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

1. For every n, the application f 7! I n (f) is linear.
2. For every n, one has E (I n (f)) = 0 and I n (f) = I n ( e f):
3. For every n 2 and m 1, for every f 2 E ( n ) and g 2 E ( m ) (if m = 1, one can take
g 2 L 2 ()),
0
if n 6= m
E [I n (f) I m (g)] =
n!hf; e egi L 2 ( n ) if n = m:
(4.12)
The proof of Proposition 4.2 follows almost immediately from the defnition (4.11); see e.g.
[65, Section 1.1.2] for a complete discussion. By combining (4.12) with (4.8), one infers that I n
can be extended to a linear continuous operator, from L 2 ( n ) into L 2 (P), verifying properties
1, 2 and 3 in the statement of Proposition 4.2. Moreover, the second line on the RHS of (4.12)
yields that the application
I n : L 2 s ( n ) ! L 2 (P) : f 7! I n (f)
(that is, the restriction of I n to L 2 s ( n )) is an isomorphism from L 2 s ( n ), endowed with the
modi…ed scalar product n! h; i L 2 ( n ) , into L2 (P). For every n 2, the L 2 (P)-closed vector
space
C n (G) = I n (f) : f 2 L 2 ( n ) (4.13)
is called the nth Wiener chaos associated with G. One conventionally sets
C 0 (G) = R. (4.14)
Note that (4.12) implies that C n (G) ? C m (G) for n 6= m, where “ ? ”indicates orthogonality
in L 2 (P) :
Remark (The case of Brownian motion). We consider the case where (Z; Z) = (R + ; B (R + ))
and is equal to the Lebesgue measure. As already observed, one has that the process t 7! W t =
G ([0; t]), t 0, is a standard Brownian motion started from zero. Also, for every f 2 L 2 (),
Z
Z 1
I 1 (f) = f (t) G (dt) = f (t) dW t ; (4.15)
R + 0
where the RHS of (4.15) indicates a standard Itô integral with respect to W . Moreover, for
every n 2 and every f 2 L 2 ( n )
I n (f) = n!
Z 1
Z t1
Z t2
0
0
0
Z tn 1
0
ef (t 1 ; :::; t n ) dW tn dW t2
dW t1 ; (4.16)
where the RHS of (4.16) stands for a usual Itô-type stochastic integral, with respect to W , of
the stochastic process
Z t Z t2
Z tn 1
t 7! ' (t) = n! ef (t 1 ; :::; t n ) dW tn dW t2 , t 1 0.
0 0 0
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