Exponential mixing of the 3D stochastic Navier-Stokes equations driven by mildly degenerate noises

0.37 MB·English

by  Sergio Albeverio

EXPONENTIAL MIXING OF THE 3D STOCHASTIC
NAVIER-STOKES EQUATIONS DRIVEN BY MILDLY
DEGENERATE NOISES
0
1
0 SERGIOALBEVERIO,ARNAUDDEBUSSCHE,ANDLIHUXU
2

n
a
J
Abstract. We prove the strong Feller property and exponential mixing for
0
3D stochastic Navier-Stokes equation drivenby mildlydegenerate noises (i.e.
3
all but finitely many Fourier modes areforced) via Kolmogorov equation ap-

proach.
]
R
P
.
h
1. Introduction
t
a
m TheergodicityofSPDEsdrivenbydegenerate noiseshavebeenintensivelystud-
ied in recent years (see for instance [7],[13], [6], [14], [21]). For the 2D stochastic
[
Navier-Stokesequations(SNS),thereareseveralresultsonergodicity,amongwhich

4 the most remarkable one is by Hairer and Mattingly ([13]). They proved that the
v
2D stochastic dynamics has a unique invariant measure as long as the noise forces
4
atleasttwolinearlyindependentFouriermodes. Asforthe3DSNS,mostofergod-
1
6 icity results are about the dynamics driven by non-degenerate noises (see [3], [11],
0 [19],[21], [18]). Inthe respectofthe degeneratenoise case,as noisesareessentially
. elliptic setting of which all but finite Fourier modes are driven, [23] obtained the
0
1 ergodicitybycombiningMarkovselectionandMalliavincalculus. Asthenoisesare
9 truly hypoelliptic ([13]), ergodicity is still open.
0
:
v Inthis paper,weshallstillstudythe3DSNSdrivenbyessentiallyellipticnoises
i as above, but our approach is essentially different from that in [23]. Rather than
X
Markov selection and cutoff technique, we prove the strong Feller property by
r
a studying some Kolmogorov equations with a large negative potential, which was
developed in [3]. Comparing with the method in [3] and [5], we cannot apply the
Bismut-Elworthy-Li formula ([8]) due to the degeneracy of the noises. To fix this
problem, we follow the ideas in [7] and split the dynamics into high and low fre-
quency parts, applying the formula to the dynamics at high modes and Malliavin
calculus to those at low ones. Due to the degeneracy of the noises again, when
applying Duhamel formula as in [3] and [5], we shall encounter an obstruction of
not integrability (see (5.1)). Two techniques are developed in Proposition 5.1 and
5.2 to conquer this problem, and the underlying idea is to trade off the spatial
regularity for the time integrability. Using the coupling method of [17], in which
2000 Mathematics Subject Classification. Primary 76D05; Secondary 60H15, 35Q30, 60H30,
76M35.
Key words and phrases. stochastic Navier-Stokes equation (SNS), Kolmogorov equation,
Galerkinapproximation,strongFeller,ergodicity,mildlydegenerate noise,Malliavincalculus.
1
2 S.ALBEVERIO,A.DEBUSSCHE,ANDL.XU
the noises have to be non-degenerate, we prove the exponential mixing and find
that the construction of the coupling can be simplified. Finally, we remark that
thelargecoefficientK infrontofthe potential(see(2.11)), besidessuppressingthe
nonlinearity B(u,u) as in [3] and [5], also conquers the crossing derivative flows
(see (3.9) and (3.10)).
Let us discuss the further application of the Kolmogorov equation method in
[3], [5] and this paper. For another essentially elliptic setting where sufficiently
large(but still finite) modes areforced([13], section4.5),due to the largenegative
potential, it is easy to show the asymptotic strong Feller ([13]) for the semigroup
Sm (see (2.13)). There is a hope to transfer this asymptotic strong Feller to the
t
semigroup Pm (see (2.14)) using the technique in Proposition 5.2. If Pm satisfies
t t
asymptotic strong Feller, then we can also provethe ergodicity. This is the further
aim of our future research in 3D SNS.
