Weak order in averaging principle for stochastic wave equations with a fast oscillation
Hongbo Fu Li Wan Jicheng Liu Xianming Liu







































hbfuhust@gmail.com

wanlinju@aliyun.com
Research Center of Nonlinear Science, College of Mathematics and Computer Science, Wuhan Textile University, Wuhan, 430073, PR China

   
jcliu@hust.edu.cn

xmliu@hust.edu.cn
School of Mathematics and Statistics, Huazhong University of Science and
Technology, Wuhan, 430074, PR China 






[cor1]Corresponding author at: School of Mathematics and
  Statistics, Huazhong University of Science and Technology, Wuhan,
      430074, China. 



This article deals with the weak errors for averaging principle for
a stochastic wave equation in a bounded interval [0,L], perturbed
by a oscillating term 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 of weak
convergence to the averaged effective dynamics is of order 1 via
an asymptotic expansion approach.









Stochastic wave equations, averaging principle, invariant measure
  weak convergence, asymptotic expansion.


MSC: primary 60H15, secondary 70K70















 

§ INTRODUCTION

Let D=[0, L]⊂ℝ be a bounded open interval.  In the
article, for fixed T>0, we consider the following class of
stochastic wave equation with fast oscillating perturbation,

    ∂^2/∂ t^2 U_t^ϵ(ξ)=Δ
    U_t^ϵ(ξ)+F(U_t^ϵ(ξ), Y_t/ϵ
    (ξ))
    +σ_1Ẇ_̇ṫ^̇1̇(ξ),  t∈ [0, T],ξ∈ D, 
    
    U_t^ϵ(ξ)=0, (ξ, t)∈∂ D× (0, T], 
    
    U_0^ϵ(ξ)=x_1(ξ),  ∂
    U_t^ϵ(ξ)/∂ t|_t=0=x_2(ξ), ξ∈ D,

where ϵ is   positive parameter, Y_t is governed by the
stochastic reaction-diffusion equation:

    ∂/∂ tY_t (ξ)= Δ Y_t (ξ)+
     g( Y_t (ξ))+ σ_2Ẇ_̇ṫ^̇2̇(ξ),  t∈ [0, T],ξ∈ D,
    
    Y_t (ξ)=0, (ξ, t)∈∂ D× (0, T], 
    
    Y_0 (ξ)=y(ξ),

Assumptions on the smoothness of the drift f and g will be given
below. The stochastic perturbations are of additive type and
W^1_t(ξ) and W^2_t(ξ) are mutually independent
L^2(D)-valued Wiener processes on a complete stochastic basis
(Ω, ℱ, ℱ_t, ℙ), which will be
specified later. The noise strength coefficients σ_1 and
σ_1 are positive constants and the parameter ϵ is
small, which describes the ratio of time scale between the process
X^ϵ_t(ξ) and Y_t/ϵ(ξ).  With this time scale
the variable X_t^ϵ(ξ) is referred as slow component and
Y_t/ϵ (ξ) as the fast component.

The equation (<ref>) is an abstract model for a random
vibration of a elastic string with  a fast oscillating perturbation.
More generally, the nonlinear coupled wave-heat equations with fast
and slow time scales may describe a thermoelastic wave propagation
in a random medium <cit.>, the interactions of fluid motion
with other forms of waves <cit.>, wave phenomena which
are heat generating or temperature related <cit.>,
magneto-elasticity <cit.> and biological problems <cit.>.

Averaging principle plays an important role in the study of
asymptotic behavior for slow-fast dynamical systems. It was first
studied by Bogoliubov<cit.> for deterministic
differential equations. The theory of averaging for stochastic
ordinary equations may be found in <cit.>, the works of
Freidlin and Wentzell <cit.>,
Veretennikov <cit.>, and  Kifer <cit.>. Further progress on averaging for stochastic dynamical
systems  with non-Gaussian noise in finite dimensional space was
studied in <cit.>. Concerning the infinite
dimensional case, it is worth quoting the paper by Cerrai
<cit.>, Bréhier <cit.>, Wang
<cit.>,  Fu <cit.> and Bao <cit.>.












In our  previous article <cit.>, the asymptotic limit
dynamics (as ϵ tends to 0) of system
(<ref>) was explored within averaging
 framework. Under suitable conditions, it can be  shown that a reduced stochastic
wave equation, without the  fast component, can be constructed  to
characterize the essential  dynamics of (<ref>) in a
pathwise sense,   as it is done in <cit.> for stochastic partial equations of parabolic type and
for stochastic ordinary differential equations <cit.>.

In the present paper, we are interested in  the rate of weak
convergence of the averaging dynamics to the true solution of slow
motion U^ϵ_t(ξ). Namely, we will determine the order,
with respect to timescale parameter ϵ, of weak deviations
between original solution of slow equation and  the solution  of the
corresponding averaged equation. To our knowledge, up to now this
problem has been treated only in the case of deterministic reaction
diffusion equations  in dimension d=1  subjected with a random
perturbation evolving with respect to the fast time t/ϵ (to
this purpose we refer to the  paper by Bréhier <cit.>).
Once the noise is included in slow variable, the method in
<cit.> used to obtain the weak order 1-ε 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 <cit.>. Under
dissipative assumption on Eq. (<ref>), the
perturbation process Y_t admits a unique invariant measure μ
with mixing property. Then, by averaging the drift coefficient  of
the slow motion Eq. (<ref>) with respect to the
invariant measure μ, the effective equation with following form
can be established:

    ∂^2 /∂ t^2U̅_t(ξ)=ΔU̅_t(ξ)+F̅(U̅_t(ξ))+σ_1Ẇ_̇ṫ^̇1̇(ξ), 
    U̅_t(ξ)=0, (ξ, t)∈∂ D× (0, T],
    U̅_0(ξ)=x_1(ξ), ∂U̅_t(ξ)/∂
    t|_t=0= x_2(ξ), ξ∈ D,

where for any u,y∈ H:=L^2(D),

    F̅(u):=∫_HF(u,y)μ(dy), u∈ H.

We prove that, under a smoothness assumption on drift coefficient in
the slow motion equation, an error estimate of the following form

    |𝔼ϕ( U^ϵ_t)-𝔼ϕ(U̅_t)|≤
    Cϵ

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 <cit.> to decompose
𝔼ϕ(U^ϵ_t) with respect to the scale parameter
ϵ in form of

    𝔼ϕ(U^ϵ_t)=u_0+ϵ u_1+r^ϵ,

where the functions u_0 has to coincide with
𝔼ϕ(U̅_t) by  uniqueness discuss, as it can be
shown that they are governed by the same Kolmogorov equation via
identification the powers of ϵ. Due to solvability of the
Poisson equation associated with generator of perturbation process
Y_t, an explicit expression of u_1 can be constructed such that
its boundedness is based on a priori estimates for the Y_t and
smooth dependence on initial data for averaging equation. The next
step consist in identifying r^ϵ as the solution of a
evolutionary equation and showing that |r^ϵ|≤ Cϵ.
The proof of bound for r^ϵ is based on estimates on
du_1/dt and ℒ_2u_1, where ℒ_2 is
the  Kolmogorov operator associated  with the slow motion 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 analysis on higher order
differentiability of 𝔼ϕ(U̅_t) with respect to the
initial datum. Let us also remark that asymptotic expansion of the
solutions of Kolmogorov equations was studied in <cit.>
and <cit.>.

The rest of the paper is arranged as follows.  Section 2 is devoted
to the general notation and framework. The ergodicity of fast
process and the averaging dynamics of system (<ref>)
is introduced in Section 3. Then the main results of 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
ϵ, whose value may  change from one line to another.












§ PRELIMINARY 

To rewrite  the systems (<ref>) and
(<ref>) 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:=L^2(D)
for the space of square integrable real-valued functions on D. The
scalar product and norm on H are denoted by (·, ·)_H and
·, respectively.

