Weak order in averaging principle for stochastic wave equations with a fast oscillation

0.25 MB·

by  Hongbo Fu

Weak order in averaging principle for stochastic wave
equations with a fast oscillation
Hongbo Fu, Li Wan
7
1 Research Centerof Nonlinear Science, College of Mathematics and Computer Science,
0 Wuhan TextileUniversity,Wuhan, 430073, PR China
2
Jicheng Liu, Xianming Liu
n ∗
a School of Mathematics and Statistics, Huazhong Universityof Science and Technology,
J
Wuhan, 430074, PR China

7
2

]
R
P
Abstract
.
h
t Thisarticledealswiththeweakerrorsforaveragingprincipleforastochastic
a
wave equation in a bounded interval [0,L], perturbed by a oscillating term
m
arising as the solution of a stochastic reaction-diffusion equation evolving with
[
respect to the fast time. Under suitable conditions, it is proved that the rate

1 of weak convergence to the averaged effective dynamics is of order 1 via an
v asymptotic expansion approach.
4
8 Keywords: Stochastic wave equations, averaging principle, invariant measure
9 weak convergence, asymptotic expansion.
7
MSC: primary 60H15, secondary 70K70
0
.
1
0
1. Introduction
7
1
: Let D = [0,L] R be a bounded open interval. In the article, for fixed
v ⊂
T > 0, we consider the following class of stochastic wave equation with fast
i
X oscillating perturbation,
r
a ∂2 Uǫ(ξ)=∆Uǫ(ξ)+F(Uǫ(ξ),Y (ξ))+σ W˙1(ξ), t [0,T],ξ D,
∂t2 t t t tǫ 1 t ∈ ∈
Utǫ(ξ)=0,(ξ,t)∈∂D×(0,T],
U0ǫ(ξ)=x1(ξ),∂U∂tǫt(ξ) t=0 =x2(ξ),ξ ∈D,
(cid:12) (1.1)
 (cid:12)(cid:12)
∗Correspondingauthor at: School ofMathematics andStatistics, Huazhong Universityof
ScienceandTechnology, Wuhan, 430074, China.
Email addresses: [email protected] (HongboFu),[email protected] (LiWan),
[email protected] (JichengLiu),[email protected] (XianmingLiu)
Preprint submitted toArxiv March 21, 2017
whereǫispositiveparameter,Y isgovernedbythestochasticreaction-diffusion
t
equation:
∂ Y (ξ)=∆Y (ξ)+g(Y (ξ))+σ W˙2(ξ), t [0,T],ξ D,
∂t t t t 2 t ∈ ∈
Yt(ξ)=0,(ξ,t) ∂D (0,T], (1.2)
∈ ×
Y0(ξ)=y(ξ),
Assumptions on the smoothness of the drift f and g will be given below. The
stochasticperturbationsareofadditivetypeandW1(ξ)andW2(ξ)aremutually
t t
independent L2(D) valued Wiener processes on a complete stochastic basis
(Ω,F,F ,P), which−will be specified later. The noise strength coefficients σ
t 1
and σ are positive constants and the parameter ǫ is small, which describes
1
the ratio of time scale between the process Xǫ(ξ) and Y (ξ). With this time
t t/ǫ
scale the variable Xǫ(ξ) is referred as slow component and Y (ξ) as the fast
t t/ǫ
component.
The equation (1.1) is an abstract model for a random vibration of a elastic
stringwithafastoscillatingperturbation. Moregenerally,thenonlinearcoupled
wave-heatequationswithfastandslowtimescalesmaydescribeathermoelastic
wavepropagationinarandommedium[9],theinteractionsoffluidmotionwith
other forms of waves [23, 38], wave phenomena which are heat generating or
temperature related [22], magneto-elasticity [26] and biological problems [8, 4,
32].
Averaging principle plays an important role in the study of asymptotic be-
havior for slow-fast dynamical systems. It was first studied by Bogoliubov[2]
for deterministic differential equations. The theory of averaging for stochastic
ordinary equations may be found in [16], the works of Freidlin and Wentzell
[11, 12], Veretennikov [27, 28], and Kifer [19, 20, 21]. Further progress on
averaging for stochastic dynamical systems with non-Gaussian noise in finite
dimensional space was studied in [33, 34, 35, 36, 37]. Concerning the infinite
dimensional case, it is worth quoting the paper by Cerrai [5, 6, 7], Br´ehier [3],
Wang [30], Fu [13] and Bao [1].
