8 Examples
In this section, we apply the abstract framework developed in this article to two concrete examples: the stochastic Navier-Stokes equations on a sphere and a class of stochastic reaction-diffusion
equations. The examples are chosen in order to highlight the techniques that can be used to verify the assumptions of our results and to get some idea of their scope of applicability. In particular, the
Navier-Stokes equations provide an example where bounds on the Jacobian are not very uniform, so that an initial condition dependent control is required in Assumption C.1. The stochastic reaction-
diffusion system on the other hand satisfies very strong a priori bounds, but Assumption A.1 is not verified with the usual choice
H = L
2
, so that one has to work a bit more to fit the equations into the framework presented here. Our strategy is as follows: in a first section, we provide a simplified
version of our results. We tried to find a formulation that strikes a balance between powerful results and easily verifiable assumptions. This general formulation will then be used by both of the examples
mentioned above.
8.1 A general formulation
The ‘general purpose’ theorem formulated in this section allows to obtain the asymptotic strong Feller property for a large class of semilinear SPDEs under a Hörmander-type bracket condition. Our first
assumption ensures that all the stability conditions of the previous sections can be verified.
Assumption D.1 The operator L has compact resolvent. Furthermore, there exists a measurable function V :
H → R
+
such that there exist constants c 0 and α 0 such that the bound
V u ≥ ckuk
α
, holds for all u
∈ H and such that the following bounds hold: There exists a constant C
0 and η
′
∈ [0, 1 such that
E expV u
1
≤ C exp η
′
V u .
105 We also require the following bounds on the Jacobian, as well as the second variation on the dynamic.
For every p 0 and every δ 0, there exists a constant C such that the bounds
sup
t ∈[0,1]
E
ku
t
k
p
≤ C exp δV u ,
106a
E sup
s,t ∈[0,1]
kJ
s,t
k
p
≤ C exp δV u ,
106b sup
s,t ∈[0,1]
E
kJ
2 s,t
k
p
≤ C exp δV u ,
106c hold for every u
∈ H . Our next assumption is simply a restatement of the Hörmander bracket condition considering only
constant ‘vector fields’, with the additional condition that the g
i
belong to H
∞
. This ensures that all the relevant brackets are in
H
∞
and hence admissible in the sense of Section 6.2.
718
Assumption D.2 The forcing directions g
i
belong to H
∞
. Furthermore, define a sequence of subsets of H recursively by A
= {g
j
: j = 1, . . . , d } and
A
k+1
def
= A
k
∪ {N
m
h
1
, . . . , h
m
: h
j
∈ A
k
} . Then, the linear span of A
∞
def
= S
n
A
n
is dense in H .
With these assumptions in hand, a simplified, yet sufficiently powerful for many uses, formulation of our main results is as follows:
Theorem 8.1 Consider the setting of equation 1 and assume that Assumptions A.1, D.1, and D.2 hold.
Then, there exist constants C, κ 0 and γ ∈ 0, 1 such that the Markov semigroup P
t
generated by 1 satisfies the bound
kDP
2n
ϕuk ≤ C e
κV u
p P
2n
ϕ
2
u + γ
n
p P
2n
kDϕk
2
u ,
107 for every integer n
0. In particular, it satisfies the asymptotic strong Feller property. Furthermore, if
β
⋆
a − 1, then for every m 0, every u ∈ H , and every linear map T : H → R
m
, the projections of the time-2 transition probabilities
T
∗
P
2
u, · have C
∞
densities with respect to Lebesgue measure on
R
m
.
Remark 8.2 The final times 1 and 2 appearing in the statement are somewhat arbitrary since it suffices to rescale the equation in time, which does not change any of our assumptions. We chose to
keep them in this way in order to avoid awkward notations in the proof.
In this result, the Hörmander-type assumption, Assumption C.2 is verified by using constant vector fields only. Before we turn to the proof of Theorem 8.1, we therefore present the following useful
little lemma:
Lemma 8.3 Let H be a separable Hilbert space and {g
i
}
∞ i=1
⊂ H a collection of elements such that its span is dense in
H . Define a family of symmetric bilinear forms Q
n
on H by 〈h, Q
n
h 〉 =
P
n i=1
〈g
i
, h 〉
2
. Let Π :
H → H be any orthogonal projection on a finite-dimensional subspace of H . Then, there exists N
0 and, for every α 0 there exists c
α
0 such that 〈h, Q
n
h 〉 ≥ c
α
kΠhk
2
for every h ∈ H with
kΠhk ≥ αkhk and every n ≥ N. Proof. Assume by contradiction that the statement does not hold. Then, there exists
α 0 and a sequence h
n
in H such that kΠh
n
k = 1, kh
n
k ≤ α
−1
, and such that lim
n →0
〈h
n
, Q
n
h
n
〉 → 0. Since kh
n
k ≤ α
−1
is bounded, we can assume modulo extracting a subsequence that there exists h ∈ H
such that h
n
→ h in the weak topology. Since Π has finite rank, one has kΠhk = 1. Furthermore, since the maps h
7→ 〈h,Q
n
h 〉 are continuous in the weak topology and since n 7→ 〈h,Q
n
h 〉 is increasing for
every n, one has 〈h,Q
n
h 〉 = lim
m →∞
〈h
m
, Q
n
h
m
〉 ≤ lim
m →∞
〈h
m
, Q
m
h
m
〉 = 0 , so that
〈h, g
i
〉 = 0 for every i 0. This contradicts the fact that the span of the g
i
is dense in H .