We recall the definition of the Wiener process in infinite space.
For more details, see <cit.>. Let {q_i,k(ξ)}_k∈ℕ be H-valued eigenvectors of a nonnegative, symmetric
operator Q_i with corresponding eigenvalues {λ_i,
k}_k∈ℕ, for i=1, 2, such that

    Q_iq_i,k(ξ)=λ_i, k q_i, k(ξ), λ_i, k>0, k∈ℕ.

For i=1, 2, let W_t^i(ξ) be an H-valued Q_i-Wiener
process with operator Q_i satisfying

    TrQ_i=∑_k=1^+∞λ_i, k< +∞.

Then

    W^i_t(ξ)=∑_k=1^+∞λ^1/2_i, kβ_i, k(t)q_i,k(ξ), t≥ 0,

where {β_i, k(t)}^i=1, 2_k∈ℕ are mutually
independent real-valued Brownian motions on a probability base
(Ω, ℱ, ℱ_t, ℙ). For the
abbreviation, we will sometimes omit the spatial variable ξ in
the sequel.



Let { e_k(ξ)}_k∈ℕ denote the complete
orthornormal system of eigenfunctions in H such that, for k =
1,2,…,

    -Δ
    e_k=α_ke_k,  e_k(0)=e_k(L)=0 ,

with 0<α_1≤α_2≤⋯α_k≤⋯.










Here we would like to recall the fact that
e_k(ξ)=sinkπξ/L and
α_k=-k^2π^2/L^2 for k = 1,2,⋯.

Let A be the realization in H of the Laplace operator Δ
with zero Dirichlet boundary condition, which generates a strong
continuous semigroups {E_t}_t≥ 0, defined by, for any h∈
H,

    E_th=∑_k=1^+∞ e^-α_kte_k(e_k,
    h)_H.

It is straightforward to check that {E_t}_t≥0 are
contractive semigroups on H.


For s∈ℝ, we introduce the space H^s:=D((-A)^s/2),
which equipped with inner product

    ⟨
    g,h⟩_s:=((-A)^s/2g,(-A)^s/2h)_H=∑_k=1^+∞α^s_i(g,e_k)(h,e_k)_H, g,h∈
    H^s

and the norm

    φ_s={∑_k=1^+∞α_k^s(φ,
    e_k)_H^2}^1/2

for φ∈ H^s. It is obvious that H^0=H and
H^α⊂ H^β for β≤α. We note that in
the case  of s>0, H^-s can be identified with the dual space
(H^s)^*, i.e. the space of the  linear functional on H^s
which are continuous with respect to the topology induced by the
norm ·_s. We shall denote by ℋ^α the
product space H^α× H^α-1, α∈ℝ,
endowed with the scalar product

    (x, y)_ℋ^α=⟨
    x_1,y_1⟩_α+⟨ x_2,y_2⟩_α-1,   
    x=(x_1, x_2)^T, y=(y_1, y_2)^T,

and the corresponding norm

    |x|_α={x_1^2_α+x_2^2_α-1}^1/2,  
    x=(x_1,x_2)^T.

If α=0 we abbreviate H^0× H^-1=ℋ and
|·|=|·|_0. To consider (<ref>) as an
abstract evolution equation, we set
V^ϵ_t=d/dtU^ϵ_t and let X^ϵ_t=[
[ U^ϵ_t; V^ϵ_t;       ]] with X_0^ϵ :=x=[
[ x_1; x_2;     ]]. The systems (<ref>) and
(<ref>) can be rewritten as an abstract form

    dX^ϵ_t=𝒜 X^ϵ_tdt+F(X^ϵ_t,Y^ϵ_t)dt+BdW^1_t,
    
    dY^ϵ_t=1/ϵA Y^ϵ_tdt+1/ϵg(Y^ϵ_t)dt+σ/√(ϵ)dW^2_t, 
    
    X^ϵ_0=x, Y^ϵ_0=y,

where

    𝒜:=[
    [ 0 I; A 0 ]], F(x, y):=[
    [           0; F(Π_1∘ x,y) ]], B:=[
    [ 0; I ]],

with

    𝒟(𝒜)={X=(x_1,x_2)^T∈ℋ:𝒜X=[
    [  x_2; Ax_1 ]]∈ℋ}=ℋ^1,
 here A is
regarded as an operator from H^1 to H^-1, and Π_1 denotes
the canonical projection ℋ→ H. It is well
known that the operator 𝒜 is the generator of a strongly
continuous semigroup {𝒮_t}_t≥0 on ℋ
with the explicit form

    𝒮_t=e^𝒜t=[
    [           C(t)  (-A)^-1/2S(t); - (-A)^1/2S(t)           C(t) ]],  t≥ 0,

where C(t)=cos((-A)^1/2 t) and
S(t)=sin((-A)^1/2 t) are so-called cosine and sine
operators  with the expression in
 term of the orthonormal eigenpairs {α_1, e_i}_i∈ℕ of A:


    C(t)h=cos((-A)^1/2 t)h=∑_k=1^+∞cos{√(α_k)t}(e_k,
      h)_H· e_k,
           S(t)h=sin((-A)^1/2 t)h=∑_k=1^+∞sin{√(α_k)t}(e_k,
      h)_H· e_k.



































Moreover, it is easy to check that |𝒮_t
x|≤|x| for t≥ 0, x∈ℋ. In order to ensure
existence and uniqueness of the perturbation process Y_t we shall
assume throughout this paper that:

(Hypothesis 1) For the mapping g:H → H, we
require that there exists a constant L_g>0 such that

    g(u_1)-g(u_2)≤ L_g(u_1-u_2),  u, v∈ H.

moreover, we assume that L_g< α_1.

Concerning the coefficient F we impose the following
conditions:

(Hypothesis 2) For the mapping F:H× H → H,
we assume that there exists a constant L_F>0 such that

    F(u_1,v_1)-F(u_2,v_2)≤
    L_F(u_1-u_2+v_1-v_2), u_1,u_2, v_1,v_2∈
    H.

Also suppose that for any u∈ H, the mapping F(u,·):
H→ H is of class C^2, 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_uF(u,y)· w≤ L w,
          D^2_uuF(u,y)· (v,w)≤ L v·w.
          [D_uF(u,y)-D_uF(u,y')]· w≤ L y-y'·w
          D^2_uu[F(u,y)-F(u,y')]· (v,w)≤ L
    y-y'·v·w.



A simple example of the dirft coefficient  F is given by

    F(u,y)=F_1(u)+F_2(y),

here F_1, F_2: H→ H are of class C^2 with   uniformly
bounded derivatives up to order 2.





According to conditions (<ref>) and (<ref>),
system (<ref>) admits a unique mild solution. Namely,
as discussed in <cit.>, for  any  y∈ H there exists a
unique adapted process Y (y)∈ L^2(Ω,C([0, T];H) such that


    Y _t(y)=E_t y + ∫_0^tE_t-s g( Y_s (y))ds+ σ_2∫_0^tE_t-s dW^2_s,

By arguing as in the proof of <cit.>, Theorem 7.2,  it is
possible to show that there exists a constant C>0 such that

    𝔼Y_t(y)^2≤ C(1+y^2), t>0,

and in correspondence of such Y_t(y), for any ϵ>0 and
x=(x_1, x_2)^T∈ℋ there exists a unique adapted
process X^ϵ(x,y)∈ L^2(Ω,C([0,T ];ℋ)) such
that

    X^ϵ_t(x,y)=𝒮_tx+∫_0^t𝒮_t-sF(X_s^ϵ(x,y), Y
    _s/ϵ(y))ds+σ_1∫_0^t𝒮_t-sBdW^1_s.

We point out   that if x=(x_1,x_2)^T is taken in
D(𝒜)=ℋ^1, then X^ϵ_t values in ℋ^1 for t>0 (see <cit.>) and satisfies

    𝔼|X^ϵ_t(x,y)|^2_1≤ C(1+y^2+x^2_1)

for some constant C>0. Moreover, we present an estimate for the
ℋ-norm of 𝒜X_t^ϵ, which is uniform
with respect to ϵ>0.