In our previous article [14], the asymptotic limit dynamics (as ǫ tends to
0) of system (1.1) was explored within averaging framework. Under suitable
conditions, it can be shown that a reduced stochastic wave equation, without
thefastcomponent,canbeconstructedtocharacterizetheessentialdynamicsof
(1.1)inapathwisesense,asitisdonein[5,6,7]forstochasticpartialequations
of parabolic type and for stochastic ordinary differential equations [15, 24, 29].
Inthepresentpaper,weareinterestedintherateofweakconvergenceofthe
averagingdynamics to the true solution of slow motion Uǫ(ξ). Namely, we will
t
determine the order, with respect to timescale parameter ǫ, of weak deviations
betweenoriginalsolutionofslowequationandthesolutionofthecorresponding
averagedequation. Toour knowledge,up to nowthis problemhas beentreated
onlyinthe caseofdeterministic reactiondiffusionequationsindimensiond=1
subjected with a random perturbation evolving with respect to the fast time
t/ǫ (to this purpose we refer to the paper by Br´ehier [3]). Once the noise is
included inslow variable,the method in[3] usedto obtainthe weak order1 ε
−
2
for arbitrarily small ε > 0 will be more complicated due to the lack of time
regularity for slow solution.
In the situation we are considering, an additive time-space white noise is
included in the slow motion and the main results show that order 1 for weak
convergence can be derived, which can be compared with the order 1 ε in
−
[3]. Under dissipative assumption on Eq. (1.2), the perturbation process Y
t
admits a unique invariantmeasureµ withmixing property. Then, by averaging
the drift coefficient of the slow motion Eq. (1.1) with respect to the invariant
measure µ, the effective equation with following form can be established:
∂2 U¯ (ξ)=∆U¯ (ξ)+F¯(U¯ (ξ))+σ W˙1(ξ),
∂t2 t t t 1 t
U¯t(ξ)=0,(ξ,t) ∂D (0,T],
∈ ×
U¯0(ξ)=x1(ξ), ∂U¯∂tt(ξ) t=0 =x2(ξ),ξ ∈D,
(cid:12)
where for any u,y H :=L2(D), (cid:12)(cid:12)
∈
F¯(u):= F(u,y)µ(dy),u H.
∈
ZH
We prove that, under a smoothness assumption on drift coefficient in the slow
motion equation, an error estimate of the following form
Eφ(Uǫ) Eφ(U¯ ) Cǫ
| t − t |≤
for any function φ with derivatives bounded up to order 3. In order to prove
the validity of above bound, we adopt asymptotic expansion schemes in [3] to
decompose Eφ(Uǫ) with respect to the scale parameter ǫ in form of
t
Eφ(Uǫ)=u +ǫu +rǫ,
t 0 1
where the functions u has to coincide with Eφ(U¯ ) by uniqueness discuss, as
0 t
it can be shown that they are governed by the same Kolmogorov equation via
identificationthe powersofǫ. Due to solvabilityofthe Poissonequationassoci-
ated with generatorof perturbation process Y , an explicit expressionof u can
t 1
be constructed such that its boundedness is based on a priori estimates for the
Y andsmoothdependenceoninitialdataforaveragingequation. Thenextstep
t
consist in identifying rǫ as the solution of a evolutionary equation and showing
that rǫ Cǫ. The proof of bound for rǫ is based on estimates on du1 and
| | ≤ dt
u , where is the Kolmogorov operator associated with the slow motion
2 1 2
L L
equation. We would like to stress that this procedure is quite involved, as it
concerns a system with noise in infinite dimensional space, and the diffusion
term leading to quantitative analysison higher orderdifferentiability of Eφ(U¯ )
t
withrespecttotheinitialdatum. Letusalsoremarkthatasymptoticexpansion
of the solutions of Kolmogorovequations was studied in [17, 18] and [31].
Therestofthe paperisarrangedasfollows. Section2isdevotedtothe gen-
eral notation and framework. The ergodicity of fast process and the averaging
dynamics of system (1.1) is introduced in Section 3. Then the main results of
3
this article, which is derived via the asymptotic expansions and uniform error
estimates, is presented in Section 4. In the final section, we state and prove
technical lemmas applied in the preceeding section.
Throughout the paper, the letter C below with or without subscripts will
denote generic positive constants independent of ǫ, whose value may change
from one line to another.
