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

Maxima. Suppose that X (h i ), i = 1; :::; m and h i 2 H, is a …nite subset of some isonormal
Gaussian process, such that the span of h i has dimension m. Consider
F =
max X (h i) :
i=1;:::;m
Then, F 2 D 1;2 (indeed, (z 1 ; :::; z m ) 7! max z i is Lipschitz), the random variable
I 0 = arg max X (h i)
i=1;:::;m
is well de…ned, and one has that
DF = h I0 .
This kind of results can also be extended to continuous-time Gaussian processes. For
isntance, if W = fW t : t 2 [0; 1]g is a standard Brownian motion initialized at zero, then
M = sup t2[0;1] W t 2 D 1;2 , and
D t M = 1 [0;T ] (t) ,
where T is the unique random point where W attains its maximum.
7.2 Divergences
7.2.1 De…nition and characterization of the domain
Let X = fX (h) : h 2 Hg be an isonormal Gaussian process over some real separable Hilbert
space H. We will now study the divergence operator , which is de…ned as the adjoint of the
derivative D. Recall that D is a closed and unbounded operator from D 1;2 L 2 ( (X)) into
L 2 ( (X) ; H), so that the domain of the operator will be some suitable subset of L 2 ( (X) ; H).
De…nition 7.3 The domain of the divergence operator , denoted by dom (), is the collection
of all random elements u 2 L 2 ( (X) ; H) such that, for every F 2 D 1;2 ,
E hu; DF i H
cE F 2 1=2
, (7.14)
where c is a constant depending on u (and not on F ). For every u 2 dom (), the random
variable (u) is therefore de…ned as the unique element of L 2 ( (X)) verifying
E hu; DF i H
= E [F (u)] ; (7.15)
for every F 2 D 1;2 (note that the existence of (u) is ensured by (7.14) and by the Riesz
Representation Theorem). Relation (7.15) is called an integration by parts formula.
Remark. By selecting F = 1 in (7.15), one deduces that E [ (u)] = 0, for every u 2 dom ().
Example. Fix h 2 H. Since, by Cauchy-Schwarz, E
hh; DF i H
khk H
E F 2 1=2 , we
deduce that h 2 dom () and also, thanks to (7.6), that (h) = X (h).
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