Diffusion in a locally stationary random environment

Diffusion in a Locally Stationary Random

Environment

Rémi Rhodes

December 18, 2007

Abstract : This paper deals with homogenization of diffusion processes in a lo-cally stationary random environment. Roughly speaking, such an environment possesses two evolution scales: both a fast microscopic one and a smoothly vary-ing macroscopic one. The homogenization procedure aims at givvary-ing a macro-scopic approximation that takes into account the micromacro-scopic heterogeneities.

1

Introduction

In this paper, we aim at describing the asymptotic behavior (homogenization), as the parame-ter ε tends to 0, of the solution Xε of the following Stochastic Differential Equation (SDE)

(1) X_tε= x + 1 ε Z t 0 b(ω, X_rε/ε, X_rε) dr + Z t 0 c(ω, X_rε/ε, X_rε) dr + Z t 0 σ(ω, X_rε/ε, X_rε) dBr,

where B is a standard d-dimensional Brownian motion and the parameter ω evolves in a ran-dom medium Ω, that is a probability space with suitable stationarity and ergodicity properties. For each fixed value of the parameter y ∈ Rd_{, the coefficients b(ω, ·, y), c(ω, ·, y) and σ(ω, ·, y)}

are stationary random fields and for this reason are said to be locally stationary. The generator Lε_{of the process X}ε_{can be written in divergence form as}

(2) Lε = (1/2) d X i,j=1 ∂ ∂xi [a + H](ω, x/ε, x) ∂ ∂xj

∗_{Address: Ceremade, Université Paris-Dauphine, Place du maréchal De Lattre de Tassigny, 75775 Paris}

for an antisymmetric matrix H and a = σσ∗.

Roughly speaking, these diffusions stand for particles moving in a medium with two evo-lution scales: particles encounter smooth macroscopic variations, represented by the variable x, and fast microscopic ones, represented by the variable x/ε. It is a quite natural assumption that occurs in many modeling problems. That is the reason why, since the end of the sixties, this problem has been subject to many works, both by means of analytical tools (see [2] among others) or probabilistic tools (see [1]), in the locally periodic setting, that is when the matrices a or H are periodic with respect to the variable x/ε.

Thereafter, random media were introduced and homogenization of diffusions in random medium have been widely studied in the case when the random medium possesses micro-scopic variations only (cf. [4], [8], [9], and many others). However, as far as we know, the first attempt to generalize the results obtained in the locally periodic setting to the locally sta-tionary one is due to Olla and Siri [12] (in a slightly more general framework). But the authors restrict themselves to the dimension 1 for technical reasons.

Let us try to briefly explain. If we look at a functional of the type (3)

Z t

0

f (ω, X_rε/ε, X_rε) dr,

for some locally stationary random field f , its asymptotic behavior, as ε goes to 0, corresponds to a very intuitive argument. At a small scale, the oscillations due to the smallness of the parameter ε make the functional average with respect to its first variable thanks to suitable ergodicity assumptions. In other words, this functional behaves, as ε goes to 0, as

(4)

Z t

0

¯

f (X_rε) dr,

where ¯f (y) is the mean over the medium of the random function f (ω, x, y). As guessed by the reader, this approximation is the key tool to pass to the limit in (1). This technique has been already used in the case of locally periodic coefficients. As explained in the already existing literature on this topic, the main difficulty actually lies in getting rid of the highly oscillating term 1_εR₀tb(ω, X_rε/ε, X_rε) dr. The method consists in constructing the so-called correctors, that is solutions of auxiliary problems stated on the random medium. They have to make up for the microscopic oscillations of this functional.

From this remark on, our approach differs from Olla and Siri [12]. Their method consists in adapting the technics of the (non locally) stationary random setting. To sum up, they solve the equation Lε_u

ε = [ε−1b + c](ω, ·/ε, ·) and then tackle the description of the asymptotic

behavior, as ε tends to 0 of the solution uε. However, because of interactions between the

round this difficulty, they choose the dimension as equal to one. Indeed, in this case only, they are provided with an explicit formula for the solution uε, expressed in terms of the coefficients.

This formula is essential to conclude their work.

Our work is based on a separation of the microscopic and macroscopic scales, as already used in the locally periodic setting. But new difficulties arise due to the particular geome-try/topology of a random medium in comparison with the torus (in the case of locally periodic coefficients). Let us develop this point. To construct the correctors, we make the following observation. From the microscopic point of view, the slowly varying macroscopic evolu-tion of the medium can be seen as frozen. So the correctors are constructed by fixing the macroscopic evolution: for a fixed parameter y ∈ Rd, they are defined as the (parameterized) solution u(ω, ·, y) of the equation

(5) (1/2) d X i,j=1 ∂ ∂xi [a + H](ω, x, y) ∂ ∂xj u(ω, x, y)
= b(ω, x, y), _{x ∈ R}d.

Applying the Itô formula then provides us with the following decomposition of the process dX_tε+ εdu(ω, X_tε/ε, X_tε) = b∂yu(ω, Xtε/ε, X ε t) dt + D(ω, X ε t/ε, X ε t) dt + (σ + ∂xu∗σ)(ω, Xtε/ε, Xtε) dBt+ dRεt, (6)

where the process Rε_{reasonably converges to 0 a ε tends to 0 and the function D is a locally}

stationary random field. The main advantage of the correctors is that their contribution is small, that is εu(ω, X_tε/ε, Xε

t) converges to 0 as ε goes to 0, but they permit to remove the

highly oscillating term from (6). In the locally periodic setting, the proof of the homogeniza-tion property is almost done. Indeed, in this context, it is possible to find a locally periodic solution u of (5). As a consequence, the gradients ∂xu and ∂yu are also locally stationary

and the approximation of (3) by (4) holds for each corresponding term in (6). The limit of Xε is then easily identified as the solution X of the following SDE with some deterministic coefficients A, B (7) Xt= x + Z t 0 B(Xr) dr + Z t 0 A(Xr) dBr.