Let X^ϵ_t(x, y)=(U_t^ϵ(x, y), V_t^ϵ(x,
y))^T  be the solution to the problem (<ref>), where
the initial value  satisfies X^ϵ_0=x=(x_1, x_2)^T∈ℋ^1, and the function  F satisfies (<ref>).
Then it holds that

    𝔼|𝒜X^ϵ_t(x, y)|^2≤ C
    (1+y^2+|x|_1^2).


We have

    𝒜X^ϵ_t(x,y)=(
    [ 0 I; A 0 ]) (
    [ U_t^ϵ(x,y); V_t^ϵ(x,y) ])=
     (
    [    V_t^ϵ(x,y); A(U_t^ϵ(x,y)) ]),

so that

    |𝒜X^ϵ_t(x,
    y)|^2   =   V_t^ϵ(x,y)^2+A(U_t^ϵ(x,y))^2_-1
       =   V_t^ϵ(x,y)^2+A^1/2(U_t^ϵ(x,y))^2
    .

Let us start to  estimate  the norm of
A^1/2(U_t^ϵ(x,y)) and consider the expression

    A^1/2U_t^ϵ(x,y)   =   A^1/2C(t)x_1-S(t)x_2-∫_0^tS(t-s)F(U_s^ϵ(x,y),Y _s/ϵ(y))ds
       +   σ_1∫_0^tS(t-s)dW^1_s.

Directly, we have

    A^1/2C(t)x_1^2+S(t)x_2^2 ≤
    C(x_1^2_1+x_2^2).

In view of the assumptions on F given in (<ref>), we
obtain

    𝔼∫_0^tS(t-s)F(U_s^ϵ(x,y),Y_s/ϵ(y))ds^2
          ≤ C_1+C_2∫_0^t𝔼
    [U_s^ϵ(x,y)^2+Y_s/ϵ(y)^2]ds
          ≤ C_1+C_2∫_0^t𝔼
    [A^1/2U_s^ϵ(x,y)^2+Y_s/ϵ(y))^2]ds,

and then, thanks to (<ref>), we have

    𝔼∫_0^tS(t-s)F(U_s^ϵ(x,y),Y_s/ϵ(y))ds
          ≤ C_1(1+y^2)+C_2∫_0^t
    𝔼A^1/2U_s^ϵ(x,y)^2ds.

Notice that in view of Ito's isometry,  we have

    𝔼∫_0^tS(t-s)dW^1_s^2≤ C_3

and then, combining this estimate with (<ref>) and (<ref>),
we have

    𝔼A^1/2U_t^ϵ(x,y)^2   ≤   
    C_1(1+y^2+x_1_1^2+x_2^2)
       +   C_2∫_0^t𝔼A^1/2U_s^ϵ(x,y)^2ds.

From the Gronwall's lemma, this gives

    𝔼A^1/2U_t^ϵ(x,y)^2   ≤   
    C_1(1+y^2+x_1_1^2+x_2^2).

In an analogous way,  we can prove that

    𝔼V_t^ϵ(x,y)^2≤
    C_1(1+y^2+x_1_1^2+x_2^2).

Thanks to (<ref>), the two inequalities above yield
(<ref>).












If 𝒳 is a Hilbert space equipped with inner product
(·,·)_𝒳, we denote by C^1(𝒳,ℝ) the space of all real function ϕ:𝒳→ℝ with continuous Fréchet derivative
and  use the notation Dϕ(x) for the differential of a C^1
function on 𝒳 at the point x. Thanks to Riesz
representation theorem, we may get the identity for x,h∈𝒳:

    Dϕ(x)· h=(Dϕ(x), h)_𝒳.

We define C_b^2(𝒳, ℝ) to be the space of all
real-valued, twice Fréchet differential function on 𝒳, whose first and second derivatives are continuous and bounded.
For ϕ∈ C_b^2(𝒳, ℝ), we will identify
D^2ϕ(x) with a bilinear operator from 𝒳×𝒳 to ℝ such that

    D^2ϕ(x)· (h,k)=(D^2ϕ(x)h,k)_𝒳,   x,h,k∈𝒳.

On some occasions, we also use the notation ϕ',ϕ” instead
of Dϕ or D^2ϕ.



§ ERGODICITY OF Y_T AND AVERAGING DYNAMICS

Now, we consider the transition semigroup P_t associated with
perturbation process Y_t(y) defined by equation
(<ref>), by setting for any ψ∈ℬ_b(H) the space of bounded functions on H,

    P_tψ(y)=𝔼ψ(Y_t(y)).

By arguing as <cit.>, we can show that

    𝔼Y_t(y)^2≤
    C(e^-(α_1-L_g)ty^2+1),  t>0

for some constant C>0. This implies that there exists an invariant
measure μ for the Markov semigroup P _t associated with
system (<ref>) in H such that

    ∫_HP_t ψ dμ =∫_Hψ dμ ,   
    t≥ 0

for any  ψ∈ℬ_b(H) (for a proof, see, e.g.,
<cit.>, Section 2.1). Then by repeating the standard
argument as in the proof of Proposition 4.2 in <cit.>,
the invariant measure has finite 2-moments:

    ∫_Hy^2μ(dy)≤ C.

Let Y_t(y') be the solution of  (<ref>) with initial
value Y_0=y', it can be check  that for any t≥0,

    𝔼Y_t(y)-Y_t( y')^2≤y-y'^2e^-η t

with η=(α_1-L_g)>0, which implies that μ is the
unique invariant measure for P _t. Then, by averaging the
coefficient F with respect to the invariant measure μ, we can
define a H-valued mapping

    F̅(u):=∫_HF(u,y)μ(dy), u∈ H,

and then, due to condition (<ref>), it is easily to check
that

    F̅(u_1)-F̅(u_2)≤ Lu_1-u_2,   u_1, u_2∈ H.


Now we  will consider the  effective dynamics system

    ∂^2/∂
    t^2U̅_t(ξ)=ΔU̅_t(ξ)+ F̅(U̅_t(ξ))+σ_1Ẇ_t^1, (ξ,t)∈ D×[0, T],
    U̅_t(ξ)=0, (ξ, t)∈∂
    D× [0, +∞), 
    U̅_0(ξ)=x_1(ξ), ∂/∂
    tU̅_t(ξ)|_t=0=x_2(ξ),  ξ∈ D.

Following the same notation as in Section 2,  the problem
(<ref>) can be transferred to a stochastic
evolution equation:

    dX̅_t=𝒜X̅_tdt+ F̅(X̅_t)dt+BdW^1_t,
    
    X_0=x,

where
 X̅_t=[
[ U̅_t; V̅_t;      ]] with V̅_t=d/dtU̅_t and
F̅(x):=[
[          0; F̅(Π_1∘ x) ]]=[
[     0; F̅(u) ]]. The mild form for system
 (<ref>) is given by

    X̅_t(x)=𝒮_tx+∫_0^t𝒮_t-sF̅(X̅_s(x))ds+σ_1∫_0^t𝒮_t-sBdW^1_s.

By arguing as before, for any x=(x_1, x_2)^T∈ℋ the
above integral equation admits a unique mild solution in L^2(Ω,C([0,T ];ℋ)) such that

    𝔼|X̅_t(x)|≤ C(1+|x|), t∈ [0,
    T].





§ ASYMPTOTIC EXPANSIONS


Let ϕ∈ C_b^2( H , ℝ) and define a function
u^ϵ: [0, T]×ℋ× H→ℝ by

    u^ϵ(t, x,y)=𝔼ϕ(U_t^ϵ(x,y)).

