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

Cp
1 (R n ) is the class of in…nitely di¤erentiable functions f on R n such that f and its
derivatives have polynomial growth;
Cb
1 (R n ) is the class of in…nitely di¤erentiable functions f on R n such that f and its
derivatives are bounded;
C0
1 (Rn ) is the class of in…nitely di¤erentiable functions f on R n such that f has compact
support.
7.1 Derivatives
7.1.1 De…nition and characterization of the domain
Let X = fX (h) : h 2 Hg be an isonormal Gaussian process. The Malliavin derivative operator
of order k transforms elements of L 2 ( (X)) into elements of L 2 (X) ; H k : Formally, one
starts by de…ning the class S (X) L 2 ( (X)) of smooth functionals of X, as the collection of
random variables of the type
F = f (X (h 1 ) ; :::; X (h m )) , (7.2)
where f 2 C 1 p (R n ) and h i 2 H.
De…nition 7.1 Let F 2 S (X) be as in (7.2).
1. The derivative DF of F is the H-valued random element given by
mX @
DF = f (X (h 1 ) ; :::; X (h m )) h i ; (7.3)
@x i
i=1
we shall sometimes use the notation DF = D 1 F .
2. For k 2, the kth derivative of F , denoted by D k F , is the element of L 2 (X) ; H k
given by
D k F =
kX
i 1 ;:::;i k =1
@ k
@x i1 @x ik
f (X (h 1 ) ; :::; X (h m )) h i1 h ik . (7.4)
Example. Let h 2 H be such that khk H
= 1. Then, for every q 1, I q (h) = H q (X (h)),
where H q is the qth Hermite polynomial, and one has therefore that
D k q (q 1) (q k + 1) Hq
I q (h) =
k (X (h)) h k , if k q
0, if k > q:
In particular, DX (h) = h.
Remarks. (a) The polynomial growth condition implies that for every F 2 S (X), every
k 1 and every h 2 H k , the real-valued random variables F and D k F; h have …nite
H k
moments of all orders.
(b) If, F; J 2 S (X), then F J 2 S (X) and
D (F J) = J DF + F DJ: (7.5)
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