However, the random context raises additional issues. Indeed, it is possible to find a so-lution u of (5) with a locally stationary gradient ∂xu but it turns out that u is not locally

stationary. Thus, the term ∂yu is not locally stationary either in such a way that the

approx-imation of (3) by (4) fails for the term R₀tb∂yu(ω, Xrε/ε, Xrε) dr. This is the main issue of

this paper. Up to introducing a new type of correctors for the function b∂yu, we prove that it

latter part can be approximated by a family (Eε)ε, which may be divergent in the L2sense but

asymptotically not seen by the process Xε, that is

(8) Z t 0 Eε(ω, Xrε/ε, X ε r) dr → 0, as ε goes to 0.

This convergence is established provided that we can control a sort of "Poincaré inequality" of the functions (Eε)ε, which connects both microscopic and macroscopic variations of the

medium. This connection is of the utmost importance. But for it, corrections at a small scale would turn out to cause undesirable effects at a large scale and vice versa.

We are then in position to pass to the limit in (1) and prove that the limit process solves (7) for some deterministic coefficients A, B. We should point out that our result is an annealed convergence result, that is our convergence result is not stated for each realization ω of the random medium but in probability with respect to the measure of the random medium.

The organization of the paper is the following. In Section 2, we set out precisely the framework, in particular we define the random medium and give a complete description of the coefficients involved in (1). Our main result is stated in Section 3. The correctors are constructed in Section 4 and Section 5 explains how to apply the Itô formula to the correctors. Ergodic theorems are stated in Section 6, in particular the approximation of (3) by (4) is given a rigorous sense and conditions for (8) to hold are explained. The new correctors for Rt

0 b∂yu(ω, X ε

r/ε, Xrε) dr are introduced in Section (7), which also describes how to pass to

the limit in (1) and get (7).

2

Setup and Assumptions

Random medium. From now on, d ≥ 1 is a fixed integer. Following [7], we introduce the following

Definition 2.1. Let (Ω, G, µ) be a probability space and τx; x ∈ Rd a group of measure

preserving transformations acting ergodically onΩ: 1)_{∀A ∈ G, ∀x ∈ R}d,µ(τxA) = µ(A),

2) If for any_{x ∈ R}dτxA = A, then µ(A) = 0 or 1,

3) For any measurable functiong on (Ω, G, µ), the function (x, ω) 7→ g(τxω) is

measur-able on_(Rd_{× Ω, B(R}d) ⊗ G).

The expectation with respect to the random medium is denoted by M. Denote by L2_(Ω)

the space of square integrable functions, by |.|2 the corresponding norm and by (., .)2 the

strongly continuous group of unitary maps in L2(Ω). For every function f ∈ L2(Ω), let f (ω, x) = f (τxω). Each function f in L2(Ω) defines in this way a stationary ergodic random

field on Rd_{. In what follows we will use the bold type to denote an element f ∈ L}2_(Ω)

and the normal type f (ω, x) (or even f (x)) to distinguish from the associated stationary field. The group possesses d generators (throughout this paper, ei stands for the i-th vector of the

canonical basis of Rd₎

(9) Dig = lim

h→0

Theig − g

h if exists, which are closed and densely defined. Setting

(10) C = Spang ? ϕ; g ∈ L∞(Ω), ϕ ∈ C_c∞_(Rd) , with g ? ϕ(ω) = Z

Rd

g(τxω)ϕ(x) dx,

the space C is dense in L2_{(Ω) and C ⊂ Dom(D}

i) for all 1 ≤ i ≤ d, with Di(g ? ϕ) =

−g ? ∂ϕ/∂xi. If g ∈ Dom(Di), we also have Di(g ? ϕ) = Dig ? ϕ. For f ∈

Td

i=1Dom(Di),

we define the divergence operator Div by Divf = Pd

i=1Dif . We distinguish this latter

operator from the usual divergence operator on Rddenoted by the small type div.

Locally stationary random fields. Following the notations introduced just above, for a measurable function f : Ω × Rd _{→ R}n_{, (n ≥ 1), we can consider the associated locally}

stationary random field (x, y) 7→ f (τxω, y) = f (ω, x, y) (or even f (x, y)).

Structure of the coefficients. The coefficients σ : Ω × Rd_{→ R}d×d_{, H : Ω × R}d_{→ R}d×d

denote measurable functions with respect to the underlying product σ-fields and thus define locally stationary random fields. H is antisymmetric and a = σσ∗. Furthermore, for some positive constant Λ (independent of ω), σ and H satisfy

Assumption 2.2. (Regularity). For each fixed ω ∈ Ω, the coefficients σ(ω, ., .) and H(ω, ., .) are two times continuously differentiable with respect to each variable and are, as well as their derivatives up to order two,Λ-Lipschitzian and bounded by Λ.

Assumption 2.3. (Uniform ellipticity)

There exists a constantM such that the matrix-valued function a(ω, y) = σσ∗(ω, y) satisfies for every_{y ∈ R}d

M−1I ≤ a(., y) ≤ M I, whereI stands for the d × d identity matrix.

Remark.Assumptions 2.2 may appear restrictive and can surely be relaxed (see [3] for results in this direction). In particular, the statement of the homogenization property only involves the

derivatives of order 1 with respect to y ∈ Rd(see Theorem 3.1). However, it avoids dealing with heavy regularizing procedure that are not the purpose of this work.

Diffusion in a locally stationary random environment. For j = 1, . . . , d, we define the coefficients (11) bj(ω, y) = 1 2 d X i=1 Di(a + H)ij(ω, y), cj(ω, y) = 1 2 d X i=1 ∂yi a + H
ij(ω, y).

From Assumption 2.2, the coefficients bj(ω, ., .) and cj(ω, ., .) are Lipschitzian so that, for a

starting point x ∈ Rdand ε > 0, we can consider the strong solution Xεof the following SDE with locally stationary coefficients:

(12) X_tε= x + 1 ε Z t 0 b Xε_r, X_rε
dr + Z t 0 c Xε_r, X_rε
dr + Z t 0 σ Xε_r, X_rε
dBr,

where we have set Xε_t ≡ Xε

t/ε and B is a standard d-dimensional Brownian motion. We

should point out that the generator of this diffusion could be written in divergence form as

(13) Lε= (1/2) d X i,j=1 ∂ ∂xi [a + H](ω, x/ε, x) ∂ ∂xj
.

