In order to obtain the second bound, we write kK
T −δ,T
ϕ − ϕk
−1
≤ kK
T −δ,T
ϕ − e
−Lδ
ϕk
−1
+ ke
−Lδ
ϕ − ϕk
−1
≤ kK
T −δ,T
ϕ − e
−Lδ
ϕk + Cδ , where the last inequality is again a consequence of standard analytic semigroup theory. The claim
then follows from 78a.
6.7 The smallness of
M
T
implies the smallness of Q
N
u
T −δ
In this section, we show that if 〈M
T
ϕ, ϕ〉 is small then 〈Q
N
u
t
K
t,T
ϕ, K
t,T
ϕ〉 must also be small with high probability for every t
∈ I
δ
. The precise statement is given by the following result:
Lemma 6.17 Let the Malliavin matrix M
T
be defined as in 36 and assume that Assumptions A.1 and C.1 are satisfied. Then, for every N
0, there exist r
N
0, p
N
0 and q
N
0 such that, provided that r
≤ r
N
, the implication 〈ϕ, M
T
ϕ〉 ≤ ǫkϕk
2
=⇒ sup
Q ∈A
N
sup
t ∈I
δ
|〈K
t,T
ϕ, Qu
t
〉| ≤ ǫ
p
N
kϕk , holds modulo some Ψ
q
N
-dominated negligible family of events. Proof. The proof proceeds by induction on N and the steps of this induction are the content of the
next two subsections. Since A
1
= {g
1
, . . . , g
d
}, the case N = 1 is implied by Lemma 6.18 below, with p
1
= 14, q
1
= 8, and r
1
= 18¯ p
β
. The inductive step is then given by combining Lemmas 6.21 and 6.24 below. At each step, the values
of p
n
and r
n
decrease while q
n
increases, but all remain strictly positive and finite after finitely many steps.
6.8 The first step in the iteration
The “priming step” in the inductive proof of Lemma 6.17 follows from the fact that the directions which are directly forced by the Wiener processes are not too small with high probability.
Lemma 6.18 Let the Malliavin matrix M be defined as in 36 and assume that Assumptions A.1 and
C.1 are satisfied. Then, provided that r ≤ 18¯p
β
, the implication 〈ϕ, M
T
ϕ〉 ≤ ǫkϕk
2
=⇒ sup
k=1...d
sup
t ∈I
δ
|〈K
t,T
ϕ, g
k
〉| ≤ ǫ
1 4
kϕk , holds modulo some Ψ
8
-dominated negligible family of events. Here, ¯ p
β
is as in 72b and β was fixed in
73. Proof. For notational compactness, we scale
ϕ to have norm one by replacing ϕ with ϕkϕk. We will still refer to this new unit vector as
ϕ. Now assume that 〈ϕ, M
T
ϕ〉 ≤ ǫ. It then follows from 36 that
sup
k=1...d
Z
I
δ
〈g
k
, K
t,T
ϕ〉
2
d t ≤ ǫ .
707
Applying Lemma 6.14 with f t = R
t T
2
|〈g
k
, K
s,T
ϕ〉| ds and α = 1, it follows that there exists a constant C
0 such that, for every k = 1 . . . d, either sup
t ∈I
δ
|〈g
k
, K
t,T
ϕ〉| ≤ ǫ
1 4
, or
|||〈g
k
, K
·,T
ϕ〉|||
1
≥ Cǫ
−14
. 80
Therefore, to complete the proof, we need only to show that the latter events form a Ψ
4
-dominated negligible family for every k. Since
|||〈g
k
, K
·,T
ϕ〉|||
1
≤ kg
k
k
−β
|||K
·,T
ϕ|||
1, β
, the bound 80 implies that sup
ϕ∈H : kϕk=1
|||K
t,T
ϕ|||
1, β
≥ C
ǫ
−14
g
∗
, 81
where g
∗
= max
k
kg
k
k
−β
which is finite since we have by assumption that −β ≤ γ + 1 γ
⋆
+ 1 and since g
k
∈ H
γ
⋆
+1
for every k . This event depends only on the initial condition u and on the model
under consideration. In particular, it is independent of ϕ.
The claim now follows from the a priori bound 72b and Lemma 6.13 with q =
1 4
and b = ¯ p
β
.
6.9 The iteration step
