(hk(x, z)) and f = (fk(z)) are assumed to be periodic with period 1 in all components. 2000 Mathematics Subject Classifi
HOMOGENIZATION FOR STOCHASTIC PARTIAL DIFFERENTIAL EQUATIONS DERIVED FROM NONLINEAR FILTERINGS WITH FEEDBACK NAOYUKI ICHIHARA
Abstract. We discuss the homogenization of stochastic partial differential equations whose coefficients are rapidly oscillating and are perturbed by a diffusion process. Such class of equations appear in nonlinear filtering problems with feedback. We specify the constant coefficients of the limit equation. The constants are essentially different from the case where the coefficients do not contain perturbed factors.
1. Introduction In this paper, we deal with the following stochastic partial differential equations (SPDEs) with small parameter > 0 dp (t, x) p (0, x)
= L (t) p (t, x) dt + M (t) p (t, x) dYt ,
0≤t≤T,
= p0 (x) ∈ L2 (Rd ),
(1.1)
where Y = (Yt )t∈[0,T ] is an n-dimensional standard Brownian motion, and L = L (t) and M = M (t) = (M1 (t), · · · , Mn (t)) represent the linear differential operator and the multiplicative operator acting on a function on Rd defined by L (t) = ∇xi (aij (x/, Zt /) ∇xj · ) ,
Mk (t) = hk (x/, Zt /) · .
Note that ∇xi (i = 1, . . . , d) are the partial derivatives with respect to xi and that we use the summation convention throughout the paper. The symbol Z = (Zt )t∈[0,T ] stands for a solution to the following stochastic differential equation (SDE) on Rn dZ t
Z 0
= f (Zt /) dt + Q dYt ,
= z ∈ Rn ,
0≤t≤T,
(1.2)
where Q = (Qkl ) denotes an (n × n)-matrix. All coefficients a = (aij (x, z)), h = (hk (x, z)) and f = (f k (z)) are assumed to be periodic with period 1 in all components. 2000 Mathematics Subject Classification. Primary 60H15; Secondary 35R60, 93E11. Short title. Homogenization for stochastic PDEs. Key words and phrases. homogenization, stochastic partial differential equations, Zakai equations, nonlinear filtering. 1
Our aim is to show that as goes to zero the family of solutions to (1.1) converges in law to the solution of an SPDE having both spatially and temporally homogeneous coefficients. It turns out that the limit equation satisfies the SPDE dp0 (t, x) p0 (0, x)
= L0 p0 (t, x) dt + M 0 p0 (t, x) dYt ,
0≤t≤T,
(1.3)
= p0 (x) ∈ L2 (Rd ) ,
where ¯k · , Mk0 = h
L0 = c¯ij ∇xi ∇xj · − g¯i ∇xi · ,
(1.4)
¯ = (h ¯ k ) (k = 1, . . . , n) are and the constants c¯ = (¯ cij ), g¯ = (¯ g i ) (i, j = 1, . . . , d) and h characterized by 0 0
c¯ij = hh(δii0 + ∇xi0 χi ) ai j (δjj0 + ∇xj0 χj )ii + hh∇zk χi Akl ∇zl χj ii , i
kl
i
g¯ = hhhk Q ∇zl χ ii ,
Z
¯k = h
Tn
Td
!1
2
Z
hk (x, z) dx
2
dz
,
with the notation hh · ii := Td ×Tn · dxdz , where Td and Tn represent the d-dimensional and n-dimensional unit torus respectively, and ∇zk (k = 1, . . . , n) denote the partial derivatives with respect to zk . The symbols (δji ) (i, j = 1, . . . , d) and (Akl ) (k, l = 1, . . . , n) stand for Kronecker’s delta and the (n × n)-matrix defined by A = Q Q∗ /2 respectively, and we denote by χm = (χm (x, z)) (m = 1, . . . , d) periodic functions with period 1 in all components which satisfy hhχm ii = 0 and the following auxiliary partial differential equations (PDEs) on Rd × Rn R