We are now in a position to turn to the proof of our general result. 719
Proof of Theorem 8.1. We show first that the supremum in 106a can easily be pulled under the expectation. Indeed, it follows from the variation of constants formula that we have the bound
ku
t
k ≤ kStu k + C
Z
s
t − s
−a
kNu
s
k
−a
ds + kW
L
tk , where W
L
is the stochastic convolution of G W with the semigroup S generated by L. It follows immediately from Hölder’s inequality that there exists a constant C and an exponent p
0 such that sup
t ≤1
ku
t
k ≤ ku k + C
Z
1
1 + ku
s
k
np
ds
1 p
+ sup
t ≤1
kW
L
tk . Combining this with 106a, we conclude immediately that for every p
0 and every δ 0 there exists C
0 such that
E sup
t ∈[0,1]
ku
t
k
p
≤ C exp δV u .
108 We now verify that Assumptions C.1 and C.2 are satisfied for our problem. It follows from 108 and
106b that for every δ 0, Assumption C.1 holds with the choice Ψ
u = exp δV u .
Furthermore, Assumption C.2 holds for every finite-rank orthogonal projection Π: H → H by
Assumption D.2 and Lemma 8.3. Note that the function Λ
α
is then constant, so that the condition on its moments is trivially satisfied. We can therefore apply Theorem 6.7 which states that for every
α ∈ 0, 1, every δ 0, every finite-rank projection Π, and every p ≥ 1 there exists a constant C such that the bound
P inf
ϕ∈S
α
〈ϕ, M
1
ϕ〉 kϕk
2
≤ ǫ ≤ C exp δV u
ǫ
p
, 109
holds for every u ∈ H and every ǫ ≤ 1.
Combining this statement with 105, we see that Assumption B.1 is satisfied with ¯ q = 8 for example
and Uu = exp δV u
with every δ ≤
1 8
. The bound 105 is nothing but a restatement of Assumption B.2. Since we assume that 106b
and 106c hold for every δ 0, we infer that Assumption B.3 holds with ¯p = 20 and η sufficiently
small. It remains to verify that, for every C
Π
0 there exists a finite-rank projection Π such that 40 is satisfied. This ensures that the required relation C
Π
C
J
+ 2ηC
L
1 − η
′
can be satisfied by a suitable choice of Π.
Because L has compact resolvent by assumption, it has a complete system of eigenvectors with the corresponding eigenvalues
{λ
n
} satisfying lim
n →∞
λ
n
= ∞. Therefore, if we denote by Π
N
the projection onto the subspace of
H spanned by the first N eigenfunctions, we have the identity ke
−Lt
Π
⊥ N
k = e
−λ
N +1
t
. This allows us to get a bound on J
0,1
Π
⊥
as follows. It follows from 17 and the variation of constants formula that
kJ
0,t
Π
⊥
k ≤ ke
−Lt
Π
⊥
k + Z
t
Cs
−a
kDNu
s
k
−a
ds ≤ e
−λ
N +1
t
+ C t
1 −a
sup
s ≤t
ku
s
k
k
,
720
so that, for every δ 0, we have by 108 the bound
E
kJ
0,t
Π
⊥
k
p
≤ C
δ,p
e
−λ
N +1
t
+ t
1 −a
exp δV u
, for some family of constants C
δ,p
independent of t ∈ [0, 1]. Since a 1, it follows that for every
ǫ, δ 0 and p 0, we can find N sufficently large and t sufficiently small such that
E kJ
0,t
Π
⊥
k
p
≤ ǫ exp δV u .
Combining this with 106b and the fact that kJ
0,1
Π
⊥
k ≤ kJ
t,1
kkJ
0,t
Π
⊥
k, we obtain
E kJ
0,1
Π
⊥
k
¯ p
≤ E
kJ
0,t
k
2¯ p
E kJ
t,1
k
2¯ p
1 2
≤ Cǫ exp 2δV u ,
provided that N is sufficiently large. By choosing δ sufficiently small, it follows that Assumption B.4
with arbitrary values for ¯ p
and C
Π
can always be satisfied by choosing for Π the projection onto the first N eigenvectors of L for some large enough value of N . The bound 107 now follows from a
simple application of Theorem 5.5. It remains to prove the statement about the smoothness of
T
∗
P
2
u, ·, which will be a consequence of 109 by [Nua95, Cor. 2.1.2]. The reason why we consider the process at time 2 is that, in order to
avoid the singularity at the origin, we consider the solution u
2
as an element of the probability space with Gaussian structure given by the increments of W over the interval [1, 2]. The increments of
W over [0, 1] are then considered as some “redundant” randomness, which is irrelevant by [Nua95, Ch. 1]. With this slightly tweaked Gaussian structure, the Malliavin matrix of Πu
2
is given almost surely by Π
M
1
u
1
Π, where M
1
is defined as before, but over the interval [1, 2]. The claim now follows from 109 and 105, provided that the random variable Πu
2
belongs to the space D
∞
of random variables whose Malliavin derivatives of all orders have moments of all orders.
Recall now see for example [BM07, Section 5.1] that for any n-tuple of elements h
1
, . . . , h
n
∈ L
2
[1, 2], R
d
, the nth Malliavin derivative of u
2
in the directions h
1
, . . . , h
n
is given by D
n
u
2
h = Z
1 ≤s
1
···s
n
≤2
J
n s,1
Gh
s
ds . 110
Applying 70a in Proposition 6.2 we see that, for every u ∈ H , every γ γ
⋆
+ 1, and every p 0, one has the bound
E sup
t ∈[1,2]
ku
t
k
p γ
∞ . We conclude from Proposition 3.11 that
E sup
1 ≤s
1
···s
n
≤2
sup
kϕ
j
k≤1
kJ
k s,t
ϕ
1
, . . . , ϕ
k
k
p
≤ ∞ , so that, by 110, u
2
does indeed have Malliavin derivatives of all orders with bounded moments of all orders. This concludes the proof.
721
8.2 The 2D Navier-Stokes equations on a sphere