Notations. Note that the law of the process Xε depends onω even if this parameter does not appear in the notationXε_{. For the sake of simplicity, we indicate the starting point x of X}ε

by writing, when necessary, Pε

x (and Eεxfor the corresponding expectation), this avoids heavy

notations asXε,x. We can then consider the probability measure ¯_Pε_{x ≡ M[P}ε_x(.)] (the so-called annealed law of the processXε_{) and its expectation ¯}

Eεx. In the sequel, the generic notations

“C” and “D” stand for constants that only refer to M and Λ. Dependencies on additional parameters are always mentioned.

3

Main Results

We are now in position to state the main result of this paper.

Theorem 3.1. Homogenization. For each fixed x ∈ Rd_{, the law of the processX}ε_converges

in_{C([0, T ]; R}d), in probability with respect to µ, towards the law of the process X that solves the following SDE with deterministic coefficients (they do not depend on the mediumΩ):

(14) Xt= x + Z t 0 B(Xr) dr + Z t 0 A1/2(Xr) dBr.

The coefficientsA and B are of class C2 and are defined, for_{y ∈ R}d, by A(y) = lim λ→0 M[(I + Duλ) ∗ a(I + Duλ)(., y)], (15a) H(y) = lim λ→0 M[(I + Duλ) ∗ H(I + Duλ)(., y)], (15b) B(y) = (1/2)∂y(A + H)(y). (15c)

Formally speaking, for each _{y ∈ R}d andλ > 0, the entries ui λ(., y)

1≤i≤d of the function

uλ(., y) : Ω → Rd solve the following so-called auxiliary problems, which are stated on the

random medium

λui_λ(., y) − 1 2

X

j,k

Dj(ajk+ Hjk)Dkuiλ(., y) = bi(., y).

A rigorous description ofuλ(., y) is given in Section 4.

Remark 1.Since the matrix a is uniformly elliptic, the homogenized coefficient A also is. Indeed, because of the stationarity of the measure µ, we have M[Duλ(., y)] = 0. Then, for

any X ∈ Rd_{and y ∈ R}d_{, it is readily seen that}

|X|2

≤ M|X + X · Duλ(., y)|2 ≤ M X∗M(I + Duλ)∗a(I + Duλ)(., y)X.

By passing to the limit as λ → 0 in the above expression and using (15a), the uniform ellip-ticity of ¯A follows.

Remark 2. As mentioned in Section 1, our result is an annealed convergence result. The reader may wonder if a quenched result holds. In other words, can we establish that the law of the process Xε converges for (almost) every fixed realization ω of the random medium? The main difficulty to prove such a result actually lies in establishing a quenched version of Theorem 6.3.

4

Auxiliary Problems

Setup and notations. Let us now introduce the different tools we will use on the medium. We aim at extending the following unbounded operators on L2_{(Ω) defined on C by}

(16) Sy ≡ (1/2) d X i,j=1 Di aij(·, y)Dj.
, Ly ≡ (1/2) d X i,j=1 Di [a + H]ij(., y)Dj.

by following the construction of [5, Ch. 3, Sect 3.] or [10, Ch. 1, Sect 2.]. For any ϕ, ψ ∈ C, we define

(ϕ, ψ)1,y ≡ −(ϕ, S(·, y)ψ)2 = (1/2) a(·, y)Dϕ, Dψ

2, (ϕ, ψ)1 ≡ (1/2) Dϕ, Dψ
2, (17)

and the associated seminorms kϕk2_1,y ≡ (ϕ, ϕ)1,y and kϕk21 ≡ (ϕ, ϕ)1, which are all

equiva-lent. Then we can define for any λ > 0 and ϕ, ψ ∈ C

E_λy(ϕ, ψ) ≡ λ(ϕ, ψ)2+ (ϕ, ψ)1,y, E (ϕ, ψ) ≡ (ϕ, ψ)2+ (ϕ, ψ)1.

All these define inner products on C ×C and we will denote by H1the closure of C with respect

to the resulting norms, which are equivalent too. For any y ∈ Rd _{and λ > 0, we introduce a}

new (nonsymmetric) bilinear form on C × C by

(18) By_λ(ϕ, ψ) = λ(ϕ, ψ)2− (Lyϕ, ψ)2.

From Assumptions 2.2 and 2.3, it is clearly continuous with respect to E so that it extends to H1 × H1 (this extension is still denoted Bλy). Now, according to [5, Ch. 3, Sect 3.]

and [10, Ch. 1, Sect 2.], we can extend Sy and Ly as follows. We define Dom(Sy) = {ϕ ∈ H1; E_λy(ϕ, ·) is L2continuous} and Dom(Ly) = {B_λy(ϕ, ·) is L2 continuous}. For

ϕ ∈ Dom(Sy) (resp. ϕ ∈ Dom(Ly)), we can find f ∈ L2_{(Ω) such that E}y

λ(ϕ, ·) = (f , ·)2

(resp. By_λ(ϕ, ·) = (f , ·)2). Then Syϕ (resp. Lyϕ) is defined as λϕ − f (resp. λϕ − f ).

It can be proved that this definition does not depend on λ > 0. Moreover, Sy is self-adjoint and Ly = ((Ly)∗)∗, where the adjoint (Ly)∗ of Ly can be described as follows: Dom((Ly)∗_{) = {ϕ ∈ H}1; Byλ(·, ϕ) is L2continuous} and for ϕ ∈ Dom((L

y₎∗_{), we can find}

f ∈ L2(Ω) such that B_λy(·, ϕ) = (f , ·)2. Then (Ly)∗ϕ exactly matches λϕ − f .

Through-out this paper, we will widely use the following relation withThrough-out referring to it anymore: for y ∈ Rd_{and ϕ ∈ Dom(L}y ), ψ ∈ H1, (Lyϕ, ψ)2 = − 1 2 (a + H)(·, y)Dϕ, Dψ
2.

If a function b satisfy the property:

(19) ∃C > 0, ∀ϕ ∈ C, (b, ϕ)2 ≤ Ckϕk1,

then we will say that b ∈ H−1 and we will define kbk−1 as the smallest constant C that

satisfies this property.

Finally, we define the space D as the closure in (L2(Ω))dof the set {Dϕ; ϕ ∈ C}. By this way, each function ϕ ∈ H1 admits a "gradient" that we still denote by Dϕ.

