For example, if N u = u − u
3
then N
3
v
1
, v
2
, v
3
= v
1
v
2
v
3
and if the forced directions {g
1
, · · · , g
d
} are
C
∞
then N
3
h
1
, h
2
, h
3
∈ C
∞
for h
i
∈ {g
1
, · · · , g
d
}. As observed in [BM07], to obtain a simple sufficient criteria for the brackets to be dense, suppose that Λ
⊂ C
∞
is a finite set of functions that generates, as a multiplicative algebra, a dense subset of the phase space. Then, if the forced modes
A = {g
1
, · · · , g
d
} contain the set {h, h¯h : h,¯h ∈ Λ}, the set A
∞
constructed as in Meta-Theorem 1.5 will span a dense subset of phase space.
1.6 Probabilistic and dynamical view of smoothing
Implicit in 3 is the “transfer of variation” from the initial condition to the Wiener path. This is the heart of “probabilistic smoothing” and the source of ergodicity when it is fundamentally probabilistic
in nature. The unique ergodicity of a dynamical system is equivalent to the fact that it “forgets its initial condition” with time. The two terms appearing on the right-hand side of 2 represent two
different sources of this loss of memory. The first is due to the randomness entering the system. This causes nearby points to end up at the same point at a later time because they are following
different noise realisations. The fact that different stochastic trajectories can arrive at the same point and hence lead to a loss of information is the hallmark of diffusions and unique ergodicity due to
randomness. From the coupling point of view, since different realizations lead to the same point yet start at different initial conditions, one can couple in finite time.
The second term in 2 is due to “dynamical smoothing” and is one of the sources of unique ergodicity in deterministic contractive dynamical systems. If two trajectories converge towards each other over
time then the level of precision needed to determine which initial condition corresponds to which trajectory also increases with time. This is another type of information loss and equally leads to
unique ergodicity. However, unlike “probabilistic smoothing”, the information loss is never complete at any finite time. Another manifestation of this fact is that the systems never couples in finite time,
only at infinity. In Section 5.1.1 about the strongly dissipative setting, the case of pure dynamical smoothing is considered. In this case one has 2 with only the second term present. When both terms
exist, one has a mixture of probabilistic and dynamical smoothing leading to a loss of information about the initial condition. In Section 2.2 of [HM08] it is shown how 2 can be used to construct a
coupling in which nearby initial conditions converge to each other at time infinity. The current article takes a “forward in time” perspective, while [EMS01, BM05] pull the initial condition back to minus
infinity. The two points of view are essentially equivalent. One advantage to moving forward in time is that it makes proving a spectral gap for the dynamic more natural. We provide such an estimate in
Section 8.4 for the stochastic Ginzburg-Landau equation.
1.7 Structure of the article
The structure of this article is as follows. In Section 2, we give a few abstract ergodic results both proving the results in the introduction and expanding upon them. In Section 3, we introduce the
functional analytic setup in which our problem will be formulated. This setup is based on Assump- tion A.1 which ensures that all the operations that will be made later differentiation with respect to
initial condition, representation for the Malliavin derivative, etc are well-behaved. Section 4 is a follow-up section where we define the Malliavin matrix and obtain some simple upper bounds on it.
We then introduce some additional assumptions in Section 6.1 which ensure that we have suitable control on the size of the solutions and on the growth rate of its Jacobian.
669
In Section 5, we obtain the asymptotic strong Feller property under a partial invertibility assumption on the Malliavin matrix and some additional partial contractivity assumptions on the Jacobian.
Section 6.3 then contains the proof that assumptions on the Malliavin matrix made in Section 5 are justified and can be verified for a large class of equations under a Hörmander-type condition. The
main ingredient of this proof, a lower bound on Wiener polynomials, is proved in Section 7. Finally,
we conclude in Section 8 with two examples for which our conditions can be verified. We consider the Navier-Stokes equations on the two-dimensional sphere and a general reaction-diffusion equation
in three or less dimensions.
Acknowledgements
We are indebted to Hakima Bessaih who pushed us to give a clean formulation of Theorem 8.1.
2 Abstract ergodic results
We now expand upon the abstract ergodic theorems mentioned in the introduction which build on the asymptotic strong Feller property. We begin by giving the proof of Corollary 1.4 from the
introduction and then give a slightly different result but with the same flavour which will be useful in the investigation of the Ginzburg-Landau equation in Section 8.4. Throughout this section,
P
t
will be a Markov semigroup on a Hilbert space
H with norm k · k. Proof of Corollary 1.4.
Since P
t
is Feller, we know that for any u ∈ H and open set A with P
t
u, A 0 there exists an open set B containing u so that
inf
u ∈B
P
t
u, A 0 . Combining this fact with the weak topological irreducibility, we deduce that for all u
1
, u
2
∈ H there exists v
∈ H so that for any ε 0 there exists a δ, t
1
, t
2
0 with inf
z ∈B
δ
u
i
P
t
i
z, B
ε
v 0 6
for i = 1, 2. Now assume by contradiction that we can find two distinct invariant probability measures
µ
1
and µ
2
for P
t
. Since any invariant probability measure can be written as a convex combination of ergodic measures, we can take them to be ergodic without loss of generality. Picking u
i
∈ suppµ
i
, by assumption there exists a v so that for any
ε 0 there exists t
1
, t
2
and δ 0 so that 6 holds. Since
u
i
∈ suppµ
i
we know that µ
i
B
δ
u
i
0 and hence µ
i
B
ε
v = Z
H
P
t
i
z, B
ε
vµ
i
dz ≥ Z
B
δ
u
i
P
t
i
z, B
ε
vµ
i
dz ≥ µ
i
B
δ
u
i
inf
z ∈B
δ
u
i
P
t
i
z, B
ε
v 0 . Since
ε was arbitrary, this shows that v ∈ suppµ
1
∩ suppµ
2
, which by Theorem 1.2 gives the required contradiction.
670
We now give a more quantitative version of Theorem 1.2. It shows that if one has access to the quantitative information embodied in 2, as opposed to only the asymptotic strong Feller property,
then not only are the supports of any two ergodic invariant measures disjoint but they are actually separated by a distance which is directly related to the function C from 2.
Theorem 2.1 Let {P
t
} be a Markov semigroup on a separable Hilbert space H such that 2 holds for some non-decreasing function C. Let
µ
1
and µ
2
be two distinct ergodic invariant probability measures for
P
t
. Then, the bound ku
1
− u
2
k ≥ 1Cku
1
k ∨ ku
2
k holds for any pair of points u
1
, u
2
with u
i
∈ supp µ
i
. Proof. The proof is a variation on the proof of Theorem 3.16 in [HM06]. We begin by defining for
u, v ∈ H the distance d
n
u, v = 1 ∧ p