2. Preliminary
To rewrite the systems (1.1) and (1.2) as the abstract evolution equations,
we present some notations and some well-known facts for later use.
For a fixed domain D =[0,L], we use the abbreviation H :=L2(D) for the
space of square integrable real-valued functions on D. The scalar product and
norm on H are denoted by (, ) and , respectively.
H
· · k·k
We recall the definition of the Wiener process in infinite space. For more
details, see [25]. Let qi,k(ξ) k N be H-valued eigenvectors of a nonnegative,
{ } ∈
symmetric operator Qi with corresponding eigenvalues λi,k k N, for i = 1,2,
{ } ∈
such that
Q q (ξ)=λ q (ξ), λ >0,k N.
i i,k i,k i,k i,k
∈
For i = 1,2, let Wi(ξ) be an H-valued Q -Wiener process with operator Q
t i i
satisfying
+
∞
TrQ = λ <+ .
i i,k
∞
k=1
X
Then
+
∞ 1
Wi(ξ)= λ2 β (t)q (ξ), t 0,
t i,k i,k i,k ≥
k=1
X
where β (t) i=1,2 aremutuallyindependentreal-valuedBrownianmotionson
a prob{abiil,kity b}ak∈seN(Ω,F,F ,P). For the abbreviation, we will sometimes omit
t
the spatial variable ξ in the sequel.
Let ek(ξ) k N denote the complete orthornormal system of eigenfunctions
{ } ∈
in H such that, for k =1,2,...,
∆e =α e , e (0)=e (L)=0,
k k k k k
−
with 0 < α α α . Here we would like to recall the fact that
1 2 k
e (ξ)=sinkπ≤ξ and≤α··=· k≤2π2··f·or k=1,2, .
k L k − L2 ···
Let A be the realizationin H of the Laplace operator ∆ with zero Dirichlet
boundary condition, which generates a strong continuous semigroups E ,
t t 0
{ } ≥
defined by, for any h H,
∈
+
∞
Eth= e−αktek ek,h .
H
Xk=1 (cid:16) (cid:17)
4
It is straightforwardto check that E are contractive semigroups on H.
t t 0
For s R, we introduce the spa{ce H}s≥:=D(( A)s/2), which equipped with
∈ −
inner product
+
hg,his := (−A)s2g,(−A)s2h H = ∞αsi g,ek h,ek H, g,h∈Hs
(cid:16) (cid:17) kX=1 (cid:16) (cid:17)(cid:16) (cid:17)
and the norm
1
+ 2
∞ 2
ϕ = αs ϕ,e
k ks ( k k H)
Xk=1 (cid:16) (cid:17)
for ϕ Hs. It is obvious that H0 =H and Hα Hβ for β α. We note that
∈ ⊂ ≤
in the case of s > 0, H s can be identified with the dual space (Hs) , i.e. the
− ∗
space of the linear functional on Hs which are continuous with respect to the
topology induced by the norm . We shall denote by α the product space
s
Hα Hα 1,α R, endowed wkit·hkthe scalar product H
−
× ∈
x,y = x ,y + x ,y , x=(x ,x )T,y =(y ,y )T,
1 1 α 2 2 α 1 1 2 1 2
α h i h i −
(cid:16) (cid:17)H
and the corresponding norm
1
x = x 2 + x 2 2 , x=(x ,x )T.