The paper is organized as follows. Section 2 gives a detailed description of the
problem, the assumptions on the noise and the main results (Theorems 2.4 and
2.5). Section3provesthe crucialestimate inTheorem3.1,while the fourthsection
4 applies Malliavin calculus to prove the important Lemma 3.5. Section 5 gives a
sketch proof for the main theorems, and the last section contains the estimate of
Malliavin matrices and the proof of some technical lemmas.
Acknowledgements: We wouldlike to thank Prof. MartinHairer for pointing
out a serious error in the original version and some helpful suggestions on how to
correct it. We also would like to thank Dr. Marco Romito for the stimulating
discussions and some helpful suggestions on correcting several errors.
2. Preliminary and main results
2.1. Notations and assumptions. Let T3 = [0,2π]3 be the three-dimensional
torus, let
H = x L2(T3,R3): x(ξ)dξ =0, divx(ξ)=0 ,
{ ∈ T3 }
Z
and let
:L2(T3,R3) H
P →
be the orthogonalprojection operator. We shall study the equation
dX +[νAX +B(X,X)]dt=QdW ,
t
(2.1)
(X(0)=x,
where
A= ∆ D(A)=H2(T3,R3) H.
• −P ∩
The nonlinear term B is defined by
•
B(u,v)= [(u )v], B(u)=B(u,u) u,v H1(T3,R3) H.
P ·∇ ∀ ∈ ∩
W isthecylindricalBrownianmotiononH andQisthecovariancematrix
t
•
to be defined later.
We shall assume the value ν = 1 later on, as its exact value will play no
•
essential role.
EXPONENTIAL MIXING OF NSE WITH MILDLY DEGENERATE NOISE 3
Define Z3 = k Z3;k >0 k Z3;k =0,k >0 k Z3;k =0,k =
+ { ∈ 1 }∪{ ∈ 1 2 }∪{ ∈ 1 2
0,k >0 , Z3 = Z3 and Z3 =Z3 Z3, for any n>0, denote
3 } − − + ∗ +∪ −
Zl(n)=[−n,n]3\(0,0,0), Zh(n)=Z3∗\Zl(n).
Let k⊥ = η R3;k η =0 , define the projection :R3 k⊥ by
k
{ ∈ · } P →
k η
(2.2) η =η · k η R3.
Pk − k 2 ∈
| |
Let e (ξ) = coskξ if k Z3, e (ξ) = sinkξ if k Z3 and let e ,e be an
k ∈ + k ∈ − { k,1 k,2}
orthonormalbasis of k⊥, denote
e1(ξ)=e (ξ)e , e2(ξ)=e (ξ)e k Z3;
k k k,1 k k k,2 ∀ ∈ ∗
ei;k Z3,i=1,2 isaFourierbasisofH (uptotheconstant√2/(2π)3/2). With
{ k ∈ ∗ }
this Fourier basis, we can write the cylindrical Brownian motion W on H by
2
W = w (t)e = wi(t)ei
t k k k k
kX∈Z3∗ kX∈Z3∗Xi=1
where each w (t)=(w1(t),w2(t))T is a 2-d standardBrownian motion. Moreover,
k k k
B(u,v)= B (u,v)e
k k
kX∈Z3∗
where B (u,v) is the Fourier coefficient of B(u,v) at the mode k. Define
k
B˜(u,v)=B(u,v)+B(v,u), B˜ (u,v)=B (u,v)+B (v,u).
k k k
We shall calculate B˜ (a e ,a e ) with a j⊥,a l⊥ in Appendix 6.1.
k j j l l j l
∈ ∈
Furthermore, given any n > 0, let π : H H be the projection from H to
n
−→
the subspace π H := x H :x= x e .
n { ∈ k∈Zl(n) k k}
Assumption 2.1 (Assumptions forPQ). We assume that Q :H H is a linear
−→
bounded operator such that
(A1) (Diagonality)Thereareasequenceoflinearmaps{qk}k∈Z3∗ withqk :k⊥ −→
k⊥ such that
Q(ye )=(q y)e y k⊥.
k k k
∈
(A2) (Finitely Degeneracy) There exists some nonempty sublattice Zl(n0) of Z3∗
such that
qk =0 k Zl(n0).
∈
(A3) (Id π )ArQ is boundedinvertible on (Id π )H with 1<r <3/2 and
− n0 − n0
moreover Tr[A1+σQQ∗]< for some σ >0.