Let Π_1 be the canonical projection ℋ→ H. Define the function Φ: ℋ→ℝ by
Φ(x):=ϕ(Π_1  x)=ϕ(x_1) for x=(x_1,
x_2)^T∈ℋ. Clearly, we have

    u^ϵ(t, x,y)=𝔼Φ(X_t^ϵ(x,y)).

 We now introduce  two differential operators associated with the
   systems (<ref>) and (<ref>), respectively:

    ℒ_1φ(y)   =   (Ay+g(y),
    D_yφ(y))_H
          +1/2σ_2^2Tr(D^2_yyφ(y)Q_2(Q_2)^*), φ(y)∈ C_b^2(H,ℝ),


    ℒ_2Ψ(x)   =   (𝒜x+F(x,y),
    D_xΨ(x))_ℋ
          +1/2σ_1^2Tr(D^2_xxΨ(x)BQ_1(BQ_1)^*), Ψ(x)∈ C_b^2(ℋ,ℝ ).

It is known that u^ϵ is a solution to the forward
Kolmogorov equation:

    d/dtu^ϵ(t, x, y)=ℒ^ϵ u^ϵ(t, x, y),
    
    u^ϵ(0, x,y)=Φ(x),

where ℒ^ϵ=1/ϵℒ_1+ℒ_2.

 Also recall the Kolmogorov operator for the averaging system is
defined as

    ℒ̅Ψ(x)   =   (𝒜x+F̅(x),
    D_xΨ(x))_ℋ
          +1/2σ_1^2Tr(D^2_xxΨ(x)BQ_1(BQ_1)^*), Ψ(x)∈ C_b^2(ℋ, ℝ ).

If we set 
    u̅(t,
    x)=𝔼ϕ(U̅_t(x))=𝔼Φ(X̅_t(x)),
 we
have

    d/dtu̅(t, x)=ℒ̅u̅(t, x),
    u̅(0, x)=Φ(x).

Then the weak difference at time T is equal to

    𝔼ϕ(U̅_T)-𝔼ϕ(U^ϵ_T)=u^ϵ(T,
    x,y)-u̅(T,x).

Henceforth, when there is no confusion, we often omit the temporal
variable t and spatial variables x and y. For example, for
u^ϵ(t, x, y), we often write it as u^ϵ. Our aim is
to seek matched asymptotic expansions for the u^ϵ(T, x,y)
of the form


    u^ϵ=u_0+ϵ u_1+r^ϵ,

where  u_0 and u_1 are smooth functions which will be
constructed below , and r^ϵ is the remainder term. With the
above assumptions and notation we have the following   result, which
is a direct consequence of Lemma <ref>, Lemma
<ref> and Lemma <ref>.

Assume that x∈ℋ^1, y∈ D(A). Then, under Hypotheses
1
 and 2, for any any T>0 and ϕ∈ C_b^3(H), there exist a
 constant C_T,ϕ,x,y such that
 
    |𝔼ϕ(U^ϵ_T(x,y))-𝔼ϕ(U̅_T(x))|≤
     C_T,ϕ,x,yϵ.






 §.§ The leading term

Let us first   determine the leading terms. Now, substituting
(<ref>) into (<ref>) yields

    du_0/dt+ϵdu_1/dt+dr^ϵ/dt   =   1/ϵℒ_1u_0+ℒ_1u_1+1/ϵℒ_1r^ϵ
          +ℒ_2u_0+ϵℒ_2u_1+ℒ_2r^ϵ.

By comparing coefficients of  powers of ϵ, we obtain

    ℒ_1u_0=0, 
          du_0/dt=ℒ_1u_1+ℒ_2u_0.

It follows from (<ref>) that u_0 is independent of y,
which means 
    u_0(t,x, y)=u_0(t,x).
 We also impose the initial
condition u_0(0,x)=Φ(x). Since μ is the invariant measure
of a Markov process with generator ℒ_1, we have

    ∫_Hℒ_1u_1(t,x,y)μ(dy)=0,

which, by invoking (<ref>), implies

    du_0/dt(t,x)   =   ∫_Hdu_0/dt(t,x)μ(dy)
       =   ∫_Hℒ_2u_0(t,x)μ(dy)
       =   (𝒜u_0(t,x)+∫_HF(x,y)μ(dy),
    D_xu_0(t,x))_ℋ
          +1/2σ_1^2Tr(D^2_xxu_0(t,x)BQ_1(BQ_1)^*)
       =   ℒ̅u_0(t,x),

so that u_0 and u̅ satisfies the same evolution equation.
By using   a uniqueness argument, such u_0 has to coincide with
the solution u̅  and we have the following lemma:

Assume Hypotheses 1 and 2. Then for any x∈ D(𝒜), y∈
D(A) and T>0, we have u_0(T,x,y)=u̅(T,x).




 §.§ Construction of u_1

Let us proceed to carry out the construction of u_1. Thanks to
Lemma <ref> and (<ref>), the equation
(<ref>) can be rewritten

    ℒ̅u̅=ℒ_1u_1+ℒ_2u̅,

and hence we get an elliptic equation for u_1 with form

    ℒ_1u_1(t,x,y)=(F̅(x)- F(x,y),
    D_xu̅(t,x))_ℋ:=-ρ(t,x,y),

where ρ is of class C^2 with respect to y, with uniformly
bounded derivative. Moreover, it satisfies for any t≥ 0 and
x∈ℋ^1,

    ∫_Hρ(t,x,y)μ(dy)=0.

For any y∈ D(A) and s>0 we have

    d/dsP_sρ(t,x,y)   =   (Ay+g(x,y),D_y(P_sρ(t,x,y)))_H
       +   1/2σ_2^2Tr[D^2_yy(P_sρ(t,x,y))Q_2Q_2^*],

here 
    P_sρ(t,x,y)=𝔼ρ(t, x,Y_s(y))
 satisfying

    lim_s→+∞𝔼ρ(t,
    x,Y_s(y))=∫_Hρ(t,x,z)μ(dz)=0.
   Indeed, by
the invariant property of μ and Lemma <ref>,

    |𝔼ρ(t,
    x,Y_s(y))-∫_Hρ(t,x,z)μ(dz)|
          =|∫_H𝔼[ρ(t, x,Y_s(y))- ρ(t,
    x,Y_s(z))μ(dz)]|
          ≤∫_H|𝔼(F(x,Y_s(z)-
    F(x,Y_s(y), D_xu̅(t,x))_ℋ|μ(dz)
          ≤ C∫_H 𝔼Y_s(z)- Y_s(y)μ(dz).

This, in view of (<ref>) and (<ref>),
yields

    |𝔼ρ(t,
    x,Y_s(y))-∫_Hρ(t,x,z)μ(dz)|
          ≤ Ce^-η/2s,

which implies the equality (<ref>). Therefore, we get

    (Ay+g(x,y),D_y∫_0^+∞P_sρ(t,x,y)
    ds)_H
          +1/2σ_2^2Tr[D^2_yy∫_0^+∞(P_sρ(t,x,y))Q_2Q_2^*]ds
          =∫_0^+∞d/dsP_sρ(t,x,y)ds
          =lim_s→+∞𝔼ρ(t,
    x,Y_s(y))-ρ(t,x,y)
          =∫_Hρ(t,x,y)μ(dy)-ρ(t,x,y)
          =-ρ(t,x,y),

which means ℒ_1(∫_0^+∞P_sρ(t,x,y)
ds)=-ρ(t,x,y). Therefore, we can set

    u_1(t,x,y)=∫_0^+∞𝔼ρ(t,x,Y_s(y))ds.


Assume Hypotheses 1  and 2. Then for any x∈ D(𝒜),
y∈ D(A) and T>0, we have

    |u_1(t,x,y)|≤ C_T(1 +y), t∈[0, T].


As known from (<ref>), we have

    u_1(t,x,y)=∫_0^+∞𝔼(F̅(x)-
    F(x,Y_s(y)), D_xu̅(t,x))_ℋds.