∇xi (aij (x, z) ∇xj χm (x, z)) + Akl ∇zk ∇zl χm (x, z) + (∇xi aim )(x, z) = 0 . (1.5) The limit equation (1.3) does not depend on f . The study of homogenization for PDEs has been largely developed for the last two decades, and numerous publications can be found at present. The books [3], [9] give us large numbers of results obtained before the nineties with an extensive bibliography. The former book treats the homogenization of linear, second-order PDEs with periodic coefficients by two different approaches, that is, analytic and probabilistic (see also [12], [14] and references therein). The latter one is concerned with the homogenization on stationary random fields (we refer to [5], [10] for more recent results). The papers [4], [15] deal with another sort of homogenization in random environment; they consider second order PDEs whose coefficients are periodic function of the space variable, and perturbed by an ergodic diffusion process. 2
On the other hand, few studies are found on the homogenization problem of SPDEs. The literature [1] consider the homogenization of the SPDE having the operators L = ∇xi (aij (x/)∇xj · ) − ∇xi (g i (x/) · ) ,
Mk = hk (x) · ,
(1.6)
under the assumption of pointwise convergence : lim↓0 hk (x) = hk (x) for all k = 1, . . . , n. However, this assumption is rather strong since it forbids an oscillatory behavior of hk written as hk (x) = hk (x/) by periodic functions hk . Motivated by this problem, our previous paper [8] deals with the case where hk allows such oscillation by taking, in place of (1.6), the operators L = aij (x/)∇xi ∇xj · + −1 bi (x/)∇xi · ,
(Mk u)(x) = Bk (x, x/, u(x)) ,
and studies its homogenization. The principal interest of the present paper is to know how L and M are homogenized when we add random factors in the coefficients. In fact, we get different limit operators from that obtained in [8] because of the presence of Z . Besides, contrary to [8], the limit operator L0 is determined not only by L but also by M since the constants g¯i contain the functions hk in their integrand. Remark that this term does not appear in the case where the coefficients do not depend on Z . The reason why L0 does not depend on f will be revealed at the end of Section 4. Roughly speaking, the constant g¯i should involve intrinsically the term of the form hhf l ∇zl χi ii, but it can be shown that this term is equal to zero by the particularity of χm . Finally, we point out that the SPDEs (1.1) often appear in certain nonlinear filtering problems. Take f = 0, Q = I and σ such that σσ ∗ = 2a, and consider the following nonlinear filtering problem with feedback terms dX
= −1 b(Xt /, Yt /) dt + σ(Xt /, Yt /) dWt , R ˆ t , Yt = t h(X /, Ys /) ds + W s 0 t
X0 = ξ,
ˆ = (W ˆ t ) are mutually independent standard Brownian where W = (Wt ) and W motions with respect to the probability measure P defined by ! Z t n Z t 1X dP 2 h(Xs /, Ys /) dYs − = exp |hk (Xs /, Ys /)| ds , dP Ft 2 k=1 0 0
with Ft = σ( Ws , Ys | s ≤ t ). Note that in this case Z = (Zt ) is identical with Y = (Yt ). Then, the solution of (1.1) appears in the following representation formula for the optimal filter E
P
R
[ ψ(Xt ) | σ(Ys ; s
≤ t) ] =
Rd
ψ(x) p (t, x) dx , Rd p (t, x) dx
R
3
P -a.s.
