Some examples of absolute continuity of measures in stochastic fluid dynamics

0.24 MB·English

by  B. Ferrario

Some examples of absolute continuity of measures in
stochastic fluid dynamics
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B. Ferrario
2
Dipartimento di Matematica - Universit`a di Pavia
n
a
J
3 Abstract

Anon linear Itˆoequation in a Hilbert space is studied bymeans of Girsanov theorem.
]
R We consider a non linearity of polynomial growth in suitable norms, including that of
P quadratic type which appears in the Kuramoto–Sivashinsky equation and in the Navier–
. Stokes equation. We prove that Girsanov theorem holds for the 1-dimensional stochas-
h
tic Kuramoto–Sivashinsky equation and for a modification of the 2- and 3-dimensional
t
a stochasticNavier–Stokesequation;thismodificationconsistsinsubstitutingtheLaplacian
m −∆ with (−∆)α, where α> d +1 (d=2,3). In this way,we proveexistence and unique-
2
ness of solutions for these stochastic equations. Moreover, the asymptotic behaviour for
[
t→∞ is characterized.

1
v Key words: stochastic fluid dynamics, Girsanov theorem, existence and uniqueness of strong
6 solutions, regularity results, uniqueness of invariant measures.
9 AMS Subject Classification (2000): 60H15, 35Q35,76M35.
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. 1 Introduction
1
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8 Thestudyofnonlinearequationsrequiressomeskilltodealsuccessfullywiththenonlinearity.
0
As fas as stochastic differential equations are concerned, a possible technique to study a non
:
v linear equation is the Girsanov transform. Indeed, given a non linear stochastic Itoˆ equation
i
X
du(t)+[ Lu(t)+F(u(t)) ]dt=Gdw(t), t∈]0,T]; u(0)=x (1)
r
a
it is possible to analyze it as a perturbation of the linear equation
dz(t)+Lz(t) dt=Gdw(t), t∈]0,T]; z(0)=x (2)
bymeansofGirsanovtheorem. ItiswellknownthatthistheoremholdsifforinstanceNovikov
condition
E exp 1 T |G−1F(z(t))|2dt <∞
2 0
is satisfied. We are interestedhin the(cid:0)caRse in which Noviko(cid:1)vicondition is not fulfilled, but it is
if the non linear term F is suitably truncated. By an approximating procedure we can apply
Girsanov transform so to get that equation (1) has a weak solution having the same regularity
asz andthelawL ofuisabsolutelycontinuouswithrespecttothelawL ofz (L ≺L )and
u z u z
1
possibly the converse too, so to get the equivalence of L and L (L ∼L ). We recall that if
u z u z
L ≺ L , uniqueness for equation (2) implies uniqueness in law for equation (1). Moreover, if
u z
L ∼ L , each property holding P-a.s. for the process z must hold also for the process u and
u z
vice versa.
Our analysis to verify if Girsanov transform can be used is quite standard. We formalize it
here in order to apply it in the next sections to some models in stochastic fluid dynamics, in
which the equations are set in a infinite dimensional Hilbert space and the non linearity F is
of quadratic type; however,the result holds true for F of polynomial growth.
As to the structure of the paper,in Section2 two abstractresults arepresented; in the first
itisprovedthatL ≺L andinthesecondthatL ∼L . Then,intheothertwosectionsthese
u z u z
resultsareappliedtoastochasticKuramoto–Sivashinskyequationandtoamodifiedstochastic
Navier–Stokesequation, respectively. For these non linear equations (which have a similar non
linearity), we obtain results of existence and uniqueness of the solution; further, uniqueness of
the invariant measure is proved, so to characterize the asymptotic behaviour.
2 Absolute continuity of laws
We are given a separable Hilbert space H, equipped with a complete orthonormal system
{e }∞ , and a complete probability space (Ω,F,{F } ,P). We denote by E the expectation
j j=1 t t≥0
with respect to the measure P.