∞
Remark 2.2. Under the Fourier basis of H, Q has the following representation
2
(2.3) Q= qijei ej
k k⊗ k
k∈XZh(n0)iX,j=1
where x y : H H is defined by (x y)z = y,z x and (qij) is a matrix
⊗ −→ ⊗ h i k
representation of q under some orthonormal basis (e ,e ) of k⊥. By (A3),
k k,1 k,2
4 S.ALBEVERIO,A.DEBUSSCHE,ANDL.XU
rank(qk)=2forallk∈Zh(n0). TakeQ=(Id−πn0)A−r withsome5/4<r <3/2,
it clearly satisfies (A1)-(A3).
With the above notations and assumptions, equation (2.1) can be represented
under the Fourier basis by
dXk+[k 2Xk+Bk(X)]dt=qkdwk(t), k Zh(n0)
| | ∈
(2.4) dXk+[k 2Xk+Bk(X)]dt=0, k Zl(n0)
| | ∈
Xk(0)=xk, k∈Z3∗
where xk,Xk,Bk(X) k⊥.
∈
We further need the following notations:
(B) denotes the Borel measurable bounded function space on the given
b
• B
Banach space B. denotes the norm of a given Banach space B
B
|·|
and , denote the norm and the inner product of H respectively.
• |·| h· ·i
Given any φ C(D(A),R), we denote
• ∈
φ(x+εh) φ(x)
(2.5) D φ(x):= lim − ,
h
ǫ→0 ε
provided the above limit exists, it is natural to define Dφ(x) : D(A) R
→
by Dφ(x)h = D φ(x) for all h D(A). Clearly, Dφ(x) D(A−1). We
h
∈ ∈
call Dφ the first order derivative of φ, similarly, one can define the second
order derivative D2φ and so on. Denote Ck(D(A),R) the set of functions
b
from D(A) to R with bounded 0-th, ..., k-th order derivatives.
Let B be some Banach space and k Z , define C (D(A),B) as the
+ k
• ∈
function space from D(A) to B with the norm
φ(x)
B
φ := sup | | φ C (D(A),B).
|| ||k 1+ Axk ∈ k
x∈D(A) | |
For any γ >0 and 0 β 1, define the Ho¨lder’s norm by
2,β
• ≤ ≤ ||·||
φ(x) φ(y)
φ = sup | − | ,
|| ||2,β Aγ(x y)β(1+ Ax2+ Ay 2)
x,y∈D(A)| − | | | | |
and the function space Cβ (D(A),R) by
2,γ
(2.6) Cβ (D(A),R)= φ C (D(A),R); φ = φ + φ < .
2,γ { ∈ 2 || ||C2β,γ || ||2 || ||2,β ∞}
2.2. Main results. The following definition of Markov family follows that in [5].
Definition 2.3. Let (Ω , ,P ) be a family of probability spaces and let
x x x x∈D(A)
F
(X(,x)) be a family of stochastic processes on (Ω , ,P ) . Let
x∈D(A) x x x x∈D(A)
· F
( t) be the filtration generated by X(,x) and let be the law of X(,x)
Fx t≥0 · Px ·
under P . The family of (Ω , ,P ,X(,x)) is a Markov family if the fol-
x x x x x∈D(A)
F ·
lowing condition hold:
(1) For any x D(A), t 0, we have
∈ ≥
P (X(t,x) D(A))=1.
x
∈
EXPONENTIAL MIXING OF NSE WITH MILDLY DEGENERATE NOISE 5
(2) The map x is measurable. For any x D(A), t , ,t 0 ,
x 0 n
→ P ∈ ··· ≥
A , ,A D(A) Borel measurable, we have
0 n
··· ⊂
P (X(t+ ) t)= ( )
x · ∈A|Fx PX(t,x) A
where = (y(t ), ,y(t ));y(t ) A , ,y(t ) A .
0 n 0 0 n n
A { ··· ∈ ··· ∈ }
The Markov transition semigroup (P ) associated to the family is then defined
t t≥0
by
P φ(x)=E [φ(X(t,x))], x D(A) t 0.
t x
∈ ≥
for all φ (D(A),R).
b
∈B
The main theorems of this paper are as the following, and will be proven in
Section 5.