Solvability and Regularity of the Resolvent Equation. Fix λ > 0 and y ∈ Rd. From Assumption 2.3, the boundedness and the antisymmetry of H(., y), the bilinear form B_λy (see (18)) is clearly continuous and coercive on H1. It thus defines a resolvent operator Gyλ

as-sociated to the operator λ − Ly. In particular, for h ∈ L2(Ω), uλ(., y) ≡ G y

λh belongs to

H1∩ Dom(Ly) and satisfies λuλ(·, y) − Lyuλ(·, y) = h.

We now investigate the regularity of uλ(·, y) with respect to the parameter y.

Proposition 4.1. Let us consider h : y ∈ Rd7→ h(., y) ∈ L2

(Ω) and f : y ∈ Rd7→ f (., y) ∈ L2_{(Ω) ∩ H}

−1. Suppose that there exist two constantsC2, C−1 such that:

1) the applicationy 7→ h(., y) ∈ L2_{(Ω) is two times continuously differentiable in L}2_(Ω).

The derivatives up to order 2 are bounded byC2 inL2(Ω) and are C2-Lipschitz inL2(Ω).

2) the applicationy 7→ f (., y) ∈ L2_{(Ω) ∩ H}

−1is two times continuously differentiable in

H−1. The derivatives up to order 2 are bounded byC−1in H−1and areC−1-Lipschitz in H−1.

Then, for anyλ > 0, the solution uλ(., y) ∈ H1∩ Dom(Ly) of the equation

(20) λuλ(., y) − Lyuλ(., y) = h(., y) + f (., y)

is two times continuously differentiable in H1 with respect to the parameter y ∈ Rd.

Fur-thermore there exists a constantD4.1 > 0, which only depends on Λ, M, C−1, such that the

functionsg_λ(., y) = uλ(., y), ∂yuλ(., y), ∂yy2 uλ(., y) satisfy the property: ∀(y, h) ∈ R2,

λ|g_λ(., y)|2₂+ kg_λ(., y)k2₁ ≤ D4.1(1 + C22/λ),

(21a)

λ|g_λ(., y + h) − g_λ(., y)|2₂+ kg_λ(., y + h) − g_λ(., y)k2₁ ≤ D4.1(1 + C22/λ)|h| 2_.

(21b)

Proof: For every ϕ ∈ H1, (20) provides us with the following weak formulation

(22) λ(uλ(., y), ϕ)2+ (1/2)([a + H]Duλ(., y), Dϕ)2 = (h(., y), ϕ)2+ (f (., y), ϕ)−1,1.

Choosing ϕ = uλ(., y), (22) yields

λ|uλ(., y)|22+ M −1_ku

λ(., y)k21 ≤ |h(., y)|2|uλ(., y)|2+ kf (., y)k−1kuλ(., y)k1

≤ |h(., y)| 2 2 2λ + λ|uλ(., y)|22 2 + M kf (., y)k2 −1 2 + kuλ(., y)k21 2M ,

from which we derive λ|uλ(., y)|22+ M −1_ku

λ(., y)k21 ≤ λ −1_C2

2 + M C−12 .

Let us investigate now the difference vλ(., y, h) = uλ(., y + h) − uλ(., y), for y, h ∈ Rd.

Thanks to (22), we have for any ϕ ∈ H1

λ(vλ(., y, h), ϕ)2+ (1/2)([a + H](·, y)Dvλ(., y, h), Dϕ)2

(23)

=(h(., y + h) − h(., y), ϕ)2+ (f (., y + h) − f (., y), ϕ)−1,1

From the Lipschitz assumptions on the coefficients, for any ϕ, ψ ∈ H1,

(1/2)([a(., y) − a(., y + h) + H(., y) − H(., y + h)]Dϕ, Dψ) ≤ Λ|h|kϕk1kψk1.

Choosing ϕ = vλ(., y, h) in (23) then leads to (in what follows, C(M, Λ, C−1) stands for a

constant that only depends on M, Λ and C−1)

λ|vλ(., y, h)|22+ M −1_kv λ(., y, h)k21 ≤C2|h||vλ(., y, h)|2+ C−1|h|kvλ(., y, h)k1+ Λ|h|kuλ(., y + h)k1kvλ(., y, h)k1 ≤C2|h||vλ(., y, h)|2+ |h| Λkuλ(., y + h)k1+ C−1
kvλ(., y, h)k1 ≤(λ/2)|vλ(., y, h)|2+ (1/2M )kvλ(., y, h)k21 + |h| 2_{C(M, Λ, C} −1)(C₂2/λ + 1)

and finally gives us

λ|vλ(., y, h)|22+ M −1_kv

λ(., y, h)k21 ≤ |h|

2_{C(M, Λ, C}

−1)(C22/λ + 1).

So, for each i = 1, . . . , d (remind that ei denotes the i-th vector of the canonical basis of Rd)

and h ∈ R, the family (h−1[uλ(., y + hei) − uλ(., y)])_h∈R is bounded in the space H1 so that

this family is weakly compact in H1. Let wi(., y) be the limit of a converging subsequence.

From (23), it is plain to see that this limit is actually the unique weak solution of the equation λ(wi(., y), ϕ)2+ (1/2)([a + H](., y)Dwi(., y), Dϕ)2

=(∂yih(., y), ϕ)2+ (∂yif (., y), ϕ)−1,1+ (1/2)(∂yi[a + H](., y)Duλ(., y), Dϕ)2.

Subtracting this latter expression from h−1×(23) and choosing ϕ = h−1_{v(·, y, he}

i)−wi(., y),

we can prove that h−1[uλ(., y + hei) − uλ(., y)] → wi in H1 as h tends to 0. The same job

can be carried out for the second order derivatives.

Auxiliary problems. Let us now tackle the study of the so-called auxiliary problems. For i = 1, . . . , d, let us consider the solution ui

λ(., y) of the equation

(24) λui_λ(., y) − Lyui_λ(., y) = bi(., y),

where bi(., y) = (1/2)

Pd

k=1Dk(a + H)ki(., y) (see (11)). The weak form of the resolvent

equation (24) then reads for ϕ ∈ C

λ(ui_λ(., y), ϕ)2+ (1/2)([a + H]Duiλ(., y), Dϕ)2 = −(1/2) [a + H](., y)ei, Dϕ

2.