As far as equation (2) is concerned, we assume that L and G are linear operators in H and
G is invertible. The process w is a cylindrical Wiener process in H, defined on the probability
space (Ω,F,{F } ,P). This means that, given a sequence {β }∞ of i.i.d. one dimensional
t t≥0 j j=1
Wiener processes defined on (Ω,F,{F } ,P), we represent w(t)= β (t)e .
t t≥0 j j j
Moreover,weassumethatthereexistsauniquestrongsolutionz (inthestochasticsense)which
P
is a Markov process such that
Ekzk2p <∞ (3)
C([0,T];E)
for some p > 1, where E is a separable dense subset of H. Actually, it would be enough z to
be a weak solution; but in our applications in Sections 3 and 4, z will be a strong solution and
thus we assume it since now.
From now on, we denote by z(t;x), or simply by z(t), the solution of (1) evaluated at time
t (thus z(0;x) = x and, for t > 0, z(t;x) is a random variable) and by z the solution process
{z(t;x)} on a time interval [0,T].
0≤t≤T
The main assumption on the non linear term is that the operator G−1F : E → H is
measurable and
|G−1F(v)| ≤c(1+|v|p) ∀v ∈E, (4)
H E
where c is a suitable constant and p>1 is the same as in (3). This implies that
T
|G−1F(z(t))|2 dt≤2Tc2 1+kzk2p (5)
H C([0,T];E)
Z0 (cid:16) (cid:17)
so
T
E |G−1F(z(t))|2 dt≤2Tc2 1+E[kzk2p ] <∞. (6)
H C([0,T];E)
Z0 (cid:16) (cid:17)
2
In particular
P T |G−1F(z(t))|2 dt<∞ =1. (7)
0 H
This condition is necessary for N(cid:8)oRvikov condition (cid:9)
E exp 1 T |G−1F(z(t))|2 dt <∞ (8)
2 0 H
h (cid:0) R (cid:1)i
to hold. It is well known (see, e.g., [2] for stochastic PDE’s in Hilbert spaces) that condition
(8) implies that
ρT :=E exp T hG−1F(z(s)),dw(s)i − 1 T |G−1F(z(s))|2 ds σT(z) (9)
u/z 0 H H 2 0 H
h (cid:16)R R (cid:17)(cid:12) i
is a probability density. Here σT(z) denotes the σ-algebra generated by {z(cid:12)(t)} . The
0≤t≤T
stochastic integral in the exponent has to be understood as T hG−1F(z(s)),e i dβ (s)
j 0 H j H j
and is well defined because of (6) (see [2], Chapter 4).
As soon as we know that E[ρT ] = 1, we apply GirsanoPv thReorem to get that L ≺ L .
u/z u z
We remind it here, for reader’s convenience (see, e.g., [2], [10], [12]). Defined the probability
measure P∗ on (Ω,F) by dP∗ =ρT dP, Girsanov theorem states that
u/z
t
w∗(t)=w(t)+ G−1F(z(s))ds
Z0
isacylindricalWienerprocesson(Ω,F,{F } ,P∗). So,ifzsolvesequation(2)withWiener
t 0≤t≤T
process w, then z solves equation (1) with Wiener process w∗, since
t t
z(t)=x− Lz(s)ds+ Gdw(s)
Z0 Z0
t t t
=x− Lz(s)ds− F(z(s))ds+ Gdw∗(s).
Z0 Z0 Z0
Thus,P{u∈Λ}=P∗{z ∈Λ}foreveryBorelsetΛ⊂C([0,T];E). ThenP{z ∈Λ}=0implies
P∗{z ∈Λ}=0 and so P{u∈Λ}=0, that is L ≺L .
u z
Summing up, assuming that the solution z to equation (2) is such that E[ρT ] = 1, then
u/z
equation (1) has a weak solution having the same regularity as z and L ≺ L ; moreover,
u z
uniqueness in law for z implies uniqueness in law for u.
If L ∼L , then each property holding P-a.s. for the process z must hold also for the process
u z
u and vice versa. Also the laws of u(t;x) and z(t;x) are equivalent. In fact, P{u(t;x)∈ Γ}=
P∗{z(t;x) ∈ Γ} for every Borel set Γ ⊂ H. In this way, if we can prove easily strong Feller
propertyandirreducibilityforthelinearequation,thesepropertieswillbe inheritedbythenon
linear equation.