We refer to [2] for more information about physical and engineering aspects of the homogenization in nonlinear filtering problems. This paper is organized as follows. In the next section, we state our main result after giving standing assumptions. In Section 3, we prove tightness of the family of solutions to (1.1) on an appropriate function space. Section 4 is devoted to the identification of the limit measure. 2. Assumptions and main result Throughout this paper, we make the following standing assumptions. Assumption 2.1. (1) a ∈ C 2 (Rd+n ; Rd ⊗ Rd ), h ∈ C 1 (Rd+n ; Rn ), and f ∈ C 1 (Rn ; Rn ) are periodic functions with period 1 in all components. (2) a = (aij (x, z)) is symmetric and strictly elliptic, that is, aij (x, z) = aji (x, z) and there exists α > 0 such that α|ξ|2Rd ≤ (a(x, z) ξ, ξ)Rd ≤ α−1 |ξ|2Rd for all ξ ∈ Rd and (x, z) ∈ Rd+n . (3) A = (Akl ) is positive definite. Let H = L2 (Rd ) be q the Hilbert space with inner product (u, v)H := Rd u(x)v(x) dx and norm |u|H := (u, u)H , and let V = H 1 (Rd ) be the Sobolev space of order 1 R
q
with norm |u|V := |u|2H + di=1 |∇xi u|2H . We denote by H 0 and V 0 the dual spaces of H and V respectively. Then, under the identification H = H 0 by the Riesz representation theorem, we have the inclusions V ,→ H ,→ V 0 that are dense and continuous. P
The conditions (1) and (2) of Assumption 2.1 ensure the existence and uniqueness of solution to (1.1) (see Theorem 1.3 of [13]). Theorem 2.1. Let (Ω, F, P ; Ft , Yt ) be a filtered probability space with a standard (Ft )Brownian motion Y = (Yt ) , and let Z = (Zt ) be a solution to (1.2). Then, there exists an (Ft )-progressively measurable process p = (pt ) ∈ L2 (Ω × [0, T ] ; V ) such that (pt , v)H = (p0 , v)H +
Z t 0
V
0
hL (s) ps , viV ds +
Z t 0
(M (s) ps , v)H dYs ,
∀v ∈ V , (2.1)
for almost all (t, ω) ∈ [0, T ] × Ω, where V 0 h · , · iV stands for the duality product between V and V 0 . Such process is unique in the following sense
P pt = qt in V 0 , ∀t ∈ [0, T ] = 1 4
for all p and q satisfying (2.1). Moreover, the solution p satisfies the energy equality |pt |2H = |p0 |2H + 2
Z t 0
V
0
hL (s) ps , ps iV ds + 2 +
n Z t X k=1 0
Z t 0
(M (s) ps , ps )H dYs (2.2)
|Mk (s) ps |2H ds ,
P -a.s.
Now we are in position to state our main result. We denote by E = { v | ve−λθ ∈ V 0 } the weighted Sobolev space with norm |v|E := |ve−λθ |V 0 , where λ > 0 is a fixed number and θ = θ( · ) is a smooth and strictly positive function on Rd such that θ(x) = |x| for all |x| ≥ 1. Let S = C(0, T ; E) be the set of continuous functions taking their values in E equipped with the uniform topology, and let Π and Π0 be the laws on S of the solutions to (1.1) and (1.3) respectively. Remark that (1.3) has a unique solution since c¯ = (¯ cij ) is positive definite. Moreover, we can show the uniqueness in law of (1.3) on S by using the uniqueness theorem of Yamada-Watanabe type (see p.89 of [11]). Our main result is the following. Theorem 2.2. The family of probability measures { Π ; > 0 } converges to Π0 as goes to zero. 3. Tightness In this section, we prove the tightness of { Π ; > 0 } . To begin with, we check the following uniform estimate. Lemma 3.1. Let p = (pt ) be a solution to (1.1). Then, there exists a constant C > 0 such that h
i
sup E sup |pt |4H + sup E >0
>0
0≤t≤T
Z T 0
!2
|pt |2V dt
≤ C |p0 |4H .
(3.1)
Proof. By applying Ito’s formula to the semi-martingale |pt |2H , which satisfies (2.2), we can see |pt |4H
+ 4α
Z t 0
|ps |2H
|ps |2V
ds
≤|p0 |4H
+C
+4
Z t 0
Z t 0
|ps |4H ds
|ps |2H (M (s) ps , ps )H dYs ,
P -a.s., (3.2)
for some constant C > 0. Here and in the following, we denote by C > 0 different constants independent of > 0. Making use of Burkholder’s inequality, we can estimate the stochastic integral part as "
4E
sup 0≤t≤T
Z t |ps |2H (M (s) ps , ps )H 0
# dYs
# "Z i T 1 h 4 4 |pt |H dt . ≤ E sup |pt |H + C E 2 0 0≤t≤T 5
Thus, by considering the expectation of both sides of (3.2) after taking the supremum in t ∈ [0, T ] , we obtain by Gronwall’s lemma that sup>0 E[ sup0≤t≤T |pt |4H ] ≤ C |p0 |4H . On the other hand, the energy equality (2.2) yields 2α
Z T 0
!2
|pt |2V
dt
≤
4 |p0 |4H
form which we obtain sup>0 E[( the proof.