(25)

Lemma 4.2. The mapping y 7→ bi(., y) ∈ L2(Ω) ∩ H−1 is two times continuously

Proof: First note that bi(., y) ∈ H−1. Indeed, for each ϕ ∈ C, from Assumptions 2.2 and 2.3,

(bi(., y), ϕ)2 = −(1/2) (a + H)(., y)ei, Dϕ

2 ≤ (M + Λ)kϕk1.

From Assumptions 2.2, it is readily seen that the H−1derivatives of bicoincide, for 1 ≤ k ≤ d,

with the classical derivatives ∂ykbiand

(∂ykbi(., y), ϕ)2 = −(1/2) (∂yka + ∂ykH)(., y)ei, Dϕ

2 ≤ Λkϕk1.

The same job can be carried out for the second order derivatives. Details are left to the reader.

From Proposition 4.1 (with the functions h = 0 and f = bi), the function y 7→ uiλ(., y)

is two times continuously differentiable in H1. We now concentrate our efforts on describing

the asymptotic behavior of ui_λ, as well as its derivatives, as λ goes to 0.

Proposition 4.3. For each fixed y ∈ Rd and 1 ≤ i ≤ d, the family (Dui

λ(., y))λ converges

to a limitξ_i(., y) ∈ L2_(Ω)d_{asλ goes to 0. The same property holds for the derivatives, that}

is, the families(D∂yju

i

λ(., y))λ, (D∂y2jyku

i

λ(., y))λ (1 ≤ i, j, k ≤ d) respectively converge to

∂yjξi(., y), ∂ 2 yjykξi(., y) in L 2_(Ω)d_{. Furthermore, we have} λ|ui_λ(., y)|2₂+ λ|∂yju i λ(., y)| 2 2+ λ|∂ 2 yjyku i λ(., y)| 2 2 → 0, asλ tends to 0,

and, each functiong_λ(., y) = ui

λ(., y), ∂yku

i

λ(., y), ∂ykylu

i

λ(., y) satisfies the property:

(26a) λ|g_λ(., y)|2₂+ kg_λ(., y)k2₁ ≤ C4.3

(26b) λ|g_λ(., y + h) − g_λ(., y)|2₂+ kg_λ(., y + h) − g_λ(., y)k2₁ ≤ C4.3|h|2

for every_{y, h ∈ R}d, whereC4.3is a positive constant independent ofλ > 0 and y ∈ Rd.

Proof: From Lemma 4.2, Proposition 4.1 applies with h = 0 and f = bi and provides us

with the estimate λ|ui

λ(., y)|22+ |Duiλ(., y)|22 ≤ C. Let us denote by ξi(., y) ∈ L2(Ω)da weak

limit of the family (Dui_λ(., y))λ as λ goes to 0. By passing to the limit in (25), it is plain to

see that ∀ϕ ∈ C

(27) (1/2)([a + H]ξ_i(., y), Dϕ)2 = −(1/2) [a + H](., y)ei, Dϕ

2.

The limit is clearly unique in D because (χ, φ) ∈ D × D 7→ ([a + H]χ, φ)2 is coercive on

D × D. Comparing (25) and (27) yields

We choose ui_λ(., y) = ϕ, so that the Cauchy-Schwarz inequality yields:

λ|ui_λ(., y)|2₂+(1/2)([a+H]Dui_λ(., y), Du_λi(., y)
₂ ≤ (1/2) [a+H]ξ_i(., y), ξ_i(., y)
₂+(λ), where the function (λ) exactly matches (1/2)([a + H]ξi(., y), (Dui

λ − ξ

i_{)(., y))}

2 and so

converges to 0 as λ goes to 0. Because of the antisymmetry of H(·, y), we deduce lim sup

λ→0

aDui_λ(., y), Dui_λ(., y)
₂ ≤ aξ_i(., y), ξ_i(., y)
₂.

Since (a(., y)., .)2 is equivalent on D to the canonical inner product (Assumption 2.3), we

deduce that the convergence of (Dui_λ(., y))λ to ξiholds in the strong sense in D. In particular

λ|ui_λ(., y)|2₂+ |Dui_λ(., y) − ξ_i(., y)|2₂ −−→

λ→0 0.

This proves the first part of the statement for the function ui_λ(., y). The second part results from Proposition 4.1 statement (21a) (with C2 = 0). The same job can be carried out for the

successive derivatives up to order 2 of ui_λ(., y).

5

The Itô Formula

In this section, our objective is to apply the Itô formula to the process (Xε, Xε) and to the function (x, y) 7→ wλ(ω, x, y), where wλ is defined as the solution of the equation

λwλ − Lywλ = h(., y) + f (., y).

The functions h(., y) and f (., y) satisfy the assumptions of Proposition 4.1. Even if the func-tion (x, y) 7→ wλ(ω, x, y) is not of class C2, we will prove that the regularity properties stated

in this latter proposition are sufficient to apply the Itô formula by means of a smooth approx-imating sequence (wm_λ)m of the function wλ and estimates on the transition densities of the

process Xε. This approximation has to converge towards wλ as m goes to ∞ in a certain

sense. Actually the main difficulty lies in the convergence of the second order derivatives with respect to the random medium. That is essentially this point we are going to discuss.

For any ϕ ∈ L2(Ω), k = 1, . . . , d and r ∈ R∗, we define Γk_rϕ = 1_r(Trekϕ − ϕ). It is plain

to check that this operator satisfies the following properties:

ϕ, ψ ∈ L2(Ω), (Γk_rϕ, ψ)2 = −(ϕ, Γk−rψ)2,

(28)

∀ϕ ∈ H1, |Γkrϕ|2 ≤ kϕk1.

(29)

Proposition 5.1. For each y ∈ Rd,D2wλ is a stationary random fields and|D2wλ(., y)|2 is

bounded independently of_{y ∈ R}d. Furthermore, for any_{y, h ∈ R}d |D2_w

λ(., y + h) − D2wλ(., y)|2 ≤ C|h|,

where the constant_{C does not depend on y, h ∈ R}d.