This implies that

    |u_1(t,x,y)|   ≤   ∫_0^+∞|F̅(x)-
    𝔼[F(x,Y_s(y))]|  ·|D_xu̅(t,x)| ds.

Then, in view of Lemma <ref> and (<ref>),
this implies :

    |u_1(t,x,y)|   ≤    C_T(1  +y)∫_0^+∞e^-η/2 sds
       ≤   C_T(1  +y).







 §.§ Determination of remainder r^ϵ

Once  u_0 and u_1   have been determined, we can carry out the
construction of the remainder r^ϵ. It is known that

    (∂_t-ℒ^ϵ)u^ϵ=0,

which, together with (<ref>) and (<ref>),
implies

    (∂_t-ℒ^ϵ)r^ϵ   =   -(∂_t-ℒ^ϵ)
    u_0-ϵ(∂_t-ℒ^ϵ)u_1
       =   -(∂_t-1/ϵℒ_1-ℒ_2)u_0-ϵ(∂_t-1/ϵℒ_1-ℒ_2)u_1
       =   ϵ(ℒ_2u_1-∂_tu_1).

In order to estimate the remainder term r^ϵ  we need the
following crucial lemmas.


Assume Hypotheses 1  and 2. Then for any x∈ D(𝒜),
y∈ D(A) and T>0, we have

    |du_1/dt(t,x,y)|≤ C (1+|x|_1 )y,
     t∈ [0, T].


According to (<ref>), we have

    du_1/dt(t,x,y)=∫_0^+∞𝔼(F̅(x)-
    F(x,Y_s(y)), d/dtD_xu̅(t,x))_ℋds.

For any h=(h_1,h_2)^T∈ℋ^1,

    D_xu̅(t,x)· h   =   D_x[𝔼ϕ(Π_1∘X̅(t,x))]
       =   𝔼[ϕ'(Π_1∘X̅(t,x))·
    (Π_1∘η_t^h,x)],

and hence

    d/dt(D_xu̅(t,x)·
    h)   =   𝔼[ϕ”(Π_1∘X̅(t,x))·(Π_1∘η_t^h,x,
    d/dt(Π_1∘X̅(t,x)) )]
       +   𝔼[ϕ'(Π_1∘X̅(t,x))·d/dt(Π_1∘η_t^h,x)],

so that, due to the fact of ϕ∈ C_b^2(H, ℝ), we
obtain

    |d/dt(D_xu̅(t,x)· h)|   ≤   
    C[𝔼Π∘η_t^h,x^2]^1/2·[𝔼d/dt(Π_1∘X̅(t,x)^2]^1/2
          +
    𝔼d/dt(Π_1∘η_t^h,x
       ≤    C |h|_1·[𝔼d/dt(Π_1∘X̅(t,x)^2]^1/2
          +
    𝔼d/dt(Π_1∘η_t^h,x,

where we used the estimate (<ref>) such that

    Π_1∘η_t^h,x≤ C|h| ≤ C|h|_1.

Now, as X̅_t(x) is the  mild solution of averaging equation
with initial data x=(x_1,x_2)^T∈ℋ^1, we have

    Π_1∘X̅_t(x)   =   U̅_t(x)
       =   C(t) x_1+(-A)^-1/2S(t) x_2+∫_0^t
    (-A)^-1/2S(t-s)F̅(U̅_s(x))ds
          +σ_1∫_0^t (-A)^-1/2S(t-s)dW^1_s

with

    d/dt[Π_1∘X̅_t(x)]   =   -(-A)^1/2S(t)
    x_1+C(t)
    x_2+∫_0^t C(t-s)F̅(U̅_s(x))ds
          +σ_1∫_0^t
     C(t-s)dW^1_s,

By straightforward computation, we have

    -(-A)^1/2S(t) x_1^2≤x_1_1^2

and

    C(t) x_2^2≤x_2^2.

According to the Lipschitz continuity of F̅ and
(<ref>), we have

    𝔼∫_0^t C(t-s)F̅(U̅_s(x))ds^2   ≤   
    C_T𝔼∫_0^t(1+U̅_s(x)^2)ds
       ≤    C_T (1+|x|_1).

Now, from (<ref>)-(<ref>) it follows

    𝔼d/dt[Π_1∘X̅_t(x)]^2≤
    C(1+|x|^2_1)

Now, we prove uniform bounds for time derivative of
Π_1∘η_t^h,x with respect to x. Clearly,  we have

    d/dt(Π_1∘η_t^h,x)   =   -(-A)^1/2S(t)h_1+C(t) h_2
       +   ∫_0^t
     C(t-s)[Π_1∘(F̅'(X̅_s(x))·η_t^h,x)]ds.

Note that for any h=(h_1, h_2)^T∈ℋ^1,

    (-A)^1/2S(t)h_1^2+C(t) h_2^2≤|h|_1^2.

In view of (<ref>) and (<ref>), we obtain

    ∫_0^t
     C(t-s)[Π_1∘(F̅'(X̅_s(x))·η_t^h,x)]ds
       ≤    C∫_0^t|η_s^h,x|ds
     
       ≤    C_T |h|_1.

Then thanks to (<ref>) and (<ref>), we obtain

    d/dt(Π_1∘η_t^h,x)^2≤ C |h|_1^2.

So, if we plug the above estimate and estimate (<ref>) into
(<ref>), we get

    |d/dtD_xu̅(t,x)· h|≤
    C|h|_1(1+|x|_1 ),

which, together with (<ref>), implies

    |du_1/dt(t,x,y)|   ≤   C(1+|x|_1
    )∫_0^+∞|F̅(x)- 𝔼F(x,Y_s(y))|_1ds
       ≤   C (1+|x|_1)y∫_0^+∞e^-η/2 sds
       ≤   C (1+|x|_1 )y.

Hence the assertions is completely proved.













Assume that all conditions in Lemma <ref>  are fulfilled. Then
we have

    |ℒ_2u_1(t,x,y)|≤(1+|𝒜x|+|x|_1+y)(1+y),
     t∈ [0, T].


As known, for any x∈ D(𝒜),

    ℒ_2u_1(t,x,y)   =   (𝒜x+ F(x, y),
    D_xu_1(t,x,y)
    )_ℋ
       +   1/2σ_2^2Tr(D^2_xxu_1(t,x,y)(BQ_1)(BQ_1)^*).

We will carry out the estimate of |ℒ_2u_1(t,x,y)| in two steps.

(Step 1) Estimate of (𝒜x+ F(x, y),
D_xu_1(t,x,y) )_ℋ.

 For any k∈ℋ, we have

    D_xu_1(t,x,y)·
    k   =   ∫_0^+∞𝔼(D_x(F̅(x)-
    F(x,Y_s))· k,D_xu̅(t,x))_ℋds
          +∫_0^+∞𝔼(F̅(x)-
    F(x,Y_s),D^2_xxu̅(t,x)·
    k)_ℋds
       :=   I_1(t,x,y,k)+I_2(t,x,y,k).

According to the invariant property of measure μ, (<ref>)
and (<ref>) we have

    |I_1(t,x,y,k)|
       ≤   ∫_0^+∞|𝔼(D_x(F̅(x)-
    F(x,Y_s(y)))·
    k,D_xu̅(t,x))_ℋ| ds
       =   ∫_0^+∞|  𝔼∫_H
    (D_x[F(x,z)- F(x,Y_s(y))]·
    k,D_xu̅(t,x))_ℋμ(dz)|ds
       =   ∫_0^+∞|  𝔼∫_H
    (D_x[F(x,Y_s(z))- F(x,Y_s(y))]·
    k,D_xu̅(t,x))_ℋμ(dz)|ds
       ≤   C|k| ·|D_xu̅(t,x)|
    ·∫_0^+∞[∫_H𝔼Y_s(z)-Y_s(y)μ(dz)]ds

By making use of (<ref>) and (<ref>),
the above yields

    |I_1(t,x,y,k)|   ≤   C|k| ·|D_xu̅(t,x)|
    ·∫_0^+∞
    e^-η/2 s (1+y)ds
       ≤    C|k| ·|D_xu̅(t,x)|
       ≤    C|k|,

where we used  Lemma <ref> in the last step. By Lemma <ref>
and (<ref>),  we have

    |I_2(t,x,y,k)|   ≤   ∫_0^+∞|(F̅(x)-
    𝔼F(x,Y_s(y)),D^2_xxu̅(t,x)·
    k)_ℋ|ds
       ≤    C|k| ∫_0^+∞|F̅(x)-
    𝔼F(x,Y_s(y))|  ds
       ≤   C|k| (1  +y)∫_0^+∞ e^-η/2 s ds
       ≤   C|k| (1  +y).