Theorem2.4. ThereexistsaMarkovfamilyofmartingalesolution(Ω , ,P ,X(,x))
x x x x∈D(A)
F ·
of the equation (2.1). Furthermore, the transition semigroup (P ) is stochasti-
t t≥0
cally continuous.
Theorem 2.5. The transition semigroup (P ) in the previous theorem is strong
t t≥0
Feller and irreducible. Moreover, it admits a unique invariant measure ν supported
on D(A) such that, for any probability measure µ supported on D(A), we have
(2.7) P∗µ ν Ce−ct 1+ x2µ(dx)
|| t − ||var ≤ | |
(cid:18) ZH (cid:19)
where isthetotalvariationofsignedmeasures,andC,c>0aretheconstants
var
||·||
depending on Q.
2.3. KolmogorovequationsforGalerkinapproximation. Letusconsiderthe
Galerkin approximations of the equation (2.4) as follows
dX = [AX +B (X )]dt+Q dW
m m m m m t
(2.8) −
(Xm(0)=xm
where x π D(A), B (x) = π B(π x) and Q = π Q. The Kolmogorov
m m m m m m m
∈
equation for (2.8) is
∂ u = 1Tr[Q Q∗ D2u ] Ax+B (x),Du
(2.9) t m 2 m m m −h m mi
(um(0)=φ
where φ is some suitable test function and
1
:= Tr[Q Q∗ D2] Ax+B (x),D
Lm 2 m m −h m i
is the Kolmogorov operator associated to (2.8). It is well known that (2.9) is
uniquely solved by
(2.10) u (t,x)=E[φ(X (t,x))], x π D(A).
m m m
∈
Now we introduce an auxiliary Kolmogorov equation with a negative potential
K Ax2 as
− | |
∂ v = 1Tr[Q Q∗ D2v ] Ax+B (x),Dv K Ax2v ,
(2.11) t m 2 m m m −h m mi− | | m
(vm(0)=φ,
6 S.ALBEVERIO,A.DEBUSSCHE,ANDL.XU
which is solved by the following Feynman-Kac formula
t
(2.12) v (t,x)=E φ(X (t,x))exp K AX (s,x)2ds .
m m m
{− | | }
(cid:20) Z0 (cid:21)
Denote
(2.13) Smφ(x)=v (t,x),
t m
(2.14) Pmφ(x)=u (t,x),
t m
foranyφ (π D(A)),itisclearthatSmandPmarebothcontractionsemigroups
∈B m t t
on (π D(A)). By Duhamel’s formula, we have
m
B
t
(2.15) u (t)=Smφ+K Sm [Ax2u (s)]ds.
m t t−s | | m
Z0
For further use, denote
t
(2.16) (t)=exp K AX (s)2ds ,
m,K m
E {− | | }
Z0
which plays a very important role in section 3. The K >1 in (2.16) is a large but
fixed number, which conquers the crossing derivative flows (see (3.9) and (3.10)).
We will often use the trivial fact (t)= (t) (t) and
Em,K1+K2 Em,K1 Em,K2
t 1 1
(2.17) AX (s)2 (s)ds= (1 (t)) .
m m,K m,K
| | E K −E ≤ K
Z0
3. Gradient estimate for the semigroups Sm
t
In this section, the main result is as follows, and it is similar to Lemma 3.4 in
[5] (or Lemma 4.8 in [3]).
Theorem 3.1. Given any T > 0 and k Z , there exists some p > 1 such that
+
∈
for any max 1,r 1 <γ 1 with γ =3/4 and r defined in Assumption 2.1, we
{2 − 2} ≤ 6
have
(3.1) A−γDSmφ Ct−α φ 0<t T
|| t ||k ≤ || ||k ≤
for all φ C1(D(A),R), where C =C(k,γ,r,T,K)>0 and α=p+ 1 +r γ.
∈ b 2 −
Remark 3.2. The condition ’γ = 3/4’ is due to the estimate (6.11) about the
6
nonlinearity B(u,v).