Proof: Remind that we have: ∀ϕ ∈ C, λwλ−Lywλ = h+f . Note that the parameter y ∈ Rd

is temporarily omitted to simplify the notations. Let us consider r ∈ R∗ and k ∈ {1, . . . , d}. For any ϕ ∈ C, we obtain:

([h + f ], Γk_rϕ)2 =λ(wλ, Γrkϕ)2+ (1/2) [a + H]Dwλ, DΓkrϕ
2 =λ(wλ, Γkrϕ)2− (1/2)([a + H]DΓk−rwλ, Dϕ)2 − (1/2)([Γk −ra + Γk−rH]Dwλ, Dϕ)2, so that we deduce (1/2) [a + H]DΓk_−rwλ, Dϕ)2 ≤ |h + f |2+ |λwλ|2+ Λkwλk1
kϕk1.

From (21a), sup_r∈R∗|DΓk_rw_λ(., y)|₂is bounded (independently of y ∈ Rd). For each y ∈ Rd,

up to extracting a subsequence, the family (Γk_rwλ(., y))radmits a weak limit in H1 as r → 0,

denoted by Fk_λ(., y). As guessed by the reader, Fk_λ(., y) is actually a weak solution of the equation:

λ(Fk_λ, ϕ)2+(1/2)([a+H]DFkλ, Dϕ)2 = −(h+f , Dkϕ)2+(1/2)([Dka+DkH]Dwλ, Dϕ)2.

This proves the uniqueness of the weak limit. The weak convergence of (Γk

rwλ)r in H1 thus

holds as r → 0 (and not up to a subsequence). In particular, it results that (DjFkλ(., y), ϕ)2 =

−(Dkwλ, Djϕ)2for any ϕ ∈ Dom(Dj) and 1 ≤ j, k ≤ d. Since the operator Dj is

antisym-metric and closed, this also proves that Dwλ(., y) ∈ Dom(D) and D2wλ(., y) is a stationary

random field.

The second part of the statement is quite similar to the corresponding part of the proof of Proposition 4.1 so that details are left to the reader.

Let us now consider a regularizing sequence of mollifiers (%m)m ∈ Cc∞(Rd× Rd) (smooth

functions with compact support) and define for any m ∈ N∗, (ω, y) ∈ Ω × Rd, wm_λ(ω, y) =

Z

Rd×Rd

Of course, the Itô formula holds for wm_λ, which is a smooth function: εdw_λm(Xε_t, X_tε) =ε−1LXtεwm λ(X ε t, X ε t) dt + c · Dw m λ (X ε t, X ε t) dt (30) + Dw_λm· σ(Xε_t, X_tε) dBt+ b∂ywλm(X ε t, X ε t) dt + ε(∂ywmλ) ∗ σ(Xε_t, X_tε) dBt+ εc · ∂ywλm(X ε t, Xtε) dt + (ε/2)trace(a∂_yy2 w_λm)(Xε_t, X_tε) dt + trace(aD∂ywmλ)(X ε t, X ε t) dt.

It remains to let m go to ∞. From the regularity properties of wλ stated in Propositions

5.1 and 4.1, it is plain to check that the derivatives of wm

λ involved in (30) can be expressed

as follows: the functions gm_λ ≡ ∂yjw

m

λ (resp. ∂y2jykw

m

λ, resp. Djwmλ, resp. D2jkwmλ, resp.

Dk∂yjw m λ) and gλ ≡ ∂yjwλ (resp. ∂ 2 yjykwλ, resp. Djwλ, resp. D 2 jkwλ, resp. Dk∂yjwλ) satisfy gm_λ(ω, y) = Z Rd×Rd g_λ(τx0ω, y − y0)%_m(x0, y0) dx0dy0.

The convergence of the derivatives of wm

λ towards the corresponding derivatives of wλ(Lemma

5.3 below) and estimates on the process Xε(Lemma 5.2 below) enable to pass to the limit, a m goes to ∞, in (30).

Lemma 5.2. There exists a positive constant A, only depending on M, Λ, d, such that the function f5.2 : R × Rd× Rd → R defined by f5.2(t, x, y) = _td/2A e

−|y−x|2_{At satisfies, for any}

measurable positive function_{ϕ defined on Ω × R}dandt > 0, ¯ Eεx[ϕ(X ε t, X ε t)] ≤ M Z Rd ϕ(ω, y)f5.2(t, x, y) dy.

Notations. To relieve the notations, for each t > 0 and x ∈ Rd, the expectations with respect to the probability measuresf5.2(t, x, y) dµ ⊗ dy and

Rt

0f5.2(r, x, y)dr
dµ ⊗ dy on Ω × R d

are respectively denoted by Mt5.2 and eMt5.2. Note that the parameter x is omitted from these

notations.

Lemma 5.3. Suppose that a measurable function w : Ω × Rd → R satisfies ∀(y, h) ∈ Rd × Rd_{, |w(·, y)|} 2 ≤ C, and |w(·, y + h) − w(·, y)|2 ≤ C|h|, and definewm_{(ω, y) =}R Rd×Rdw(τx0ω, y − y 0_)%

m(x0, y0) dx0dy0. Then, for anyt > 0

(31) ¯_Eε_x[|wm(Xε_t, X_tε) − w(Xε_t, X_tε)|] + ¯_Eε_x[ Z t 0 |wm_(Xε s, X ε s) − w(X ε s, X ε s)| ds] m→∞ −−−→ 0.

Proof of Lemma 5.2: It easily results from the Aronson estimates (cf. [13]): there exists a constant A that only depends on M, Λ, d such that ∀ε > 0, ∀ω ∈ Ω, ∀(t, x, y) ∈ R∗+× Rd× Rd

(32) 1 Atd/2e −A|y−x|2 t ≤ pε,ω(0, x, t, y) ≤ A td/2e −|y−x|2 At ,

where pε,ω denote the transition densities of the Markov process Xε. Proof of Lemma 5.3: From Lemma 5.2, we have

¯ Eεx[|w m_(Xε t,X ε t) − w(X ε t, X ε t)|] ≤ Mt 5.2|w m_{− w|} ≤ Z (Rd₎3

|w(τx0ω, y − y0) − w(ω, y)|₂f_5.2(t, x, y)%_m(x0, y0) dydx0dy0

≤ Z (Rd₎3 |w(τx0ω, y − y0) − w(ω, y − y0)|₂f_5.2(t, x, y)%_m(x0, y0) dydx0dy0 + Z (Rd₎3

|w(ω, y − y0) − w(ω, y)|2f5.2(t, x, y)%m(x0, y0) dydx0dy0.