Together with (<ref>) , this yields

    |D_xu_1(t,x,y)· k|   ≤   C|k| (1+ y)

which means

    |( 𝒜x+ F(x,
    y), D_xu_1(t,x,y) )_ℋ|
          ≤ C(1+|𝒜x|+|x|_1+y)(1+y).

(Step 2) Estimate of
Tr(D^2_xxu_1(t,x,y)(BQ_1)(BQ_1)^*). Note that we have

    D_xxu_1(t,x,y)· (h,
    k)
          =∫_0^+∞𝔼(D^2_xx(F̅(x)-
    F(x,Y_s(y)))· (h,k),D_xu̅(t,x))_ℋds
          +∫_0^+∞𝔼(D_x(F̅(x)-
    F(x,Y_s(y)))· h,D^2_xxu̅(t,x)·
    k)_ℋds
          +∫_0^+∞𝔼(D_x(F̅(x)-
    F(x,Y_s(y)))· k,D^2_xxu̅(t,x)·
    h)_ℋds
          +∫_0^+∞𝔼(F̅(x)-
    F(x,Y_s(y)), D^3_xxxu̅(t,x)·
    (h,k))_ℋds
          :=∑_i=1^4J_i(t,x,y,h,k).

In view of (<ref>) and invariant property of measure μ
we have

    |J_1(t,x,y,h,k)|
       ≤   ∫_0^+∞|𝔼(D^2_xx(F̅(x)- F(x,Y_s(y)))·
    (h,k),D^2_xu̅(t,x))_ℋ| ds
       =   ∫_0^+∞|  𝔼∫_H
    (D^2_xx[F(x,z)- F(x,Y_s(y))]·
    (h,k),D_xu̅(t,x))_ℋμ(dz)|ds
       =   ∫_0^+∞|  𝔼∫_H
    (D^2_xx[F(x,Y_s(z))- F(x,Y_s(y))]·
    (h,k),D_xu̅(t,x))_ℋμ(dz)|ds

By taking (<ref>) and Lemma <ref>   into account, we can
deduce

    |J_1(t,x,y,h,k)|
       ≤   C|h|·|k|
    ·∫_0^+∞[∫_H𝔼Y_s(z)-Y_s(y)μ(dz)]ds
       ≤   C|h|·|k|  ·∫_0^+∞
    e^-η/2 s (1+y)ds
       ≤    C|h|·|k|  
       ≤    C|h|·|k| (1+y),

Again, by   (<ref>) and invariant property of measure μ, we
have

    |J_2(t,x,y,h,k)|
       ≤   ∫_0^+∞|  𝔼∫_H
    (D_x[F(x,Y_s(z))- F(x,Y_s(y))]·
    h,D^2_xxu̅(t,x)· k)_ℋμ(dz)|ds,

which, by Lemma <ref> and condition (<ref>), implies

    |J_2(t,x,y,h,k)|
       ≤   C|h|·|k|
    ·∫_0^+∞[∫_H𝔼Y_s(z)-Y_s(y)μ(dz)]ds
       ≤   C|h|·|k| ·∫_0^+∞
    e^-η/2 s (1+y)ds
       ≤    C|h|·|k|(1+y).

Parallel to (<ref>),  we can obtain the same estimate for
J_3(t,x,y, h,k), that is,

    |J_3(t,x,y,h,k)| ≤ C|h|·|k|(1+y).

By proceeding again as in the estimate for J_1(t,x,y,h,k) we have

    |J_4(t,x,y,h,k)|
       ≤   ∫_0^+∞|  𝔼∫_H
    (F(x,Y_s(z))- F(x,Y_s(y))
    ,D^3_xxxu̅(t,x)·(h,k))_ℋμ(dz)|ds

and then thanks to Lemma <ref> and (<ref>), we get

    |J_4(t,x,y,h,k)|
       ≤   C|h|·|k|
    ·∫_0^+∞[∫_H𝔼Y_s(z)-Y_s(y)μ(dz)]ds
       ≤    C|h|·|k|(1+y).

Collecting together (<ref>), (<ref>), (<ref>) and
(<ref>), we obtain


    |D^2_xxu_1(t,x,y)· (h, k)|≤ C |h|·|k|(1+y),

which means that for fixed y∈ H and t∈ [0, T],

    D^2_··u_1(t,·,y)
    _L(ℋ×ℋ,ℝ)≤ C(1+y),

so that, as the operator Q_1 has finite trace, we get

    Tr(D^2_xxu_1(t,x,y)(BQ_1)(BQ_1)^*)
          ≤D^2_xxu_1(t,x,y)Tr( (BQ^1)(BQ^1)^*)
          ≤ C(1+y).


Finally, by taking  inequalities (<ref>) and (<ref>) into
account, we can conclude the proof of the lemma.


As a consequence of Lemma <ref> and <ref>, we have the
following fact for the remainder term r^ϵ.

Under the conditions of Lemma <ref>, for any T>0, x∈
D(𝒜), y ∈ H, we have

    |r^ϵ(T,x,y)|≤
    Cϵ(1+|x|+y)(1+|𝒜x|+|x|_1).






By a variation of constant formula, we have

    r^ϵ(T,x,y)   =   𝔼[r^ϵ(0,X^ϵ_T(x,y),Y_T/ϵ( y)]
       +   ϵ𝔼[∫_0^T(ℒ_2u_1-∂
    u_1/∂ s)( X^ϵ_T-s(x,y),Y_T-s/ϵ(
    y)) ds].

Since u^ϵ and u_0=u̅ has the same initial condition
Φ(x), we have

    |r^ϵ(0, x,y)|   =   |u^ϵ(0,x,y)-u̅(0,x)-ϵ
    u_1(0,x,y)|
       =   ϵ |u_1(0,x,y)|,

so that, from (<ref>) and (<ref>)   we have

    𝔼[r^ϵ(0,X^ϵ_T(x,y),Y_T/ϵ(y)]≤
    Cϵ(1+ y).

Thanks to Lemma <ref> and Lemma <ref>, we have

    𝔼[(ℒ_2u_1-∂ u_1/∂ s)(
    X^ϵ_T-s(x,y),Y_T-s/ϵ(y))]
          ≤ C 𝔼[(1+|𝒜X^ϵ_T-s(x,y)|
    +|X^ϵ_T-s(x,y)|_1+Y_T-s/ϵ(y))
            ·(1+Y_T-s/ϵ(y))],

and, according to (<ref>), (<ref>), (<ref>) and
the Hölder inequality, this implies that

    𝔼[∫_0^T(ℒ_2u_1-∂
    u_1/∂ s)( X^ϵ_T-s(x,y),Y_T-s/ϵ)
    ds]
          ≤ C (1+ y)(1+|𝒜x|+|x|_1+y).

This, together with (<ref>), implies

    |r^ϵ(T,x,y)|≤ Cϵ(1+ y)
    (1+|𝒜x|+|x|_1+y),

which completes the proof.














§ APPENDIX

In this section, we state and prove some technical lemmas needed in
the former sections.