[3] and [5] proved the estimate (3.1) by applying the identity
(3.2)
1 t
D Smφ(x)= E[ (t)φ(Xm(t,x)) Q−1D Xm(s,x),dW ]
h t t Em,K h h si
Z0
t s
+2KE (t)φ(Xm(t,x)) (1 ) AXm(s,x),AD Xm(s,x) ds ,
m,K h
E − t h i
(cid:20) Z0 (cid:21)
and bounding the two terms on the r.h.s. of (3.2). Since the Q in Assumption
2.1 is degenerate, the formula (3.2) is not available in our case. Alternatively, we
apply the idea in [7] to fix this problem, i.e. splitting X (t) into the low and high
m
frequency parts, and applying Malliavin calculus and Bismut-Elworthy-Li formula
on the them respectively.
EXPONENTIAL MIXING OF NSE WITH MILDLY DEGENERATE NOISE 7
Let n N be a fixed number throughout this paper which satisfies n > n
0
∈
and will be determined later (n is the constant in Assumption 2.1). We split the
0
Hilbert space H into the low and high frequency parts by
l h
(3.3) π H =π H, π H =(Id π )H.
n n
−
(Weremarkthatthetechniqueofsplittingfrequencyspaceintotwopiecesissimilar
to the well known Littlewood-Paley projection in Fourier analysis.) Then, the
Galerkinapproximation(2.8) with m>n canbe divided into two parts as follows:
l l l l l
dX +[AX +B (X )]dt=Q dW
m m m m m t
(3.4)
h h h h h
dX +[AX +B (X )]dt=Q dW
m m m m m t
l l h h
whereX =π X , X =π X andtheothertermsaredefinedinthe sameway.
m m m m
In particular,
2 2
(3.5) Ql = qijei ej , Qh = qijei ej,
m k k⊗ k m k k⊗ k
k∈Zl(Xn)\Zl(n0)iX,j=1 k∈Zl(Xm)\Zl(n)iX,j=1
with x y :H H defined by (x y)z = y,z x
⊗ −→ ⊗ h i
With such separationfor the dynamics, it is natural to split the Frechet deriva-
tivesonH intothelowandhighfrequencyparts. Moreprecisely,foranystochastic
process Φ(t,x) on H with Φ(0,x) = x, the Frechet derivative D Φ(t,x) is defined
h
by
Φ(t,x+ǫh) Φ(t,x)
D Φ(t,x):= lim − h H,
h
ǫ→0 ǫ ∈
provided the limit exists. The map DΦ(t,x) : H H is naturally defined by
−→ l
DΦ(t,x)h = D Φ(t,x) for all h H. Similarly, one can easily define D Φ(t,x),
h
h l h h l ∈ h l h l
D Φ(t,x), D Φ (t,x), D Φ (t,x) and so on, for instance, D Φ (t,x):π H π H
→
is defined by
h l l h
D Φ (t,x)h=D Φ (t,x) h π H
h
∀ ∈
l l l
with D Φ (t,x)= lim[Φ (t,x+ǫh) Φ (t,x)]/ǫ.
h
ǫ→0 −
Recall that for any φ C1(D(A),R) one can define Dφ by (2.5), in a similar
l ∈ hb l
way as above, D φ(x) and D φ(x)) can be defined (e.g. D φ(x)h = lim [φ(x+
ε→0
l
εh) φ(x)]/ε h D(A) ).
− ∈
Lemma 3.3. Denote Z(t) = te−A(t−s)QdW , for any T > 0 and ε < σ/2 with
0 t
the σ as in Assumption 2.1, one has
R
(3.6) E sup A1+εZ(t)2k C(α)T2k(σ−2ε−2α)
| | ≤
(cid:20)0≤t≤T (cid:21)
where 0 < α < σ/2 ε and k Z . Moreover, as K > 0 is sufficiently large, for
+
− ∈
any T >0 and any k 2, we have
≥
(3.7) E sup (t)AX (t)k C(k,T)(1+ Axk).
m,K m
E | | ≤ | |
(cid:20)0≤t≤T (cid:21)
8 S.ALBEVERIO,A.DEBUSSCHE,ANDL.XU
Proof. The proofof (3.6) is standard(see Proposition3.1of[5]). Writing X (t)=
m
Y (t)+Z (t), and differentiating AY (t)2 (or seeing (3.1) in Lemma 3.1 of [3]),
m m m
| |
we have
(t)AY (t)2 Ax2+ sup AZ (t)2.