However, by (6) it does not follow that Novikov condition holds. Anyway, we can approx-
imate the non linearity F in such a way that Novikov condition holds for the approximate
equation and by this we obtain E[ρT ] = 1. The procedure is standard, but the results avail-
u/z
able in the literature do not apply here. For instance, there are similar techniques in [12] (but,
even if they deal with a stochastic Navier–Stokes equation, the important issue there is the
existenceofweaksolutions;GirsanovtheoremisprovedforotherstochasticPDE’s)or[4](but,
3
evenif they dealwith a stochastic Kuramoto–Sivashinskyequation, the Novikovconditionand
Girsanov theorem are analyzed in a finite dimensional context). We point out that in this
paper we prove Girsanov theorem for a 1D stochastic Kuramoto–Sivashinsky equation and for
a modification of the 2D and 3D stochastic Navier–Stokes equation. Further, our results give
regularity of strong solutions of equation (1) (we shall deal with a variety of spaces E ⊂ H)
and the equivalence of all its transition functions so to characterize the asymptotic behaviour
by means of Doob theorem.
We now state a first result on the absolutely continuity of the measures.
Proposition 2.1 Assume (4) holds and that for every x ∈ E there exists a unique strong
solution z of equation (2) on the time interval [0,T], safisfying (3).
Then, given u(0) = x there exists a unique weak solution u to equation (1) on the time
interval [0,T] and the law of the process u is absolutely continuous with respect to the law of
the process z solving (2), with density given by (9).
Proof. Let us define the approximating equation by
duN(t)+LuN(t)dt+χN(uN)F(uN(t))dt=Gdw(t)
t (10)
uN(0)=x
(cid:26)
where for each N =1,2,..., the truncation function χN is defined as follows:
1 if t|G−1F(v(s))|2 ds≤N
χN(v)= 0 H
t (0 othRerwise
Notice that χN(z) is a progressively measurable process. Novikov condition
·
E exp 1 T |G−1χN(z)F(z(s))|2 ds <∞
2 0 s H
now is trivially satisfied, sinhce b(cid:16)y thRe definition of χN we have(cid:17)i
t
T
|G−1χN(z)F(z(s))|2 ds≤N P −a.s..
s H
Z0
Hence, for any N =1,2,...
E[eVT,N]=1,
where VT,N = T χN(z) hG−1F(z(s)),dw(s)i − 1 T χN(z)|G−1F(z(s))|2 ds, and by Gir-
0 s H H 2 0 s H
sanov theorem we have that L ≺L with the density
R uN z R
ρT =E[eVT,N|σT(z)].
uN/z
NowwewanttoprovethatE[eVT]=1,wheretheexponentisVT = T hG−1F(z(s)),dw(s)i −
0 H H
1 T |G−1F(z(s))|2 ds.
2 0 H R
We know that E[eVT,N]=1; moreover
R
E[eVT,N]=E[χN(z)eVT,N]+E[(1−χN(z))eVT,N]
T T
=E[χN(z)eVT]+P{χN(z)=0}.
T T
4
Bymonotoneconvergence,lim E[χN(z)eVT]=E[eVT]. Ontheotherhand,lim P{χN(z)=
N→∞ T N→∞ T
0}=lim P{ T |G−1F(z(s))|2 ds>N}=0.
N→∞ 0 H
Therefore E[eVT] = 1 so that E[eVT|σT(z)] is a probability density. Then, as explained
R
before, the probability measuredP∗ =ρT dP (with ρT givenby (9)) defines a weak solution
u/z u/z
to equation (1).
Uniqueness (in law) of u is a consequence of uniqueness of z and L ≺L . 2
u z
Now, besides the previous conditions, let us assume that also equation (1) has a unique
strong solution u, enjoying the same property (3) as z. We obtain a stronger result.