+ C sup 0≤t≤T
RT 0
|pt |4H
Z T
+8
0
!2
(M
(t)pt , pt )H
dYt
,
|pt |2V dt )2 ] ≤ C |p0 |4H . Hence we have completed
Let us denote by Cc∞ (Rd ) the set of smooth functions on Rd with compact supports. The following lemma gives the equicontinuity of { (p , ψ)H ; > 0 } for every ψ ∈ Cc∞ (Rd ) . Lemma 3.2. For each ψ ∈ Cc∞ (Rd ), there exists a constant C > 0 such that h 4i sup E (pt , ψ)H − (ps , ψ)H ≤ C |p0 |4H |t − s|2 ,
0 ≤ ∀s < ∀t ≤ T .
>0
(3.3)
Proof. Remark first that there exists a constant C > 0 such that L and M satisfy V 0 hL (r)pr , ψiV ≤ C |pr |V |ψ|V and (Mk (r)pr , ψ)H ≤ C |pr |H |ψ|H for all r ∈ [0, T ] . Thus, we can easily show h E (pt
4 i − ps , ψ)H
≤ C |t − s|2 E
Z T 0
!2
|pr |2V dr
+ C |t − s|2 E
sup |pr |4H ,
0≤r≤T
which implies (3.3) by virtue of (3.1).
Proposition 3.1. The family of probability measures { Π ; > 0 } is tight in S . Proof. By the estimate (3.1) and the compactness of the injection H ,→ E (cf. Lemma 9.21 of [6]), we have only to check the tightness of the family of real valued processes { (p , ψ) ; > 0 } for each ψ ∈ Cc∞ (Rd ) (see for example [7]). But in view of (3.1), (3.3) and Kolmogorov’s tightness criterion, we can conclude that { (p , ψ) ; > 0 } is a tight family in C([0, T ]; R). Hence, we get the desired result. 4. Identification of the limit measure By Proposition 3.1, we can extract a subsequence of { Π ; > 0 } having a limit Π . Hereafter we fix such converging subsequence arbitrarily and denote it by { Π ; > 0 } again to avoid heavy notation. The goal of this section is to prove Π = Π0 as probability measures on S . For this purpose, we adopt martingale formulation for infinite dimensional diffusion processes following the notation in [11]. 6
Let X = (Xt )t∈[0,T ] and X = (Xt )t∈[0,T ] be the canonical process and the canonical filtration on S, respectively. For φ ∈ Cc∞ (R) , ψ ∈ Cc∞ (Rd ) and t ∈ [0, T ] , we define the functional Γt = Γφ,ψ : S −→ R by t Γφ,ψ t (w) := φ((wt , ψ)H ) − φ((w0 , ψ)H ) −
Z t 0
φ0 ((wr , ψ)H ) (wr , (L0 )∗ ψ)H dr
n Z t 1X ¯ k ψ)2 dr , − φ00 ((wr , ψ)H ) (wr , h H 2 k=1 0
where (L0 )∗ denotes the adjoint operator of L0 . Then, from the uniqueness in law of the limit equation (1.3), we can show the uniqueness of probability measures under ∞ ∞ d which (Γφ,ψ t ) is a (Xt )-martingale for every φ ∈ Cc (R) and ψ ∈ Cc (R ) (see p.76 − Γφ,ψ of [11]). Thus, we have only to check E Π [Ψs (Γφ,ψ t s )] = 0 for arbitrarily fixed 0 ≤ s < t ≤ T and bounded (Xs )-measurable functional Ψs , where E Π stands for the expectation with respect to the probability measure Π . Now, we set ψ (x, z) = ψ(x)+ χm (x/, z/) ψxm (x) , where ψxm := ∂ψ/∂xm . Recall that χm = χm (x, z) (m = 1, . . . , d) are bounded and periodic solutions to (1.5) that belong to C 2 (Rd+n ). Then, by Ito’s formula, we have Z t 0
φ0 ((pr , ψ ( · , Zr ))H ) (pr , (M ψ )( · , Zr ))H dYr
(4.1)
= φ((pt , ψ ( · , Zt ))H ) − φ((p0 , ψ ( · , z))H ) −
Z t 0
φ0 ((pr , ψ ( · , Zr ))H ) (pr , (L ψ )( · , Zr ))H dr
n Z t 1 X − φ00 ((pr , ψ ( · , Zr ))H ) (pr , (Mk ψ )( · , Zr ))2H dr , 2 k=1 0
where (L ψ )(x, z) = {(cij (x/, z/) + dij (x/, z/)} ψxi xj (x) + g i (x/, z/) ψxi (x) + (aij χm )(x/, z/) ψxi xj xm (x) , (Mk ψ )(x, z) = Qkl (∇zl χm )(x/, z/) ψxm (x) + hk (x/, z/) ψ(x) + (hk χm )(x/, z/) ψxm (x) , and cij = (cij (x, z)) , dij = (dij (x, z)) and g i = (g i (x, z)) are defined by cij = aij + aim (∇xm χj ) ,
dij = ∇xm (amj χi ) ,
g i = (hk Qkl + f l )(∇zl χi ) .