For any x∈ℋ and y∈ H, there exists a constant
C>0 such that

    |F̅(x)-𝔼[F(x,Y_t(y))]|^2_1 ≤
    Ce^-η t(1 +y^2),

where η= α_1-L_g >0.

According to the invariant property of μ,
(<ref>) and hypothesis (<ref>), we have

    |F̅(x)-𝔼[F(x,Y_t(y))]|^2_1
    
    
    
    
       =   F̅(Π_1x)- 𝔼[F(Π_1x,Y_t(y))]^2
       =   ∫_HF(Π_1 x,z)μ(dz)- 𝔼[F(Π_1x,Y_t(y))]^2
       =   ∫_H 𝔼[F(Π_1x,
    Y_t(z))-F(Π_1x,Y_t(y))]μ(dz)^2,

so thanks to (<ref>) and (<ref>), we
have

    |F̅(x)-𝔼[F(x,Y_t(y))]|_1^2   ≤   
    C∫_H𝔼Y_t(y)-Y_t(z)^2μ (dz)
       ≤   Ce^-η t∫_Hy-z^2μ (dz)
       ≤   Ce^-η t(1 +y^2).





Next, we introduce the following regularity results of averaging
function F̅.

For any w∈ H, the function (F̅(·), w)_H: 
H→ℝ is Gâteaux differential and for any
v∈ H, it hold that

    (DF̅(u)· v, w)_H=∫_H(D_u F(u,y)· v,
    w)_Hμ(dy), w∈ H.


For any λ≠0 we have

    (∫_H1/λ[F(u+λ
    v,y)-F(u,y)]μ(dy),w)_H-∫_H(D_uF(u,y)·
    v,w)_Hμ(dy)
          =∫_H (1/λ[F(u+λ
    v,y)-F(u,y)]-D_uF(u,y)· v,w)_Hμ(dy).

and then

    |(∫_H1/λ[F(u+λ
    v,y)-F(u,y)]μ(dy),w)_H-∫_H(D_uF(u,y)·
    v,w)_Hμ(dy)|
          ≤∫_H |(1/λ[F(u+λ
    v,y)-F(u,y)]-D_uF(u,y)· v,w)_H|μ(dy)
          ≤w∫_H  1/λ[F(u+λ
    v,y)-F(u,y)]-D_uF(u,y)· vμ(dy).

Now, since F(·, y): H→ H is Gâteaux
differentiable in H, for any h∈ H, we obtain

    lim_λ→ 01/λ[F(u+λ
    v,y)-F(u,y)]-D_uF(u,y)· v=0.

Moreover, by mean value theorem,

    1/λ[F(u+λ v,y)-F(u,y)]-D_uF(u,y)·
    v
          =∫_0^1[D_u F(u+λθ v)-D_uF(u,y)]· v
    dθ

so that, due to the boundedness of D_uF(u,y), we get

    1/λ[F(u+λ v,y)-F(u,y)]-D_uF(u,y)· v≤
    C v.

and then, by using the dominated convergence theorem, taking
(<ref>) into account, we can conclude

    lim_λ→
    0(∫_H1/λ[F(u+λ
    v,y)-F(u,y)]μ(dy),w)_H
          =∫_H(D_uF(u,y)· v,w)_Hμ(dy),

which implies that

    (DF̅(u)· v, w)_H=∫_H(D_u F(u,y)·
    v,w)_Hμ(dy).





As a as a consequence of Lemma <ref>, it is easily to
check that

    (D_xF̅(x)·  h, k)_ℋ=∫_H(D_x
    F(x,y)· h,k)_ℋμ(dy),  h,k∈ℋ,

and this yields

    |D_xF̅(x)· h|    ≤   ∫_H|D_x
    F(x,y)· h|μ(dy)
       =   ∫_HA^-1/2[(D_uF)(Π_1∘ x,y)·(Π_1∘
    h)]μ(dy)
       ≤   C∫_H (D_uF)(Π_1∘ x,y)·(Π_1∘ h) μ(dy).

Moreover, by invoking conditions (<ref>), we get

    |D_xF̅(x)· h|    ≤   C |h|,  h∈ℋ.


 As far as the higher order derivative are concerned, by proceeding
as in the proof of above lemma, we can show that

    (D_uu^2F̅(u)·(v,w), ν)_H=∫_H(D_uu
    F(u,y)·(v,w),ν)_Hμ(dy),  v,w,ν∈ H.

As a consequence, we obtain

    (D_xx^2F̅(x)·(h,k), l)_ℋ=∫_H(D_xx^2 F(x,y)·(h,k),l)_ℋμ(dy),  h,k,l∈ℋ

and

    |D_xx^2F̅(x)·(h,k)≤
    C|h|·|k|, h,k∈ℋ.









For any T>0, there exists C_T>0 such that for any
x∈ℋ and t∈ [0, T], we have

    D_xu̅(t,x)≤ C_T.



Note that for any h∈ℋ,

    D_xu̅(t,x)·
    h   =   𝔼[DΦ(X̅_t(x))·η ^h,x_t]
       =   𝔼(Φ'(X̅_t(x)),η^h,x_t)_ℋ,

where η^h,x_t is the mild solution of

    dη ^h,x_t=(𝒜η ^h,x_t+DF̅(X̅_t(x))·η ^h,x_t)dt
    η ^h,x(0)=h.

This means that  η ^h,x_t  is the  solution of the integral
equation

    η^h,x_t=𝒮_th+∫_0^t𝒮_t-s[DF̅(X̅_s(x))·η
    ^h,x_t]ds,

and then thanks to (<ref>), we get

    |η ^h,x_t| ≤|h| +C∫_0^t|η ^h,x_s| ds.

Then by  Gronwall lemma it follows that

    |η ^h,x_t| ≤ C_T|h|,  t∈ [0, T],

which means

    |D_xu̅(t,x)· h|≤ C_Tsup_z∈ℋ|Φ(z)|·|h|,

so that

    |D_xu̅(t,x)| ≤ C_T.



For any T>0, there exists C_T>0 such that for any
x,h,k∈ℋ and t∈ [0, T], we have

    |D^2_xxu̅(t,x)·(h,k)|≤ C_T,ϕ|h|·|k|.



For any h, k ∈ℋ, we have

    D^2_xxu̅(t,x)·(h,k)   =   𝔼[Φ”(X̅_t(x))·(η^h,x_t,η^k,x_t)
       +   Φ'(X̅_t(x))·ζ^h,k,x_t],

where ζ^h,k,x is the mild solution of equation

    dζ^h,k,x_t=[𝒜ζ^h,k,x_t+ F̅”(X̅_t(x))·(η^h,x_t,η^k,x_t)+F̅'(X̅_t(x))·ζ^h,k,x_t]dt
    ζ^h,k,x_0=0.

This means that  ζ^h,k,x_t  is the  solution of the integral
equation

    ζ^h,k,x_t=
    ∫_0^t𝒮_t[F̅”(X̅_s(x))·(η^h,x_s,η^k,x_s)+F̅'(X̅_s(x))·ζ^h,k,x_s]ds.

Thus, by (<ref>) and (<ref>) we have

    |ζ^h,k,x_t|   ≤    C∫_0^t(|η^h,x_s|·|η^k,x_s|+|ζ^h,k,x_s|)ds
       ≤    C|h|·|k| +C∫_0^t|ζ^h,k,x_s|ds.

By applying the Gronwall lemma we have

    |ζ^h,k,x_t|≤ C_T|h|·|k|,   t>0.

Returning to (<ref>), we can get

    |D_xxu̅(t,x)·(h,k)|≤ C |h|·|k|.



By proceeding again as in the proof of above lemma,  we have the
following result.

For any T>0, there exists C_T>0 such that for any
x,h,k,l∈ℋ and t∈ [0, T], we have

    D^3_xxxu̅(t,x)·(h,k,l)≤ C_T,ϕ|h|·|k|·|l|.


















