6.1 Bounds on the dynamic
As the previous sections have shown, it is sufficient to have control on the moments of u and J in H to control their moments in many stronger norms. This motivates the next assumption. For the
entirety of this section we fix a T 0.
Assumption C.1 There exists a continuous function Ψ :
H → [1, ∞ such that, for every T ∈ 0, T ]
and every p ≥ 1 there exists a constant C such that
E sup
T ≤t≤2T
ku
t
k
p
≤ CΨ
p
u ,
E sup
T ≤st≤2T
kJ
s,t
k
p
≤ CΨ
p
u ,
for every u ∈ H . Here, kJk denotes the operator norm of J from H to H .
Under this assumption, we immediately obtain control over the adjoint K
s,t
.
Proposition 6.1 Under Assumption C.1 for every T ∈ 0, T
] and every p ≥ 1 there exists a constant C such that
E sup
T ≤st≤2T
kK
s,t
k
p
≤ CΨ
p
u ,
for every u ∈ H .
Proof. By Proposition 3.10 we know that K
s,t
is the adjoint of J
s,t
in H . Combined with Assump-
tion C.1 this implies the result. In the remainder of this section, we will study the solution to 8 away from t = 0 and up to some
terminal time T which we fix from now on. We also introduce the interval I
δ
= [
T 2
, T − δ] for
some δ ∈ 0,
T 4
] to be determined later. Given u
t
a solution to 8, we also define a process v
t
by v
t
= u
t
− GW t, which is more regular in time. Using Assumption C.1 and the a priori estimates from the previous sections, we obtain:
Proposition 6.2 Let Assumption C.1 hold and Ψ be the function introduced there. For any fixed
γ γ
⋆
and β β
⋆
there exists a positive q so that if Ψ = Ψ
q
then the solutions to 8 satisfy the following bounds for every initial condition u
∈ H :
E sup
t ∈I
δ
ku
t
k
p γ+1
≤ C
p
Ψ
p
u ,
70a
E sup
t ∈I
δ
k∂
t
v
t
k
p γ
≤ C
p
Ψ
p
u .
70b Furthermore, its linearization J
0,t
is bounded by
E sup
t ∈I
δ
sup
kϕk≤1
kJ
0,t
ϕk
p γ+1
≤ C
p
Ψ
p
u ,
71a
E sup
t ∈I
δ
sup
kϕk≤1
k∂
t
J
0,t
ϕk
p γ
≤ C
p
Ψ
p
u .
71b
699
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
Kategori