Proposition 2.2 If (4) holds and for any x ∈ E both equations (1) and (2) have a unique
strong solution on the time interval [0,T]safisfying (3), then the laws L and L are equivalent
u z
and the densities are given, respectively, ρT by (9) and
u/z
ρT =E exp − T hG−1F(u(s)),dw(s)i − 1 T |G−1F(u(s))|2 ds σT(u) .
z/u 0 H H 2 0 H
h (cid:16) R R (cid:17)(cid:12) i
Proof. According to the previous proposition, we know that Lu ≺ Lz. On t(cid:12)he other hand,
interchangingtherˆoleofuandz,againProposition2.1providesthatL ≺L . Therefore,they
z u
are mutually absolutely continuous, i.e. equivalent. 2
As a consequence, also the laws of z(t;x) and u(t;x) are equivalent. Before stating the last
result,weneedtorecallsomedefinitions. AMarkovprocessuissaidtobestronglyFellerinEat
timet>0ifP mapsB (E)intoC (E),where(P φ)(x):=E[φ(u(t;x))]; andirreducibleinE at
t b b t
time t>0 if P(t,x,Γ)>0 for any x∈E, 06=Γ⊂E open, where P(t,x,Γ):=P{u(t;x)∈Γ}.
Corollary 2.3 Under the assumptions of Proposition 2.2, the process z is strongly Feller and
irreducible if and only if so is the process u.
In the next sections, we shall study first the linear equation so to check condition (3) and
then estimate (4).
3 The 1D stochastic Kuramoto–Sivashinsky equation
Wereferto[7]forthe abstractsetting,inwhichthestochasticKuramoto–Sivashinskyequation
in written as
du(t)+[νA2u(t)−Au(t)+B(u(t),u(t))] dt=Aγdw(t)
(11)
u(0)=x
(cid:26)
and the linear equation associated is
dz(t)+[νA2z(t)−Az(t)+az(t)] dt=Aγdw(t)
(12)
z(0)=x
(cid:26)
The unknown u can be interpreted as a one-dimensional velocity field in a compressible fluid
(see [15]).
With respect to the setting of Section 2, we have that the linear operator is
Lu=νA2u−Au+au
5
with a>0 large enough and ν >0, and the non linear operator is
F(u)=B(u,u)−au.
The operator G in front of the Wiener process is taken of the form Aγ (γ ∈ R). w is a
cylindrical Wiener process in H on a probability space (Ω,F,P); {F } is the canonical
t t∈[0,T]
filtration associated to the Wiener process.
The functional spaces are (given L>0, so the spatial domain is [−L,L])
2 2
H ={u=u(ξ)∈L2(−L,L): L/2 u dξ =0},
2 2 −L/2
E =D(Aθ) for some θ >0, R
where
Au=−u′′
D(A)=H ∩{u=u(ξ)∈H2(−L,L):u(−L)=u(L),u′(−L)=u′(L)}.
2 2 2 2 2 2
The operator A is a strictly positive unbounded self-adjoint operator in H, whose eigenvectors
{e }∞ form a complete orthonormalbasis of the space H. The powers Aθ are defined for any
j j=1
θ ∈ R: if Ae = λ e then Aθv = λθhv,e ie , D(Aθ) = {v = v e : λ2θv2 < ∞}.
j j j j j j j j j j j j j
Moreover,λ ∼j2 as j →∞.
j P P P
The operator−(νA2−A+a) generates in H (and in anyD(Aβ)) an analytic semigroupof
negative type of class C .
0
The operator B is the bilinear operator defined by
B(u,v)=uv′.
For instance, B maps D(A1/2)×D(A1/2) into H; other domains of definition of B are givenin
[7].
First, let us consider the linear equation. We are interested in the regularityofthe solution
z and in the asymptotic behaviour for t→∞. For this, we denote by R(t,x,·) the transitions
functions for (12), i.e. R(t,x,Γ) = P{z(t;x) ∈ Γ}, and by R the Markovian semigroup ,
t
i.e. (R φ)(x) = E[φ(z(t;x))]. We say that a measure m is invariant for equation (12) if
t
R φ dm = φ dm for every t ≥ 0,φ ∈ C (D(Aθ)). We collect the results in the following
t b
proposition.