k| |kα k 1kα k 2kα−1 1 2
If α = 0 we abbreviate(cid:8)H0 H 1 = a(cid:9)nd = . To consider (1.1)
− 0
× H k|·k| k|·k|
Uǫ
as anabstractevolutionequation, we setVǫ = dUǫ and let Xǫ = t with
t dt t t Vǫ
(cid:20) t (cid:21)
x
Xǫ :=x= 1 . The systems(1.1)and(1.2)canbe rewrittenas anabstract
0 x
2
(cid:20) (cid:21)
form
dXǫ = Xǫdt+F(Xǫ,Yǫ)dt+BdW1,
t A t t t t
dYǫ = 1AYǫdt+ 1g(Yǫ)dt+ σ dW2, (2.1)
 t ǫ t ǫ t √ǫ t
Xǫ =x,Yǫ =y,
0 0
where 
0 I 0 0
:= ,F(x,y):= ,B := ,
A A 0 F(Π1 x,y) I
(cid:20) (cid:21) (cid:20) ◦ (cid:21) (cid:20) (cid:21)
with
x
( )= X =(x ,x )T : X = 2 = 1,
D A 1 2 ∈H A Ax1 ∈H H
(cid:26) (cid:20) (cid:21) (cid:27)
hereAisregardedasanoperatorfromH1toH 1,andΠ denotesthecanonical
− 1
projection H. It is well known that the operator is the generator of a
H → A
strongly continuous semigroup on with the explicit form
t t 0
{S } ≥ H
1
t =eAt = C(t1) (−A)−2S(t) , t 0, (2.2)
S ( A)2S(t) C(t) ≥
(cid:20) − − (cid:21)
5
1 1
whereC(t)=cos(( A)2t)andS(t)=sin(( A)2t)areso-calledcosineandsine
− −
operatorswith the expressionin termofthe orthonormaleigenpairs α1,ei i N
{ }∈
of A:
+
1 ∞
C(t)h=cos(( A)2t)h= cos √α t e ,h e ,
k k k
− { } H ·
Xk=1 (cid:16) (cid:17)
+
1 ∞
S(t)h=sin(( A)2t)h= sin √α t e ,h e .
k k k
− { } H ·
Xk=1 (cid:16) (cid:17)
Moreover, it is easy to check that x x for t 0,x . In order to
t
k|S k| ≤ k| k| ≥ ∈ H
ensure existence anduniqueness ofthe perturbationprocessY we shallassume
t
throughout this paper that:
(Hypothesis 1) For the mapping g : H H, we require that there exists a
→
constant L >0 such that
g
g(u ) g(u ) L ( u u ), u,v H. (2.3)
1 2 g 1 2
k − k≤ k − k ∈
moreover,we assume that L <α .
g 1
Concerning the coefficient F we impose the following conditions:
(Hypothesis 2)Forthe mappingF :H H H,weassumethatthereexists
× →
a constant L >0 such that
F
F(u ,v ) F(u ,v ) L ( u u + v v ), u ,u ,v ,v H. (2.4)
1 1 2 2 F 1 2 1 2 1 2 1 2
k − k≤ k − k k − k ∈
Also suppose that for any u H, the mapping F(u, ): H H is of class C2,
∈ · →
with bounded derivatives. Moreover, we require that there exists a constant L
such that for any u,v,w,y,y H its directional derivatives are well-defined
′
∈
and satisfy
D F(u,y) w L w , (2.5)
u
k · k≤ k k
D2 F(u,y) (v,w) L v w . (2.6)
k uu · k≤ k k·k k
[D F(u,y) D F(u,y )] w L y y w (2.7)
u u ′ ′
k − · k≤ k − k·k k
kDu2u[F(u,y)−F(u,y′)]·(v,w)k≤Lky−y′k·kvk·kwk. (2.8)
Remark 2.1. A simple example of the dirft coefficient F is given by
F(u,y)=F (u)+F (y),
1 2
here F ,F : H H are of class C2 with uniformly bounded derivatives up to
1 2
→
order 2.
According to conditions (2.3) and (2.4), system (2.1) admits a unique mild
solution. Namely, as discussed in [25], for any y H there exists a unique
∈
adapted process Y(y) L2(Ω,C([0,T];H) such that
∈
6
t t
Y (y)=E y+ E g(Y (y))ds+σ E dW2, (2.9)
t t t s s 2 t s s
Z0 − Z0 −
Byarguingasintheproofof[25],Theorem7.2,itispossibletoshowthatthere
exists a constant C >0 such that
E Y (y) 2 C(1+ y 2), t>0, (2.10)
t
k k ≤ k k
and in correspondenceof such Y (y), for any ǫ>0 andx=(x ,x )T there
t 1 2
∈H
exists a unique adapted process Xǫ(x,y) L2(Ω,C([0,T]; )) such that
∈ H
t t
Xǫ(x,y)= x+ F(Xǫ(x,y),Y (y))ds+σ BdW1. (2.11)
t St Z0 St−s s s/ǫ 1Z0 St−s s
We point out that if x = (x ,x )T is taken in D( ) = 1, then Xǫ values in
1 2 A H t
1 for t>0 (see [10]) and satisfies
H
E Xǫ(x,y) 2 C(1+ y 2+ x 2) (2.12)
k| t k|1 ≤ k k k k1
for some constantC >0. Moreover,we presentan estimate for the normof
H−
Xǫ, which is uniform with respect to ǫ>0.
