T t (F ) = E (F ) +
1X
e qt I q (f q ) =
q=1
1X
e qt I q (f q ) ; (7.21)
q=0
for every t 0 and every F 2 L 2 ( (X)) as in (6.6):
The Ornstein-Uhlenbeck semigroup plays a fundamental role in our theory. Its relevance for
Stein’s method is not new: see for instance the so-called “Barbour-Götze generator approach”,
introduced in [2] and [29] (see [80] for a survey). As another example, see [57], [62] and the
discussion contained in Section 9, where it is shown that the use of the semigroup fT t g leads
31