R R
Proposition 3.1 If θ +γ < 3, then for any x ∈ D(Aθ) equation (12) has a unique strong
4
solution z such that
Ekzk2p <∞ (13)
C([0,T];D(Aθ))
for any p ≥ 1and T < ∞; this is a Markov process, strongly Feller and irreducible in D(Aθ)
for any t>0. The Gaussian measure µ =N(0,1A2γ[νA2−A+a]−1) is the unique invariant
l 2
measure, all transition functions R(t,x,·) are equivalent to µ and
l
lim R φ(x)= φ dµ , (14)
t l
t→+∞
Z
lim R(t,x,Γ)=µ(Γ)
l
t→+∞
for any x∈D(Aθ),φ∈C (D(Aθ)) and Borel set Γ⊂D(Aθ).
b
6
Proof. From (3.10) in [7], we know that, given x ∈ D(Aθ), if θ+γ < 3 equation (12) has a
4
unique strong solution z
t
z(t)=e−(νA2−A+a)tx+ e−(νA2−A+a)(t−s)Aγdw(s)
Z0
whose paths are, P-a.s., in C([0,T];D(Aθ)). This is a Markov process; many of its properties
are easy to check, since the semigroup {e−(νA2−A+a)t} and the covariance of the noise are
t≥0
diagonal operators and commute.
We recall the basic steps for checking the regularity of z (the result follows rigorously,e.g.,
from [2], Chapter 5, and is proved in [7]):
|Aθe−(νA2−A+a)tx| ≤|Aθx| ∀t≥0
H H
t ∞ t
E Aθe−(νA2−A+a)(t−s)Aγdw(s) 2H =E λθj+γ e−(νλ2j−λj+a)(t−s) dβj(s) ej 2H
Z0 j=1 Z0
(cid:12) (cid:12) (cid:12)X (cid:12)
(cid:12) (cid:12) ∞(cid:12) t (cid:12)
= λ2(θ+γ) e−2(νλ2j−λj+a)(t−s) ds
j
j=1 Z0
X
∞ λ2(θ+γ)
≤ j ∀t>0.
2(νλ2−λ +a)
j=1 j j
X
The last series is convergentif θ+γ < 3, since λ ∼j2 as j →∞.
4 j
Accordingto Burkholder-Davis-Gundyinequality,the secondestimate providesthat inequality
(13) holds for any p.
The result on the invariant measure is obtained as in [2], Chapter 11. Actually, the result
is trivial if we work first on each component z and then we recover the infinite dimensional
j
result for z (z(t)= ∞ z (t)e ). Indeed, each component z satisfies
j=1 j j j
dzP(t)+[νλ2−λ +a]z (t) dt=λγdβ (t), z (0)=x ;
j j j j j j j j
itslawisN e−(νλ2j−λj+a)txj,12νλ2jλ−2jλγj+a(1−e−2(νλ2j−λj+a)t) andfort→+∞thedensityofthis
Gaussianm(cid:0)easureconvergestothedensityoftheGaussian(cid:1)measureN 0,1 λ2jγ ,whichis
2νλ2j−λj+a
the unique stationary measure. Therefore, equation (12) has a unique invariant measure; this
(cid:0) (cid:1)
is the Gaussian measure with mean 0 and covariance operator Q = 1A2γ[νA2−A+a]−1;
∞ 2
It is easy to check that |Aθx|2 dµ (x) < ∞ and that µ (Γ) > 0 for any open and non
H l l
empty set Γ⊂D(Aθ) .
R
We expect that irreducibility and strong Feller property hold, because the noise acts on all
directions e and the semigroup e−(νA2−A+a)t makes z depending very regularly on the initial
j
data x.
As far as strong Feller property is concerned, by [2] (Chapter 9) we know that condi-
tion Ran(Q1/2) ⊃ Ran(e−(νA2−A+a)t) is equivalent to the strong Feller property, where Q
t t
is the covariance operator of the Gaussian random variable z(t;x). Since Q = 1A2γ[I −
t 2
e−2(νA2−A+a)t][νA2 −A+a]−1 and for t > 0 the range of the operator e−(νA2−A+a)t is con-
tained in any space D(Aβ) for β >0, we see that this condition is trivially satisfied.
7
According to Theorem 11.13 in [2], (14) holds and all the transition measures R(t,x,·) are
absolutely continuous with respect to µ. Irreducibility comes straightforward. Let us point
l
out that in the proof of this theorem, it is shown also that the law of z(t;x) is equivalent to
the law of z(s;y) for any t,s > 0 and x,y ∈ D(Aθ); actually, this follows by Feldman-Hajek
theorem, which is easy to verify in this case of diagonal operators. 2
To set our problem as in Section 2, we have to fix some space E =D(Aθ). The interesting
spaces are D(Aθ) for θ ≥ 0: D(A0) = H is the basic space of finite energy and, for θ > 0,
D(Aθ)isasubspaceofH. Inpractise,givenθ ≥0wechooseγ asbigaspossible(γ < 3−θ)so
4
to make to weakestassumption on the covariance of the noise. Or, givenγ < 3 (the limitation
4
is due to θ ≥ 0), we choose θ as big as possible (θ < 3 −γ). Decreasing γ, the operator Aγ is
4
”more regular” (in the sense that, for instance, Aγ is a bounded operator for γ ≤ 0) and this
stronger assumption provides a more regular solution z with paths in C([0,T];D(Aθ)).