A t
Proposition 2.1. Let Xǫ(x,y) = (Uǫ(x,y),Vǫ(x,y))T be the solution to the
t t t
problem (2.11), where the initial value satisfies Xǫ = x = (x ,x )T 1, and
0 1 2 ∈ H
the function F satisfies (2.4). Then it holds that
E Xǫ(x,y) 2 C(1+ y 2+ x 2). (2.13)
k|A t k| ≤ k k k| k|1
Proof. We have
0 I Uǫ(x,y) Vǫ(x,y)
Xǫ(x,y)= t = t ,
A t A 0 Vǫ(x,y) A(Uǫ(x,y))
(cid:18) (cid:19)(cid:18) t (cid:19) (cid:18) t (cid:19)
so that
Xǫ(x,y) 2 = Vǫ(x,y) 2+ A(Uǫ(x,y)) 2
k|A t k| k t k k t k−1
= Vǫ(x,y) 2+ A21(Uǫ(x,y)) 2. (2.14)
k t k k t k
Let us start to estimate the norm of A21(Uǫ(x,y)) and consider the expression
t
t
A21Utǫ(x,y) = A12C(t)x1−S(t)x2− S(t−s)F(Usǫ(x,y),Ys/ǫ(y))ds
Z0
t
+ σ S(t s)dW1.
1 − s
Z0
Directly, we have
kA12C(t)x1k2+kS(t)x2k2 ≤C(kx1k21+kx2k2). (2.15)
7
In view of the assumptions on F given in (2.4), we obtain
t
E S(t s)F(Uǫ(x,y),Y (y))ds 2
k − s s/ǫ k
Z0
t
C +C E[ Uǫ(x,y) 2+ Y (y) 2]ds
≤ 1 2 k s k k s/ǫ k
Z0
t
≤C1+C2 E[kA21Usǫ(x,y)k2+kYs/ǫ(y))k2]ds,
Z0
and then, thanks to (2.10), we have
t
E S(t s)F(Uǫ(x,y),Y (y))ds
k − s s/ǫ k
Z0
t
≤C1(1+kyk2)+C2 EkA12Usǫ(x,y)k2ds. (2.16)
Z0
Notice that in view of Ito’s isometry, we have
t
E S(t s)dW1 2 C
k − sk ≤ 3
Z0
and then, combining this estimate with (2.15) and (2.16), we have
EkA21Utǫ(x,y)k2 ≤ C1(1+kyk2+kx1k21+kx2k2)
t
+ C2 EkA12Usǫ(x,y)k2ds.
Z0
From the Gronwall’s lemma, this gives
EkA21Utǫ(x,y)k2 ≤ C1(1+kyk2+kx1k21+kx2k2).
In an analogous way, we can prove that
E Vǫ(x,y) 2 C (1+ y 2+ x 2+ x 2).
k t k ≤ 1 k k k 1k1 k 2k
Thanks to (2.14), the two inequalities above yield (2.13).
If is a Hilbert space equipped with inner product (, ) , we denote by
C1( ,XR) the space of all real function φ : R with·c·onXtinuous Fr´echet
X X →
derivative and use the notation Dφ(x) for the differential of a C1 function on
at the point x. Thanks to Riesz representation theorem, we may get the
X
identity for x,h :
∈X
Dφ(x) h=(Dφ(x),h) .
· X
We define C2( ,R) to be the space of all real-valued, twice Fr´echetdifferential
b X
function on , whose first and second derivatives are continuous and bounded.
Forφ C2(X,R), we willidentify D2φ(x) with a bilinearoperatorfrom
to R s∈uchbthXat X ×X
D2φ(x) (h,k)=(D2φ(x)h,k) , x,h,k .
· X ∈X
On some occasions, we also use the notation φ,φ instead of Dφ or D2φ.