m,K m m
E | | ≤| | | |
0≤t≤T
as K >0 is sufficiently large. Hence,
(t)AX (t)2 (t)AY (t)2+ (t)AZ (t)2 Ax2+2 sup AZ(t)2.
m,K m m,K m m,K m
E | | ≤E | | E | | ≤| | | |
0≤t≤T
Hence, by (3.6) and the above inequality, we immediately have (3.7). (cid:3)
The mainingredientsofthe proofofTheorem3.1arethe followingtwo lemmas,
and they will be proven in Appendix 6.3 and Section 4.2 respectively.
Lemma 3.4. Let x D(A) and let X (t) be the solution to (2.8). Then, for any
m
∈
max 1,r 1 < γ 1 with γ = 3/4, h π H and v L2 (0, ;H), as K is
{2 − 2} ≤ 6 ∈ m ∈ loc ∞
sufficiently large, we have almost surely
t
(3.8) AγD X (t)2 (t)+ A1/2+γD X (s)2 (s)ds Aγh2
h m m,K h m m,K
| | E | | E ≤| |
Z0
C
(3.9) |AγDhhXml (t)|2Em,K(t)≤ K|Aγh|2
C
(3.10) |AγDhlXmh (t)|2Em,K(t)≤ K|Aγh|2
t
(3.11) ArD X (s)2 (s)ds Ct1−2(r−γ) Aγh2
h m m,K
| | E ≤ | |
Z0
t t
(3.12) E[ (t) v(s),dW(s) ] E[ 2 (s)v(s)2ds]
Em,K h i ≤ Em,K | |
Z0 Z0
where all the C =C(γ)>0 above are independent of m and K.
Lemma 3.5. Given any φ C1(D(A)) and h πlH, there exists some p >
∈ b ∈
1 (possibly very large) such that for any k Z , we have some constant C =
+
∈
C(p,k)>0 such that
(3.13) E[Dlφ(X (t))D Xl (t,x) (t)] Ct−peCt φ (1+ Axk)h
| m h m Em,K |≤ || ||k | | | |
Proof of Theorem 3.1. Forthe notationalsimplicity,we shalldropthe index inthe
quantities if no confusion arises. For S φ(X(s)), applying Itoˆ formula to X(s)
t−s
and the equation (2.11) to S , (differentiating on s), we have
t−s
d[S φ(X(s)) (s)]= S φ(X(s)) (s)ds+DS φ(X(s)) (s)QdW
t−s K m t−s K t−s K s
E L E E
S φ(X(s)) (s)ds+K AX(s)2S φ(X(s)) (s)ds
m t−s K t−s K
−L E | | E
S φ(X(s))K AX(s)2 (s)ds
t−s K
− | | E
=DS φ(X(s)) (s)QdW
t−s K s
E
where is the Kolmogorovoperator defined in (2.9), thus
m
L
t
(3.14) φ(X(t)) (t)=S φ(x)+ DS φ(X(s)) (s)QdW
K t t−s K s
E E
Z0
EXPONENTIAL MIXING OF NSE WITH MILDLY DEGENERATE NOISE 9
h
Given any h π H, by (A3) of Assumption 2.1 and (3.14), we have y :=
(Qh)−1DhhXh(∈t) somthat t
t/2
h h
E[φ(X(t)) (t) y ,dW ]
EK h s si
Z0
t t/2
(3.15) =E[ DSt−sφ(X(s))EK(s)QhdWsh h(Qh)−1DhhXh(s),dWshi]
Z0 Z0
t/2
h h
= E[D St−sφ(X(s))DhhX (s)EK(s)]ds,
Z0
hence,
(3.16)
t/2 t/2
E[DhhSt−sφ(X(s))EK(s)]ds=E[φ(X(t))EK(t) h(Qh)−1DhhXh(s),dWshi]
Z0 Z0
t/2
l l
+ E[D St−sφ(X(s))DhhX (s)EK(s)]ds.