Now, we deal with estimate (4). We have the following result.
Lemma 3.2 Let parameters γ and θ be chosen as follows:
for 1 <γ < 3 : 3 − γ ≤θ < 3 −γ
4 4 8 2 4
for 0≤γ ≤ 1 : 5 −γ ≤θ < 3 −γ (15)
4 8 4
for γ <0: 1 −γ ≤θ < 3 −γ.
2 4
Then there exists a constant c, depending on γ,θ and a, such that
|A−γ[B(v,v)−av]| ≤c 1+|Aθv|2 ∀v ∈D(Aθ).
H H
Proof. Notice that (15) imply the bounds γ(cid:0)< 3, θ >0 a(cid:1)nd θ+γ < 3. The non linear term is
4 4
estimated as follows:
|A−γB(v,v˜)|H ≤C1|A83−γ2v|H|A83−γ2v˜|H if 1 <γ < 3 (16)
4 4
1
|A−γB(v,v˜)|H ≤C2|A85−γv|H|A58−γv˜|H if 0≤γ ≤ (17)
4
|A−γB(v,v˜)|H ≤C3|A21−γv|H|A12−γv˜|H if γ <0 (18)
ThetwofirstinequalitiescomefromtheproofofLemma2.2in[9]. ThelatterisprovedinPropo-
sition2.1in[7]. Bytheway,recallingthatB(v ,v )−B(v ,v )=B(v −v ,v )+B(v ,v −v )
1 1 2 2 1 2 1 2 1 2
by bilinearity, the above inequalities show that the operator A−γB(v,v) is continuous (hence,
measurable) in the spaces where it is defined.
Notice that if (15) are satisfied, then θ >−γ. Therefore, choosing θ as in (15) we get
|A−γ[B(v,v)−av]| ≤|A−γB(v,v)| +a|A−γv|
H H H
≤C |Aθv|2 +aC |Aθv|
4 H 5 H
≤C 1+|Aθv|2 .
6 H
2
(cid:0) (cid:1)
Remark 3.3 The case θ =0 is not included. Indeed, we have
|A−γB(v,v)| ≤c|v|2
H H
8
for γ > 3, because
4
L/2 1 1 L/2
|hB(v,v),xi|=| (v2)′xdξ|= | v2x′dξ|
2 2
Z−L/2 Z−L/2
1
≤ 2|v2|L1|x′|L∞
1
≤c|v|2 |x′| for m>
L2 D(Am) 4
1
=c|v|2 |x| for m> .
L2 D(A12+m) 4
But the condition γ > 3 is incompatible with θ+γ < 3,θ =0.
4 4
Now, we consider equation (11). Let us denote by P(t,x,·) its transitions functions.
Theorem 3.4 For every γ < 3 and choosing θ as in (15), we have the following result.
4
Given x ∈ D(Aθ) there exist unique strong solutions of equations (11) and (12) on any finite
timeinterval[0,T],withpaths inC([0,T];D(Aθ)), P-a.s.. Wehave L ∼L , withthedensities
u z
ρTu/z =E eV+T σT(z) with V+T = 0THhA−γ[B(z(s),z(s))−az(s)],dw(s)iH
h (cid:12) i R − 1 T |A−γ[B(z(s),z(s))−az(s)]|2 ds
(cid:12) 2 0 H
R
ρT =E eV−T σT(u) with VT =− T hA−γ[B(u(s),u(s))−au(s)],dw(s)i
z/u − 0H H
h (cid:12) i R − 1 T |A−γ[B(u(s),u(s))−au(s)]|2 ds
(cid:12) 2 0 H
for any T >0. R
Further, P(t,x,·)∼µ for any t>0,x∈D(Aθ), where µ =N(0,1A2γ[νA2−A+a]−1) is the
l l 2
unique invariant measure for (12). The process u is strongly Feller and irreducible in D(Aθ)
at any time t>0.
