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
While the aim of this section is to prove Theorem 6.7, we begin with some preliminary definitions which will simplify its presentation. Many of the arguments used will rely on the construction of
“exceptional sets” of small probability outside of which certain intuitive implications hold. This justifies the introduction of the following notational shortcut:
Definition 6.9 Given a collection H = {H
ǫ
}
ǫ≤1
of subsets of the ambient probability space Ω, we will say that “H is a family of negligible events” if, for every p
≥ 1 there exists a constant C
p
such that
PH
ǫ
≤ C
p
ǫ
p
for every ǫ ≤ 1.
Given such a family H and a statement Φ
ǫ
depending on a parameter ǫ 0, we will say that “Φ
ǫ
holds modulo H” if, for every ǫ ≤ 1, the statement Φ
ǫ
holds on the complement of H
ǫ
. We will say that the family H is “universal” if it does not depend on the problem at hand. Otherwise,
we will indicate which parameters it depends on. Given two families H
1
and H
2
of negligible sets, we write H = H
1
∪ H
2
as a shortcut for the sentence “H
ǫ
= H
ǫ 1
∪ H
ǫ 2
for every ǫ ≤ 1.” Let us state the following useful fact, the proof of which is immediate:
702
Lemma 6.10 Let H
ǫ n
be a collection of events with n ∈ {1, . . . , Cǫ
−κ
} for some arbitrary but fixed constants C and
κ and assume that PH
ǫ k
= PH
ǫ ℓ
for any pair k, ℓ. Then, if the family {H
ǫ 1
} is negligible, the family
{H
ǫ
} defined by H
ǫ
= S
n
H
ǫ n
is also negligible.
Remark 6.11 The same statement also holds of course if the equality between probabilities of events is replaced by two-sided bounds with multiplicative constants that do not depend on k,
ℓ, and ǫ. An important particular case is when the family H depends on the initial condition u
to 8. We will then say that H is “Ψ-controlled” if the constant C
p
can be bounded by ˜ C
p
Ψ
p
u , where ˜
C
p
is independent of u
. In this language, the conclusion of Theorem 6.7 can be restated as saying that there exists
θ 0 such that, for every
α 0, the event inf
ϕ∈S
α
〈ϕ, M
T
ϕ〉 ≤ ǫkϕk
2
is a Ψ
θ
-controlled family of negligible events. Recall that the terminal time T was fixed once and for all and that the function Ψ was defined in Proposition 6.2. We further restate this as an implication
in the following theorem which is easily seen to be equivalent to Theorem 6.7:
Theorem 6.12 Let Π be a finite rank orthogonal projection satisfying Assumption C.2. Then, there exists
θ 0 such that for every α ∈ 0, 1, the implication ϕ ∈ S
α
=⇒ 〈ϕ, M
T
ϕ〉 ǫkϕk
2
holds modulo a Ψ
θ
-controlled family of negligible events.
6.4 Basic structure and idea of proof of Theorem 6.12
