Corollary 3.8. Let the assumptions of Theorem 3.1 prevail. Let Φ
Z
: R
H
→ H be measurable and such that DZ = Φ
Z
X . Then, for almost all z in supp ρ, the density ρ of the law of Z is given by ρz =
E|Z| 2
R
∞
e
−u
E 〈Φ
Z
X , Φ
Z
e
−u
X + p
1 − e
−2u
X
′
〉
H
|Z = z du
× exp
−
Z
z
x d x R
∞
e
−v
E 〈Φ
Z
X , Φ
Z
e
−v
X + p
1 − e
−2v
X
′
〉
H
|Z = x d v
.
Now, we give several examples of application of this corollary.
3.2.1 First example: monotone Gaussian functional, finite case.
Let N ∼ N
n
0, K with K positive definite, and f : R
n
→ R be a C
1
function having bounded derivatives. Consider an isonormal Gaussian process X over the Euclidean space H = R
n
, endowed with the inner product 〈h
i
, h
j
〉
H
= EN
i
N
j
= K
i j
. Here, {h
i
}
1¶i¶n
stands for the canonical basis of H = R
n
. Without loss of generality, we can identify N
i
with X h
i
for any i = 1, . . . , n. Set Z = f N −E f N . The chain rule 2.10 implies that Z ∈ D
1,2
and that DZ = Φ
Z
N = P
n i=1
∂ f ∂ x
i
N h
i
. Therefore
〈Φ
Z
X , Φ
Z
e
−u
X + p
1 − e
−2u
X
′
〉
H
=
n
X
i, j=1
K
i j
∂ f ∂ x
i
N ∂ f
∂ x
j
e
−u
N + p
1 − e
−2u
N
′
, for N
′ i
= X
′
h
i
, i = 1, . . . , n. In particular, Corollary 3.5 combined with Proposition 3.7 yields the following.
Proposition 3.9. Let N ∼ N
n
0, K with K positive definite, and f : R
n
→ R be a C
1
function with bounded derivatives. If there exist
α
i
, β
i
¾ 0 such that
α
i
¶
∂ f ∂ x
i
x ¶ β
i
for any i ∈ {1, . . . , n} and x ∈ R
n
, if K
i j
¾ 0 for any i, j ∈ {1, . . . , n} and if
P
n i, j=1
α
i
α
j
K
i j
0, then the law of Z = f N − E f N admits a density
ρ which satisfies, for almost all z ∈ R, E|Z|
2 P
n i, j=1
β
i
β
j
K
i j
exp −
z
2
2 P
n i, j=1
α
i
α
j
K
i j
¶ ρz ¶
E|Z| 2
P
n i, j=1
α
i
α
j
K
i j
exp −
z
2
2 P
n i, j=1
β
i
β
j
K
i j
.
3.2.2 Second example: monotone Gaussian functional, continuous case.
Assume that X = X
t
, t ∈ [0, T ] is a centered Gaussian process with continuous paths, and that f : R → R is C
1
with a bounded derivative. The Gaussian space generated by X can be identified with an isonormal Gaussian process of the type X = {X h : h ∈ H}, where the real and separable
2298
Hilbert space H is defined as follows: i denote by E the set of all R-valued step functions on [0, T ], ii define H as the Hilbert space obtained by closing E with respect to the scalar product
〈1
[0,t]
, 1
[0,s]
〉
H
= EX
s
X
t
. In particular, with such a notation, we identify X
t
with X 1
[0,t]
. Now, let Z = R
T
f X
v
d v − E
R
T
f X
v
d v . Then Z ∈ D
1,2
and we have DZ = Φ
Z
X = R
T
f
′
X
v
1
[0,v]
d v. Therefore 〈Φ
Z
X , Φ
Z
e
−u
X + p
1 − e
−2u
X
′
〉
H
= ZZ
[0,T ]
2
f
′
X
v
f
′
e
−u
X
w
+ p
1 − e
−2u
X
′ w
EX
v
X
w
d vd w. Using Corollary 3.5 combined with Proposition 3.7, we get the following.
Proposition 3.10. Assume that X = X
t
, t ∈ [0, T ] is a centered Gaussian process with continuous paths, and that f : R → R is C
1
. If there exists α, β, σ
min
, σ
max
0 such that α ¶ f
′
x ¶ β for all x ∈ R and
σ
2 min
¶ EX
v
X
w
¶ σ
2 max
for all v, w ∈ [0, T ], then the law of Z = R
T
f X
v
d v − E
R
T
f X
v
d v has a density
ρ satisfying, for almost all z ∈ R, E|Z|
2 β
2
σ
2 max
T
2
e
−
z2 2
α2 σ2 min
T 2
¶ ρz ¶
E|Z| 2
α
2
σ
2 min
T
2
e
−
z2 2
β2 σ2 max T
2
.
3.2.3 Third example: maximum of a Gaussian vector.