Proposition 4.1. Let us denote the left-hand side of the equality (4.1) by Λt . Then, lim E[ Ψs (p )(Γt (p ) − Γs (p ) − Λt + Λs ) ] = 0 . ↓0
7
(4.2)
Before proving this proposition, we point out here that the convergence (4.2) implies E Π [Ψs (Γt −Γs )] = 0 , and in consequence Π = Π0 . This claim can be verified as follows. For N > 0, we define θN : S −→ S by θN (w)(t) := |wt |−1 E min{ |wt |E , N } wt , and set N Γt (w) = Γt (θN (w)) . Clearly, θN (p ) = p on the event { sup0≤t≤T |pt |E ≤ N } . Thus, taking into account that
2 sup sup E[ |ΓN t (p )| ] ≤ C 1 + E
N >0 0≤t≤T
h
sup |pt |4H
i
,
0≤t≤T
we can show that for all > 0 and N > 0, 2 N E[ Ψs (p )(Γt (p ) − Γs (p )) ] − E[ Ψs (p )(ΓN (p ) − Γ (p )) ] t s
≤ CP ( { sup |pt |E > N } ) × 1 + E
h
0≤t≤T
sup |pt |4H
i
≤
0≤t≤T
C . N2
Therefore, by using (4.2) and the martingale property of (Λt ), we obtain
N lim E[Ψs (p )(ΓN t (p ) − Γs (p ))] ≤ ↓0
C . N
Furthermore, if Ψs is continuous on S , then the left-hand side of the above inequality N N is equal to |E Π [Ψs (ΓN is bounded and continuous on S and Π t − Γs )] | since Γt converges to Π . Thus, in consideration of the fact sup0≤t≤T E Π [ |Γt | ] < ∞ , we obtain E Π [Ψs (Γt − Γs )] = 0 by letting N → ∞ . This equality is also valid for all bounded (Xs )-measurable functional Ψs by approximation. Hence it remains to prove Proposition 4.1. Proof of Proposition 4.1. We set Γt (p ) − Γs (p ) − Λt + Λs := Φ1 + Φ2 + Φ3 + Φ4 , where Φ1 = φ((pt , ψ)H ) − φ((pt , ψ ( · , Zt ))H ) − { φ((ps , ψ)H ) − φ((ps , ψ ( · , Zs ))H ) } , Φ2
=
Z t s
{ φ0 ((pr , ψ ( · , Zr ))H ) − φ0 ((pr , ψ)H ) } (pr , (L ψ )( · , Zr ))H dr ,
n Z t 1 X + { φ00 ((pr , ψ ( · , Zr ))H ) − φ00 ((pr , ψ)H ) } (pr , (Mk ψ )( · , Zr ))2H dr , 2 k=1 s
Φ3 = Φ4 =
Z t s
φ0 ((pr , ψ)H ) (pr , (L ψ − (L0 )∗ ψ)( · , Zr ))H dr ,
n Z t 1 X ¯ k ψ)2 } dr . φ00 ((pr , ψ)H ) { (pr , (Mk ψ )( · , Zr ))2H − (pr , h H 2 k=1 s
We prove lim↓0 E[ Ψs (p )Φi ] = 0 one by one for each i = 1, . . . , 4. Note first that ψ satisfies |ψ ( · , z) − ψ|H ≤ |χm |L∞ |ψxm |H for all z ∈ Rn by definition, where | · |L∞ stands for the L∞ -norm. Thus, we get | E[ Ψs (p )Φ1 ] | ≤ C E[ sup0≤t≤T |pt |H ] , which implies lim↓0 E[ Ψs (p )Φ1 ] = 0 . Similarly, we can easily 8
show that E[ Ψs (p )Φ2 ]
≤ C E[ sup |pt |2H ] + E[ sup |pt |3H ] −→ 0 0≤t≤T
↓0
0≤t≤T
since |L ψ ( · , z)|H and |Mk ψ ( · , z)|H are bounded, uniformly in > 0 and z ∈ Rn . In order to prove lim↓0 E[ Ψs (p )Φ3 ] = 0 , we prepare the following lemma. Lemma 4.1. Let α ∈ C 1 (Rd+n ) be a periodic function with period 1 in all components such that hhαii = 0 . Then, for every ϕ ∈ Cc∞ (Rd ), we have
lim E Ψs (p )
Z t
↓0
0
φ
s
((pr , ψ)H )(pr , α( · /, Zr /) ϕ)H
Proof. For each z ∈ Rn , we set α ˜ (z) :=
R
Td
dr = 0 .
