Finally, the adjoint K
t,T
to the linearization satisfies the bounds
E sup
t ∈I
δ
sup
kϕk≤1
kK
t,T
ϕk
p β+1
≤ C
p
Ψ
p
u δ
¯ p
β
p
, 72a
E sup
t ∈I
δ
sup
kϕk≤1
k∂
t
K
t,T
ϕk
p β
≤ C
p
Ψ
p
u δ
¯ p
β
p
, 72b
where ¯ p
β
is as in Proposition 3.9. In all these bounds, C
p
is a constant depending only on p and on the details of the equation 8.
Remark 6.3 One can assume without loss of generality, and we will do so from now on, that the exponent q defining Ψ is greater or equal to n, the degree of the nonlinearity. This will be useful in
the proof of Lemma 6.16 below.
Proof. It follows immediately from Assumption C.1 that
E sup
t ∈[T 4,T ]
ku
t
k
p
≤ CΨ
p
u .
Combining this with Proposition 3.6 yields the first of the desired bounds with q = p
γ
. Here, Ψ is as
in Assumption C.1 and p
γ
is as in Proposition 3.6. Turning to the bound on
∂
t
v
t
, observe that v satisfies the random PDE ∂
t
v
t
= F v
t
+ GW t = F u
t
, v
= u .
It follows at once from Proposition 3.6 and Assumption A.1.2 that the quoted estimate holds with q = p
γ+1
. More precisely, it follows from Proposition 3.6 that u
t
∈ H
α
for every α γ
⋆
+ 1. Therefore, Lu
t
∈ H
γ
for γ γ
⋆
. Furthermore, N ∈ PolyH
γ+1
, H
γ
by Assumption A.1.2, so that N u
t
∈ H
γ
as well. The claim then follows from the a priori bounds obtained in Proposition 3.6.
Concerning the bound 71a on the linearization J
0,t
, Proposition 3.7 combined with Assumption C.1 proves the result with q = ¯
q
γ
+ 1. The line of reasoning used to bound k∂
t
v
t
k
γ
also controls k∂
t
J
s,t
k
γ
for s t and s, t ∈ I
δ
, since ∂
t
J
s,t
= −LJ
s,t
+ DN u
t
J
s,t
. Since Proposition 6.1 give an completely analogous bound for K
s,t
in H as for J
s,t
the results on K follow from the a priori bounds in Proposition 3.9.
6.2 A Hörmander-like theorem in infinite dimensions
In this section, we are going to formulate a lower bound on the Malliavin covariance matrix M
t
under a condition that is very strongly reminiscent of the bracket condition in Hörmanders celebrated “sums of squares” theorem [Hör85, Hör67]. The proof of the result presented in this section will be
postponed until Section 6.3 and constitutes the main technical result of this work. Throughout all of this section and Section 6.3, we are going to make use of the bounds outlined in
Proposition 6.2. We therefore now fix once and for all some choice of constants γ ∈ [−a, γ
⋆
and β ∈ [−a, β
⋆
satisfying γ + β ≥ −1 .
73 700
From now on, we will only ever use Proposition 6.2 with this fixed choice for γ and β. This is purely
a convenience for expositional clarity since we will need these bounds only finitely many times. As a side remark, note that one should think of these constants as being arbitrarily close to
γ
⋆
and β
⋆
respectively. With
γ and β fixed as in 73, we introduce the set Poly
γ, β
def
= PolyH
γ
, H
−β−1
∩ PolyH
γ+1
, H
−β
74 for notational convenience. For integer m, Poly
m
γ, β is defined analogously. A polynomial Q
∈ Polyγ, β is said to be admissible if [Q
α
, F
σ
] ∈ Polyγ, β , for every pair of multi-indices
α, σ. Here, Q
α
and F
σ
are defined as in 11 and F is the drift term of the SPDE 8 defined in 9.
This definition allows us to define a family of increasing subsets A
i
⊂ Polyγ, β by the following recursion:
A
1
= {g
k
, k = 1, . . . , d
} ⊂ H
γ
⋆
+1
≈ Poly H
γ
⋆
+1
⊂ Polyγ, β , A
i+1
= A
i
∪ {Q
α
, [F
σ
, Q
α
] : Q ∈ A
i
, Q admissible, and α, σ multi-indices} .
Remark 6.4 Recall from 12 that Q
α
is proportional to the iterated “Lie bracket” of Q with g
α
1
, g
α
2
and so forth. Similarly, [F
σ
, Q
α
] is the Lie bracket between two different iterated Lie brackets. As such, except for the issue of admissibility, the set of brackets considered here is exactly the same as in
the traditional statement of Hörmander’s theorem, only the order in which they appear is slightly different.
To each A
N
we associate a positive symmetric quadratic form-valued function Q
N
by 〈ϕ, Q
N
uϕ〉 = X
Q ∈A
N
〈ϕ,Qu〉
2
. Lastly for
α ∈ 0, 1, and for a given orthogonal projection Π: H → H , we define S
α
⊂ H by S
α
= {ϕ ∈ H \ {0} : kΠϕk ≥ αkϕk} . 75
With this notation, we make the following non-degeneracy assumption:
Assumption C.2 For every α 0, there exists N 0 and a function Λ
α
: H → [0, ∞ such that
inf
ϕ∈S
α
〈ϕ, Q
N
uϕ〉 kΠϕk
2
≥ Λ
2 α
u , for every u
∈ H
a
. Furthermore, for every p ≥ 1, t 0 and every α ∈ 0, 1, there exists C such that
E Λ
−p α
u
t
≤ CΨ
p
u for every initial condition u
∈ H .
701
Remark 6.5 Assumption C.2 is in some sense weaker than the usual non-degeneracy condition of Hörmander’s theorem, since it only requires
Q
N
to be sufficiently non-degenerate on the range of Π. In particular, if Π = 0, then Assumption C.2 is void and always holds with Λ
α
= 1, say. This is the reason why, by choosing for Π a projector onto some finite-dimensional subspace of
H , one can expect Assumption C.2 to hold for a finite value of N , even in our situation where A
N
only contains finitely many elements.
Remark 6.6 As will be seen in Section 8, it is often possible to choose Λ
α
to be a constant, so that the second part of Assumption C.2 is automatically satisfied.
When Assumption C.2 holds, we have the following result whose proof is given in Section 6.3.
Theorem 6.7 Consider an SPDE of the type 8 such that Assumptions A.1 and C.1 hold. Let furthermore the Malliavin matrix
M
t
be defined as in 36 and S
α
as in 75. Let Π be a finite rank orthogonal projection satisfying Assumption C.2. Then, there exists
θ 0 such that, for every α ∈ 0, 1, every p ≥ 1 and every t
0 there exists a constant C such that the bound
P inf
ϕ∈S
α
〈ϕ, M
t
ϕ〉 kϕk
2
≤ ǫ ≤ CΨ
θ p
u ǫ
p
, holds for every u
∈ H and every ǫ ≤ 1.
Remark 6.8 If Π is a finite rank orthogonal projection satisfying Assumption C.2 then Theorem 6.7 provides the critically ingredient to prove the smoothness of the density of
P
∗ t
δ
x
Π
−1
with respect to Lebesgue measure. Though [BM07] contains a few unfortunate errors, it still provides the framework
needed to deduce smoothness of these densities from Theorem 6.7. In particular, one needs to prove that Πu
t
is infinitely Malliavin differentiable. Section 5.1 of [BM07] shows how to accomplish this in a setting close to ours, see also [MP06].
6.3 Proof of Theorem 6.7