Finally, there exists only one invariant measure µ for (11) which is equivalent to µ .
KS l
µ is ergodic, i.e.
KS
1 T
lim φ(u(t;x))dt = φ dµ
KS
T→+∞T Z0 Z
P-a.s. for every x∈D(Aθ),φ∈L1(µ ), and strongly mixing, i.e.
KS
lim P(t,x,Γ)=µ (Γ)
KS
t→+∞
for every x∈D(Aθ) and Borel set Γ⊂D(Aθ).
Proof. If θ and γ are chosen as in (15), from Proposition 3.1 and Lemma 3.2 we know
that the assumptions of Proposition 2.1 (with p = 2 and E = D(Aθ)) are satisfied. This
implies that for x∈ D(Aθ) equation (11) has a weak solution u living in C([0,T];D(Aθ)) and
L ≺ L ; but Theorem 4.3 in [7] provides existence and uniqueness of a strong solution u for
u z
9
any u(0) ∈ H = D(A0) and γ < 3. Thus, we have the regularity result: given x ∈ D(Aθ)
4
equation (11) has a unique strong solution u with paths in C([0,T];D(Aθ)). By Proposition
2.2 we obtain that L ∼L ; moreover, P(t,x,·) ∼ R(s,y,·)∼ µ and Corollary 2.3 holds. We
u z l
conclude our proof, bearing in mind Doob theorem for uniqueness of invariant measures (see
[3]). The existence of an invariant measure has been proved in [7]. 2
Let us notice that, as far as the regularity of solutions is concerned, this result improves
thatofProposition6.5in[7],sincenowwecanconsideranyspaceD(Aθ)withθ >0. However,
we are not able to prove the absolute continuity result in H =D(A0), as explained in Remark
3.3, even if we know from [7] that for any u(0)∈H there exists a unique solution u such that
u∈C([0,T];H) (P-a.s.).
Moreover,the results of this sectionhold true if the operatorin frontof the Wiener process
in equation (11) is of the form LAγ, where L is an isometry in H (e.g., in [7] we considered
LAγw(t)= ∞ λγβ (t)(−1)je ;thisincludesinterestingcasesfromthephysicalpoint
j=1 j j j+(−1)j+1
of view as explained in [7]).
P
4 A modified stochastic Navier–Stokes equation
SincethequadratictermintheKuramoto–SivashinskyequationissimilartothatintheNavier–
Stokes equation, the only difference being that the Navier–Stokes equation is set in spaces
of divergence free vectors, it is appealing to investigate if Girsanov transform holds for the
stochastic Navier–Stokes equation. Unfortunately, the answer is negative. Anyway, let us
analyse this problem modifying the linear part. Our issue is to determine how to modify the
Navier–Stokes equation to apply our procedure.
Therefore, instead of the stochastic Navier–Stokes equation
du(t)+ νAu(t)+B u(t),u(t) dt=Aγ dw(t)
(cid:2) (cid:0) (cid:1)(cid:3)
(studied, e.g., in [1], [16], [8]), we introduce a modification in the linear part; given any α ≥ 1
we consider
du(t)+ νAαu(t)+B u(t),u(t) dt=Aγ dw(t)
(19)
u(0)=x
(cid:26) (cid:2) (cid:0) (cid:1)(cid:3)
This corresponds to replace the Laplacian −∆ with (−∆)α in the Navier–Stokes equations in
order to seek which values of α provide the absolute continuity of L with respect to the law
u
of the linear equation associated to (19), which is the modified stochastic Stokes equation:
dz(t)+νAαz(t)dt=Aγ dw(t)
(20)
z(0)=x
(cid:26)
Inthis sense,our analysisreminds thatof [11]to investigatefor whichvalues ofα the modified
deterministic Navier–Stokes equation
du
(t)+νAαu(t)+B u(t),u(t) =f(t)
dt
(cid:0) (cid:1)
iswellposedford=3(werecallthatford=2thereisnoneedofmodificationtogetexistence
and uniqueness of a global solution).
10