Because of the stochastic continuity of the translation operators on the medium, the term |w(τx0ω, y − y0) − w(ω, y − y0)|₂ tends to 0 as x0 goes to 0 (and is bounded independently

of y − y0 ∈ Rd_{) so that classical convolution technics ensure that the first term in the above}

right-hand side converges to 0 as m goes to ∞. Concerning the second term, it also converges to 0, as m goes to ∞, because of the estimate |w(ω, y − y0) − w(ω, y)|2 ≤ min(2C, |y0|) and

classical convolution technics again.

The second term in the left-hand side of (31) raises no additional difficulty. So details are left to the reader.

We sum up this discussion in the following

Theorem 5.4. Let h, f be two functions satisfying the assumptions of Proposition 4.1. Let wλ be the solution of the resolvent equation:

λwλ(·, y) − Lywλ(·, y) = h(·, y) + f (·, y).

Then the following Itô formula holds: εdwλ(X ε t, X ε t) =ε −1_(λw λ − h − f )(X ε t, X ε t) dt + c · Dwλ(X ε t, X ε t) dt + Dwλ· σ(X ε t, X ε t) dBt+ b∂ywλ(X ε t, X ε t) dt + ε(∂ywλ)∗σ(X ε t, Xtε) dBt+ εc · ∂ywλ(X ε t, Xtε) dt + (ε/2)trace(a∂_yy2 wλ)(X ε t, X ε t) dt + trace(aD∂ywλ)(X ε t, X ε t) dt

6

Asymptotic Theorems

Classical ergodic theorem. Let us now investigate the asymptotic behavior, as ε → 0, of functionals of the typeR₀tΨ(Xε_r, X_rε) dr for a suitable locally stationary random field Ψ. This behavior is very intuitive: this functional averages with respect to its first variable, that is at a small scale, while the second variable prescribes the behavior at a large scale. More precisely Theorem 6.1. (Ergodic Theorem)

Let us consider_{Ψ : Ω × R}d_{→ R such that eM}t_{5.2[|Ψ|] < +∞. Denoting Ψ(y) = M[Ψ(·, y)],} the following convergence holds:

¯ Eεx  sup 0≤s≤t | Z s 0 Ψ(Xε_r, X_rε) − Ψ(X_rε) dr| −−→ ε→0 0.

Proof: The proof of this theorem is split up into two steps. The first one consists in proving this result for a bounded smooth function Ψ. The second step extends the result to the general case.

First step: We consider a bounded function Ψ(ω, y) = ϕ(ω)g(y), with ϕ ∈ C, g(y) ∈ C3

b(Rd) (bounded, three times continuously differentiable with bounded derivatives up to

or-der three). Obviously, Theorem 5.4 applies and provides us with the Itô formula: dvλ(X ε t, X ε t) = −ε −2 Ψ(Xε_t, X_tε) − λvλ(X ε t, X ε t)
dt (33) +ε−1c · Dvλ(X ε t, X ε t) dt + ε −1 Dvλ· σ(X ε t, X ε t) dBt +(∂yvλ)∗σ(X ε t, X ε t) dBt+ εc · ∂yvλ(X ε t, X ε t) dt +(1/2)trace[a∂_yy2 vλ](X ε t, X ε t) dt + ε −1_b∂ yvλ(X ε t, X ε t) dt +ε−1trace[aD∂yvλ](X ε t, X ε t) dt.

We deduce (the constant C may change from line to line) ¯ Eεx  sup 0≤s≤t   Z s 0 [Ψ − λvλ](X ε r, Xrε) dr   2  ≤Cε2_¯ Eεx  sup 0≤s≤t  vλ(X ε s, Xsε)   2 (34) + Ct¯_Eε_x Z t 0  εc · Dvλ(X ε r, X ε r)   2 dr + ε2∆ε,λ₁ + ε4∆ε,λ₂ .

Here ∆ε,λ₁ and ∆ε,λ₂ stand for the other terms that are not written but can be treated in the same way as the second one. Fix λ > 0. From the boundedness of Ψ, vλ is bounded ((G

y λ)λ is a

So the first term of the right-hand side vanishes as ε tends to 0. We now turn to the second term ¯ Eεx  Z t 0  εc · Dvλ(X ε r, X ε r)   2 dr Lemma 5.2 ≤ ε2_Met_5.2[|c · Dv_λ|2],

From the boundedness of c (see Assumption 2.2) and |Dvλ(., y)|2 (cf. (21a)), this latter

quantity tends to 0 as ε goes to 0.

From Lemma 6.2 below and (21a), which implies that sup_y∈Rd|λv_λ(·, y)−Ψ(y)|₂ < +∞,

we obtain: ¯ Eεx h sup 0≤s≤t   Z s 0 [λvλ− Ψ](X ε r, X ε r) dr   2i ≤ t e_Mt_5.2[|λvλ− Ψ|2] Lemma 6.2 = δ(λ)

where limλ→0δ(λ) = 0. The result follows in this case by fixing first λ small enough to make

δ(λ) small and then choosing ε small enough to make the right-hand side of (34) small too. Second step: Let us now generalize the class of considered functions Ψ. Let us only assume that e_Mt_5.2[|Ψ|] < +∞. Thanks to density arguments, we can find (ϕ_n)n ∈ CN and

(gn)n ∈ Cb3(Rd)Nsuch that eMt5.2[|Ψ − ϕngn|2] → 0 as n tends to 0. As guessed by the reader,

we obviously conclude with the help of Lemma 5.2.

Lemma 6.2. For each fixed y ∈ Rd, the solutionvλ(., y) of the equation (λ − Ly)vλ(., y) =

Ψ(., y) satisfiesλvλ(., y) − Ψ(., y)

₂ → 0 as λ goes to 0.

Proof: Apply Proposition 4.1 with the functions h = Ψ and f = 0. From (21a), the family (λvλ(., y))λis bounded in L2(Ω). So we can extract a L2(Ω)-weakly converging subsequence,

still indexed with λ > 0, and let g ∈ L2(Ω) denote its limit. The weak form of the resolvent equation reads, for ϕ ∈ H1,

(35) λ(vλ(., y), ϕ)2+ (1/2)([a + H]Dvλ(., y), Dϕ)2 = (Ψ(., y), ϕ)2.