(4.3)
α(x, z) dx , and consider the PDE on Rd
∆x η( · , z) = α( · , z) − α ˜ (z) ,
(4.4)
where ∆x stands for the Laplace operator with respect to x = (x1 , . . . , xd ). Recall that (4.4) admits a unique periodic solution η( · , z) ∈ C 2 (Rd ) with period 1 in all R components such that η˜(z) := Td η(x, z) dx = 0 . Then, by the integration by parts formula, (pr , α( · /, Zr /) ϕ)H = (pr , ϕ)H α ˜ (Zr /) −
d X
(∇xi pr , (∇xi η)( · /, Zr /) ϕ)H
i=1
−
d X
(pr , (∇xi η)( · /, Zr /) ∇xi ϕ)H .
i=1
Thus, it is sufficient for (4.3) to prove
lim E Ψs (p ) ↓0
Z t s
0
φ
((pr , ψ)H )(pr , ϕ)H
Let us take the N -partition (s, t] := Then, h Z t E Ψs (p ) φ0 ((pr , ψ)H )(pr , ϕ)H s
≤C
N Z si+1 X i=1 si
+
SN
i=1 (si , si+1 ] ,
α ˜ (Zr /) dr
α ˜ (Zr /) dr
= 0.
(4.5)
where si = s + N −1 (t − s)(i − 1) .
i
E[ |(pr − psi , ψ)H | |pr |H + |(pr − psi , ϕ)H | ] dr
Z si+1 N h X E Ψs (p )φ0 ((p , ψ)H )(p , ϕ)H si si si
i=1
i
α ˜ (Zr /) dr =: I1 + I2 .
By (3.3) and H¨older’s inequality, I1 ≤ C
h
1+ E
4
sup |pt |H3
0≤t≤T
N Z si+1 i 3 X 4 si
i=1
9
(r − si )2 dr
1 4
3 C (si+1 − si ) 4 ≤ √ . N (4.6)
On the other hand, by using lim↓0 E[( ssii+1 α ˜ (Zr /) dr)2 | Fsi ] = 0 (see p.400 of [3]), we can verify that for each fixed N > 0, R
I2
s h
≤C E
sup 0≤t≤T
|pt |2H
N i X
s Z si+1
E
si
i=1
α ˜ (Zr /) dr
2
−→ 0 . ↓0
Thus, in combination with (4.6), we obtain (4.5).