§ ACKNOWLEDGMENTS

We would like to thank Professor Jinqiao Duan for helpful
discussions and comments. Hongbo Fu is supported by CSC scholarship
(No. [2015]5104), NSF  of China (Nos. 11301403, 11405118, 11271295)
and Foundation of Wuhan Textile University 2013. Li Wan is supported
by NSF  of China (No. 11271295) and Science and Technology Research
Projects of Hubei Provincial Department of Education (No.
D20131602). Jicheng Liu is supported by NSF  of China (Nos.
11271013, 10901065). Xianming Liu is  supported by NSF  of China
(No. 11301197)




99

Bao
J. H. Bao, G. Yin, C. G. Yuan, Two-time-scale stochastic partial
differential equations driven by  α-stable noises: Averaging
principles, Bernoulli  23 (1) (2017)  645-669.

Bogoliubov
N. N. Bogoliubov, Y. A. Mitropolsky, Asymptotic Methods in the
Theory of Non-linear Oscillations, Gordon & Breach Science
Publishers, New York, 1961.

Brehier
C. E. Bréhier, Strong and weak orders in averging for SPDEs,
Stochastic Process. Appl. 122 (2012) 2553-2593.


Cardetti
F. Cardetti, Y. S. Choi,  A parabolic-hyperbolic system modelling a
moving cell, Electron. J. Differential Equations 95 (2009) 1-11.

Cerrai1
S. Cerrai, M. I. Freidlin, Averaging principle for a class of
stochastic reaction-diffusion equations, Proba. Theory Related
Fields. 144 (2009) 137-177.

Cerrai2
S. Cerrai, A Khasminkii type averaging principle for stochastic
reaction-diffusion equations, Ann. Appl. Probab. 19 (3) (2009)
899-948.

Cerrai-Siam
S. Cerrai, Averaging principle for systems of reaction-diffusion
equations with polynomial nonlinearities perturbed by multiplicative
type noise, SIAM J. Math. Anal. 43(6) (2011) 2482-2518.

Choi
 Y. S. Choi, Craig Miller, Global existence of solutions to a coupled parabolic-hyperbolic
system with moving boundary, Proc. Amer. Math. Soc. 139 (2011)
3257-3270

Chow-1
P. L. Chow, Thermoelastic wave propagation in a random medium and
some related problems, Int. J. Eng. Sci. 11 (1973) 253-971.

Chow
P. L. Chow, Stochastic Partial Differential Equations, 2nd ed.,
Chapman & Hall/CRC,    New York, 2015.




Freidlin-Wentzell1
M. I. Freidlin, A. D. Wentzell,  Random Perturbation of Dynamical
 Systems, 2nd ed., Springer-Verlag, New York, 1998.

Freidlin-Wentzell2
M. I. Freidlin, A. D. Wentzell, Long-time behavior of weakly
coupled oscillators, J. Stat. Phys 123 (2006) 1311-1337.

Fu-Liu
H. Fu, L. Wan, Y. Wang, J. Liu, Strong convergence rate in averaging
principle for stochastic FitzHug-Nagumo system with two time-scales,
J. Math. Anal. Appl. 416 (2014) 609-628.


Fu-Liu-2
H. Fu,  L. Wan, J. Liu,  Strong convergence in averaging principle
for stochastic hyperbolic-parabolic equations with two time-scales.
Stochastic Process. Appl. 125 (2015) 3255-3279.

Givon1
D. Givon, I. G. Kevrekidis, R. Kupferman, Strong convergence of
projective integration schemes for singular perturbed stochastic
differential systems, Comm. Math. Sci. 4 (2006) 707-729.



Khas
R. Z. Khasminskii, On the principle of averaging the Itô
stochastic differential equations (Russian), kibernetika 4 (1968)
260-279.

Khas2
R. Z. Khasminskii, G. Yin, Limit behavior of two-time-scale
diffusions revisited, J. Differential Equations 212 (1) (2005)
85-113.

Khas3
R.Z. Khasminskii, G. Yin, On Averaging Principles: An Asymptotic
Expansion Approach, SIAM J. Math. Anal  35(6)  1534-1560.





Kifer1
Y. Kifer, Some recent advance in averaging, Modern dynamical
systems and applications, Cambridge University Press, Cambridge, UK,
2004, pp. 385-403.

Kifer2
Y. Kifer, Diffusion approximation for slow motion in fully coupled
averaging, Proba. Theor. Relat. Fields, 129 (2004) 157-181.

Kifer3
Y. Kifer, Another proof of the averaging principle for fully
coupled dynamical systems with hyperbolic fast motions, Discrete
Contin. Dyn. Syst. 13 (2005) 1187-1201.



Leung-0
A. W. Leung, Asymptotically stable invariant manifold for coupled
nonlinear parabolic-hyperbolic partial differential equations, J.
Differential Equations 187 (2003) 184-200.

Leung
A. W. Leung, Stable invariant manifolds for coupled Navier-Stokes
and second-order wave systems, Asymptotic Analysis 43 (2005)
339-357.

LiuDi
D. Liu, Strong convergence of principle of averaging for multiscale
stochastic dynamical systems, Commun. Math. Sci. 8 (4) (2010)
999-1020.





Daprato
G. Da Prato, J. Zabczyk, Stochastic Equations in Infinite
Dimensions, 2nd ed., Cambridge University Press,  Cambridge,
2014.



Rivera
J. E. M. Rivera, R. Racke, Smoothing properties, decay and global
existence of solutions to nonlinear coupled systems of
thermoelasticity type, SIAM J. Math. Anal. 26
 (1995) 1547-1563.

Vere1
A. Y. Veretennikov, On the averaging principle for systems of
stochastic differential equations, Mathematics of the USSR-Sbornik
69 (1991) 271-284.

Vere2
A. Y. Veretennikov, On large deviations in the averaging principle
for SDEs with full dependence, Ann. Probab. 27 (1999) 284-296.

Wainrib
G. Wainrib, Double averaging principle for periodically forced
slow-fast stochastic systems, Electron. Commun. Probab. 18(51)
(2013) 1-12.

wangwei

W. Wang, A. J. Roberts, Average and deviation for slow-fast
stochastic partial differential equations. J. Differential Equations
253 (2012) 1265-1286


Weinan
E. Weinan, D. Liu, E. Vanden-Eijnden, Analysis of multiscale methods
for stochastic differential equations, Comm. Pure Appl. Math. 58
(11) (2005) 1544-1585.

S.H. Wu
S. H. Wu, H. Chen, W. X. Li, The local and global existence of the
solutions of hyperbolic-parabolic system modeling biological
phenomena, Acta Mathematica Scientia 28B(1) (2008) 101-116.

Xu
Y. Xu, J. Duan, W. Xu, An averaging principle for stochastic
dynamical systems with Lévy noise, Physica D 240(17) (2011)
1395-1401.

Xu1
Y. Xu, B. Pei, Y. Li, An averaging principle for stochastic
differential delay equations with fractional Brownian motion, Abstr.
Appl. Anal. (2014) ID 479195.

Xu2
Y. Xu, R. Guo, D. Liu, H. Zhang, J. Duan, Stochastic averaging
principle for dynamical systems with fractional Brownian motion,
Discrete Contin. Dynam. Systems (Series B) 19(4) (2014) 1197-1212.

Xu3
Y. Xu, B. Pei, Y. Li, Approximation properties for solutions to
non-Lipschitz stochastic differential equations with Lévy noise,
Math. Methods Appl. Sci. 4 (1) (2014) DOI: 10.1002/mma.3208.

Xu4
Y. Xu, R. Guo,  W. Xu, A limit theorem for the solutions of
slow-fast systems with fractional Brownian motion, Theor. Appl.
Mech. Lett. 4 (2014) ID 3-013003.


X Zhang
X. Zhang, E. Zuazua, Long-time behavior of a coupled heat-wave
system arising in fluid-structure interaction, Arch. Rational Mech.
Anal. 184(1) (2007) 49-120.


















































































































