Proposition 3.10 For every 0 ≤ s t, K
s,t
is the adjoint of J
s,t
in H , that is K
s,t
= J
∗ s,t
. Proof. Fixing 0
≤ s t and ϕ, ψ ∈ H
∞
, we claim that the expression 〈J
s,r
ϕ, K
r,t
ψ〉 , 23
is independent of r ∈ [s, t]. Evaluating 23 at both r = s and r = t then concludes the proof.
We now prove that 23 is independent of r as claimed. It follows from 20 and Proposition 3.6 that, with probability one, the map r
7→ K
r,t
ϕ is continuous with values in H
β+1
and differentiable with values in
H
β
, provided that β β
⋆
. Similarly, the map r 7→ J
s,r
ψ is continuous with values in H
γ+1
and differentiable with values in H
γ
, provided that γ γ
⋆
. Since γ
⋆
+ β
⋆
−1 by assumption, it thus follows that 23 is differentiable in r for r
∈ s, t with ∂
r
〈J
s,r
ϕ, K
r,t
ψ〉 = 〈 L + DNu
r
J
s,r
ϕ, K
r,t
ψ〉 − 〈J
s,r
ϕ, L + DN
∗
u
r
K
r,t
ψ〉 = 0 . Since furthermore both r
7→ K
r,t
ϕ and r 7→ J
s,r
ψ are continuous in r on the closed interval, the proof is complete. See for example [DL92, p. 477] for more details.
3.4 Higher order variations
We conclude this section with a formula for the higher-order variations of the solution. This will mostly be useful in Section 8 in order to obtain the smoothness of the density for finite-dimensional
projections of the transition probabilities. For integer n
≥ 2, let ϕ = ϕ
1
, · · · , ϕ
n
∈ H
⊗n
and s = s
1
, · · · , s
n
∈ [0, ∞
n
and define ∨s =
s
1
∨ · · · ∨ s
n
. We will now define the n-th variation of the equation J
n s,t
ϕ which intuitively is the cumulative effect on u
t
of varying the value of u
s
k
in the direction ϕ
k
. If I =
{n
1
. . . n
|I|
} is an ordered subset of {1, . . . , n} here |I| means the number of elements in I , we introduce the notation s
I
= s
n
1
, . . . , s
n
|I|
and ϕ
I
= ϕ
n
1
, . . . , ϕ
n
|I|
. Now the n-th variation of the equation J
n s,t
ϕ solves ∂
t
J
n s,t
ϕ = −LJ
n s,t
ϕ + DN utJ
n s,t
ϕ + G
n s,t
ut, ϕ, t
∨s, 24
J
n s,t
ϕ = 0, t
≤ ∨s, where
G
n s,t
u, ϕ =
m ∧n
X
ν=2
X
I
1
,...,I
ν
D
ν
N u J
|I
1
| s
I1
,t
ϕ
I
1
, . . . , J
|I
ν
| s
Iν
,t
ϕ
I
ν
, 25
and the second sum runs over all partitions of {1, . . . , n} into disjoint, ordered non-empty sets
I
1
, . . . , I
ν
. The variations of constants formula then implies that
J
n s,t
ϕ = Z
t
J
r,t
G
n s,r
u
r
, ϕd r ,
26 see also [BM07]. We obtain the following bound on the higher-order variations:
679
Proposition 3.11 If β
⋆
a − 1 then there exists γ γ
⋆
+ 1 such that, for every n 0, there exist exponents N
n
and M
n
such that kJ
n s,t
ϕk ≤ C sup
r ∈[0,t]
1 + ku
r
k
γ N
n
sup
≤uv≤t
1 + kJ
u,v
k
M
n
, uniformly over all n-uples
ϕ with kϕ
k
k ≤ 1 for every k. Proof. We proceed by induction. As a shorthand, we set
E M, N = sup
r ∈[0,t]
1 + ku
r
k
γ N
sup
≤uv≤t
1 + kJ
u,v
k
M
. The result is trivially true for n = 1 with M
1
= 1 and N
1
= 0. For n 1, we combine 26 and 25, and we use Assumption A.1, part 2., to obtain
kJ
n s,t
ϕk ≤ C Z
t
kJ
r,t
k
−a→0
1 + ku
r
k
n
+ X
I
kJ
|I| s
I
,r
ϕ
I
k
n
d r ≤ CE nM
n −1
, nN
n −1
+ 1 Z
t
kK
r,t
k
→a
d r . To go from the first to the second line, we used the induction hypothesis, the fact that K
r,t
= J
∗ r,t
, and the duality between
H
a
and H
−a
. It remains to apply Proposition 3.9 with
β = a to obtain the required bound.
4 Malliavin calculus
In this section, we show that the solution to the SPDE 8 has a Malliavin derivative and we give an expression for it. Actually, since we are dealing with additive noise, we show the stronger result that