Z0
By the fact S φ(x)=E[S φ(X(s)) (s)], (3.15) and (3.16), we have
t t−s K
E
2 t/2
DhhStφ(x)= t DhhE[St−sφ(X(s))EK(s)]ds
Z0
2 t/2
= tE[φ(X(t))EK(t) h(Qh)−1DhhXh(s),dWshi]
Z0
2 t/2
l l
+ t E[D St−sφ(X(s))DhhX (s)EK(s)]ds
Z0
4K t/2 s
− t E[St−sφ(X(s))EK(s) hAX(r),ADhhX(r)idr]ds
Z0 Z0
2 2 4K
= I + I I .
1 2 3
t t − t
We now fix T,γ,k,r and let C be constants depending on T,γ,k and r (whose
values can vary from line to line), then I ,I and I above can be estimated as
1 2 3
follows:
t/2
|I1|≤||φ||kE"EK/2(t)(1+|AX(t)|k)EK/2(t)Z0 h(Qh)−1DhhXh(s),dWshi#
1
≤||φ||kE(cid:18)0≤sus≤pT(1+|AX(s)|k)2EK(s)(cid:19)12 E |EK2 (t)Z0t/2h(Qh)−1DhhXh(s),dWshi|2!2
1/2
t/2
≤C||φ||k(1+|Ax|k)"E Z0 |ArDhhXh(s)|2EK(s)ds!#
Ct1/2−(r−γ) φ (1+ Axk)Aγh
k
≤ || || | | | |
10 S.ALBEVERIO,A.DEBUSSCHE,ANDL.XU
where the last two inequalities are by (3.7), (3.12) and (3.11) in order. By (3.9)
and (3.7),
C t/2
I A−γDlS φ E (1+ AX(s)k) (s) dsAγh
| 2|≤ K || t−s ||k | | EK/2 | |
Z0
C t/2 (cid:2) (cid:3)
A−γDS φ ds(1+ Axk)Aγh.
t−s k
≤ K || || | | | |
Z0
By Markov property of X(t) and (3.7), we have
t/2 s
|I3|= E E[φ(X(t))e−KRst|AX(r)|2dr|Fs]EK(s) hAX(r),ADhhX(r)idr ds
Z0 (cid:26) Z0 (cid:27)
t s
≤C||φ||k E[(1+|AX(s)|k)EK/2(s) EK/2(r)|AX(r)|·|ADhhX(r)|dr]ds,
Z0 Z0
moreover, and by Ho¨lder inequality, Poincare inequality Aγ+12x Ax, (2.17)
| | ≥ | |
and (3.8),
s
EK/2(r)|AX(r)|·|ADhhX(r)|dr
Z0
s s
≤( EK/2(r)|AX(r)|2dr)12( |ADhhX(r)|2EK/2(r)dr)12
Z0 Z0
s
≤[ EK/2(r)|A12+γDhhX(r)|2dr]12 ≤|Aγh|;
Z0
hence, by (3.7) and the above,
I Ct φ (1+ Axk)Aγh.
3 k
| |≤ || || | | | |
Collecting the estimates for I , I and I , we have
1 2 3
(3.17)
1 t/2
|DhhStφ(x)|≤C((t−12−(r−γ)+K)||φ||k+ KtZ0 ||A−γDSt−sφ||kds)(1+|Ax|k)|Aγh|
For the low frequency part, according to Lemma 3.5, we have
(3.18)
|DhlStφ(x)| =|DhlSt/2(St/2φ)(x)|
h h
≤|E[D St/2φ(X(t/2))DhlX (t/2)EK(t/2)]|
l l
+|E[D St/2φ(X(t/2))DhlX (t/2)EK(t/2)]|
t/2
+E[|St/2φ(X(t/2))|EK(t/2)K |AX(s)||ADhlX(s)|ds]
Z0
1
C A−γDS φ +t−peCt φ +K φ (1+ Axk)Aγh
≤ K|| t/2 ||k || ||k || ||k | | | |
(cid:26) (cid:27)
where the last inequality is due to (3.10), (3.7) and (3.13), and to the following
estimate (which is obtained by the same argument as in estimating I ):
3
t/2
E[St/2φ(X(t/2))EK(t/2) |AX(s)||ADhlX(s)|ds]≤C||φ||k(1+|Ax|k)|Aγh|
Z0