Now we return to the proof of lim↓0 E[ Ψs (p )Φ3 ] = 0 . First, we check hhcij ii = c¯ij , hhdij ii = 0 and hhg i ii = g¯i . By the integration by parts formula, we can see hhdij ii = 0 and 0 0
0
c¯ij − hhcij ii = −hhχi {∇xi0 (ai j ∇xj0 χj ) + Akl ∇zk ∇zl χj + ∇xi0 ai j }ii = 0 . The equality hhg i ii = g¯i can be seen as follows. For z ∈ Rn and m = 1, . . . , d, we define R χ˜m (z) := Td χm (x, z) dx . Then, in view of (1.5), χ˜m satisfy Akl ∇zk ∇zl χ˜m (z) = 0 . Since (Akl ) is positive definite and χ˜m are periodic, the strong maximum principle implies that χ˜m are constant functions; in particular, ∇zl χ˜m (z) = 0 for all R z ∈ Rn . Therefore hhf l ∇zl χm ii = Tn f (z) ∇zl χ˜m (z) dz = 0 , and we obtain hhg i ii = g¯i . Hence, applying Lemma 4.1 to α = cij − c¯ij , α = dij and α = g i − g¯i , we get lim↓0 E[ Ψs (p )Φ3 ] = 0 . Finally, we shall show lim↓0 E[ Ψs (p )Φ4 ] = 0 . Recall that Qkl ∇zl χ˜m (z) = 0 for ˜ k (z) , all z ∈ Rn . Then, by taking α(x, z) = Qkl ∇zl χ(x, z) and α(x, z) = hk (x, z) − h R ˜ k (z) := d hk (x, z) dx , we can show similarly to the proof of Lemma 4.1 that where h T E[ Ψs (p )Φ4 ] Z t n X E Ψs (p ) ≤ φ00 ((pr , ψ)H )(pr , ψ)2H k=1
s
h
i
+ ( + 2 ) C E sup |pt |4H + E
0≤t≤T
˜ k (Z /))2 − |h ¯ k |2 } dr { (h r
Z T 0
!2
|pt |2V dt
.
Since the first term of the right-hand side converges to zero by the same argument as in the proof of (4.5), we obtain lim↓0 E[ Ψs (p )Φ4 ] = 0 , and the proof of Proposition 4.1 has been completed. Acknowledgment The author would like to express his sincere thanks to the referee. He pointed out the errors in the first version of this paper. References [1] Bensoussan,A., Homogenization of a class of stochastic partial differential equations, Composite media and homogenization theory (Trieste, 1990). Progr. Nonlinear Differential Equations Appl., 5 (1991), 47-65. 10
[2] Bensoussan,A.-Blankenship,G.L., Nonlinear filtering with homogenization , Stochastics, 17 (1986), 67-90. [3] Bensoussan,A.-Lions,J.L.-Papanicolaou,G., Asymptotic analysis for periodic structures, Studies in Mathematics and its Applications. 5, North-Holland, New York, 1978. [4] Bouc,R.-Pardoux,E., Asymptotic analysis of P.D.E.s with wide-band noise disturbances, and expansion of the moments, Stochastic Analysis and Applications., 2 (1984), 369-422. [5] Castell,F., Homogenization of random semilinear PDEs, Probab. Theory Relat. Fields., 121 (2001), 492-524. [6] Funaki,T., The scaling limit for a stochastic PDE and the separation of phases, Probab. Theory Relat. Fields., 102 (1995), 221-288. [7] Holley,R.A.-Stroock,D.W., Generalized Ornstein-Uhlenbeck processes and infinite particle branching Brownian motions, Publ. RIMS, Kyoto Univ., 14 (1978), 741-788. [8] Ichihara,N, Homogenization problem for stochastic partial differential equations of Zakai type, to appear in Stochastics and Stochastics Reports. [9] Jikov,V.V. , Kozlov,S.M., Oleinik,O.A., Homogenization of Differential Operators and Integral Functionals, Springer-Verlag, 1994. [10] Lejay,A., Homogenization of divergence-form operators with lower-order terms in random media, Probab. Theory Relat. Fields., 120 (2001), 255-276. ´tivier,M., Stochastic partial differential equations in infinite dimensional spaces, Quaderni, [11] Me Scuola normale superiore, 1988. [12] Papanicolaou,G.-Stroock,D.-Varadhan,S.R.S., Martingale approach to some limit theorems Papers from the Duke Turbulence Conference (Duke Univ., Durham, N.C.), no. 6, 1977. [13] Pardoux,E., Stochastic partial differential equations and filtering of diffusion processes, Stochastics, 3 (1979), 127-167. [14] , Homogenization of linear and semilinear second order parabolic PDEs with periodic coefficients, Journal of Functional Analysis., 167 (1999), 498-520. [15] Pardoux,E.-Piatnitski,A.L., Homogenization of a nonlinear random parabolic partial differential equation, Stochastic Processes and their Applications, 104 (2003), 1-27. ´matiques, Universite ´ de Bretagne Occidentale, Laboratoire de Mathe Victor Le Gorgeu, B.P. 809, 29285 BREST CEDEX, FRANCE. E-mail address:
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