′ ′′
8
3. Ergodicity of Yt and averaging dynamics
Now, we consider the transition semigroup P associated with perturbation
t
processY (y) definedby equation(2.9), by setting for anyψ (H) the space
t b
∈B
of bounded functions on H,
P ψ(y)=Eψ(Y (y)).
t t
By arguing as [13], we can show that
E Yt(y) 2 C e−(α1−Lg)t y 2+1 , t>0 (3.1)
k k ≤ k k
(cid:16) (cid:17)
for some constant C >0. This implies that there exists an invariant measure µ
for the Markov semigroup P associated with system (2.9) in H such that
t
P ψdµ= ψdµ, t 0
t
≥
ZH ZH
for any ψ (H) (for a proof, see, e.g., [6], Section 2.1). Then by repeating
b
∈ B
the standard argument as in the proof of Proposition 4.2 in [7], the invariant
measure has finite 2 moments:
−
y 2µ(dy) C. (3.2)
k k ≤
ZH
Let Y (y ) be the solution of (2.9) with initial value Y = y , it can be check
t ′ 0 ′
that for any t 0,
≥
E Y (y) Y (y ) 2 y y 2e ηt (3.3)
t t ′ ′ −
k − k ≤k − k
withη =(α L )>0,whichimpliesthatµistheuniqueinvariantmeasurefor
1 g
−
P . Then, by averaging the coefficient F with respect to the invariant measure
t
µ, we can define a H-valued mapping
F¯(u):= F(u,y)µ(dy),u H,
∈
ZH
and then, due to condition (2.4), it is easily to check that
F¯(u ) F¯(u ) L u u , u ,u H. (3.4)
1 2 1 2 1 2
k − k≤ k − k ∈
Now we will consider the effective dynamics system
∂2 U¯ (ξ)=∆U¯ (ξ)+F¯(U¯ (ξ))+σ W˙ 1, (ξ,t) D [0,T],
∂t2 t t t 1 t ∈ ×
U¯ (ξ)=0,(ξ,t) ∂D [0,+ ), (3.5)
 t
∈ × ∞
U¯0(ξ)=x1(ξ), ∂∂tU¯t(ξ)|t=0 =x2(ξ), ξ ∈D.
FollowingthesamenotationasinSection2,theproblem(3.5)canbetransferred
to a stochastic evolution equation:
dX¯ = X¯ dt+F¯(X¯ )dt+BdW1,
t A t t t (3.6)
(X0 =x,
9
U¯ 0 0
where X¯ = t with V¯ = dU¯ and F¯(x) := = .
t V¯ t dt t F¯(Π x) F¯(u)
t 1
(cid:20) (cid:21) (cid:20) ◦ (cid:21) (cid:20) (cid:21)
The mild form for system (3.6) is given by
t t
X¯ (x)= x+ F¯(X¯ (x))ds+σ BdW1.
t St Z0 St−s s 1Z0 St−s s
By arguing as before, for any x = (x ,x )T the above integral equation
1 2
∈ H
admits a unique mild solution in L2(Ω,C([0,T]; )) such that
H
E X¯ (x) C(1+ x ), t [0,T]. (3.7)
t
k| k|≤ k| k| ∈
4. Asymptotic expansions
Let φ C2(H,R) and define a function uǫ :[0,T] H R by
∈ b ×H× →
uǫ(t,x,y)=Eφ(Uǫ(x,y)).
t
Let Π be the canonical projection H. Define the function Φ: R by
1
H→ H→
Φ(x):=φ(Π x)=φ(x ) for x=(x ,x )T . Clearly, we have
1 1 1 2
∈H
uǫ(t,x,y)=EΦ(Xǫ(x,y)).
t
We now introduce two differential operators associated with the systems (2.9)
and (2.11), respectively:
ϕ(y) = Ay+g(y),D ϕ(y)
1 y
L H
(cid:16) 1 (cid:17)
+ σ2Tr(D2 ϕ(y)Q (Q ) ), ϕ(y) C2(H,R),
2 2 yy 2 2 ∗ ∈ b
Ψ(x) = x+F(x,y),D Ψ(x)
2 x
L A
(cid:16) 1 (cid:17)H
+ σ2Tr(D2 Ψ(x)BQ (BQ ) ), Ψ(x) C2( ,R).
2 1 xx 1 1 ∗ ∈ b H
It is known that uǫ is a solution to the forward Kolmogorovequation:
duǫ(t,x,y)= ǫuǫ(t,x,y),
dt L (4.1)
(uǫ(0,x,y)=Φ(x),
where ǫ = 1 + .
L ǫL1 L2
Also recall the Kolmogorovoperator for the averaging system is defined as
¯Ψ(x) = x+F¯(x),D Ψ(x)
x
L A
(cid:16) 1 (cid:17)H
+ σ2Tr(D2 Ψ(x)BQ (BQ ) ), Ψ(x) C2( ,R).
2 1 xx 1 1 ∗ ∈ b H
10