Remind that Ly∗ stands for the adjoint operator of Ly in L2(Ω, µ). If ϕ ∈ Dom((Ly)∗), (35) also reads λ(vλ(., y), ϕ)2 + (vλ(., y), (Ly)∗ϕ)2 = (Ψ, ϕ)2. Multiplying this equality by λ

and passing to the limit as λ goes to 0 leads to (g, (Ly)∗ϕ)2 = 0 for any ϕ ∈ Dom((Ly)∗).

In particular (see Section 4), g ∈ Dom(Ly) ⊂ H1 and Lyg = 0. From Assumption 2.3,

kgk2

1 ≤ M kgk21,y = −(g, L y_g)

2 = 0. So we get Dig = 0, ∀i = 1, . . . , d. As a consequence,

g is invariant under space translations. From Definition 2.1, g must be constant µ almost surely and then necessarily equal to

g = lim

λ→0M[λvλ(., y)] = limλ→0M[λvλ(., y) − L y_v

λ(., y)] = lim

This proves the uniqueness of the weak limit. Choosing ϕ = λvλ in (35) leads to

lim

λ→0[λ 2_|v

λ(., y)|22+(λ/2)(aDvλ(., y), Dvλ(., y))2] = lim

λ→0(Ψ(., y), λvλ(., y))2 = M[Ψ(., y)] 2_.

In particular, lim sup_λ→0λ2|vλ(., y)|22 ≤ |g|22, from which the strong convergence in L2(Ω)

results.

Asymptotic theorem for highly oscillating functionals. Theorem 6.1 settles the issue of the asymptotic behavior of functionals of the formR₀tΨ(Xε_r, Xε

r) dr for a locally random

field Ψ ∈ L1(Ω × Rd_{; e}

Mt5.2). As explained in Section 1 and unlike the locally periodic case

(see [2] or [1]), the locally stationary framework raises the issue of describing the asymptotic behavior of functionals of the typeR₀tΨε(X

ε

r, Xrε) dr for a family (Ψε)εthat is not convergent

in L1_{(Ω × R}d_{; e}

Mt5.2). Theorem 6.3 states that these functionals converge to 0 as ε goes to 0

provided that we can suitably control a sort of Poincaré inequality for the family (Ψε)ε. More

precisely

Theorem 6.3. (Ergodic theorem II) For each ε > 0, let us consider a function Ψε such

thatR

Rd|Ψε(·, y)| 2

2dy < +∞. Let us additionally assume that, for each ε > 0, we can find

a positive constantCε such that the following type of "Poincaré inequality" holds: for any

ϕ(ω, y) = χ(ω)%(y), (χ, %) ∈ C × C_c∞_(Rd_), (36) _M Z Rd Ψεϕ(., y) dy ≤ Cε  M Z Rd |(D + ε∂y)ϕ(., y)|2dy 1/2 , IfεCε → 0 as ε → 0 then ∀t > 0, ¯ Eεx| Z t 0 Ψε(X ε r, X ε r) dr| 2_{ → 0, asε → 0.}

Proof: The proof of this result is divided in three steps:

First step (setup). Equip the set Ω × Rd_{with the product measure}

dΠ(ω, y) = dµ ⊗ dy
(ω, y).

Denote (·, ·)2,Πthe canonical inner product associated to the space L2(Ω × Rd, Π), | · |2,Π the

corresponding norm and CΠthe following subspace of L2(Ω × Rd, Π) of smooth functions:

CΠ= {χ(ω)%(x); (χ, %) ∈ C × Cc∞(R d

Let us now define the following unbounded operators on CΠ⊂ L2(Ω × Rd, Π):

∀ϕ ∈ CΠ, Lεϕ =

1

2(Div + εdivy)(a + H)(D + ε∂y)ϕ. (37)

We now proceed as in [5, Ch. 3, Sect. 3] or [10, Ch. 1, Sect. 2] to extend this operator. We define, for any ϕ, ψ ∈ CΠ,

(38) (ϕ, ψ)1,ε= 1 2 Z Ω×Rd a(D + ε∂y)ψ · (D + ε∂y)ϕ(ω, y) dΠ(ω, y),

and Θε(ϕ, ψ) = (ϕ, ψ)2,Π + (ϕ, ψ)1,ε. For each ε > 0, we can define HεΠ as the closure

of CΠ in L2(Ω × Rd, Π) with respect to the norm k · k1,εassociated to the inner product Θε.

Although the gradients Dϕ and ∂yϕ need not separately exist for ϕ ∈ HεΠ, it makes sense to

consider the "2-scale gradient" (D + ε∂y)ϕ thanks to (38) and Assumption 2.3.

We now extend the operator Lεas follows. For any α > 0, we define the following bilinear form on Hε Π× HεΠ: Bα,ε_L (ϕ, ψ) = α(ϕ, ψ)2,Π+ 1 2 Z Ω×Rd (a + H)(D + ε∂y)ψ · (D + ε∂y)ϕ(ω, y) dΠ(ω, y).

This form is clearly continuous and coercive on HεΠ× HεΠ. If, for some function ϕ ∈ HεΠ,

the application ψ ∈ HεΠ 7→ B α,ε

L (ϕ, ψ) is L2(Ω × Rd, Π) continuous, then ϕ ∈ Dom(L ε_).

Moreover, we can find f ∈ L2_{(Ω × R}d_{, Π) such that B}α,ε

L (ϕ, ·) = (f , ·)2,Π. We then define

Lεϕ = αϕ − f . This definition does not depend on α and extends definition (37).

Second step (asymptotic control). We now aim at solving the resolvent equation, for each  > 0,

(39) ε2ϕ_ε− Lε_ϕ

ε = Ψε,

and at describing the asymptotic behavior, as ε tends to 0, of the solution ϕ_ε. From now on, Bε

Ldenotes the bilinear form B ε2_,ε

L . Of course, from the coerciveness of B ε

L, we can find ϕε ∈

HεΠ∩ Dom(L

ε_{) that satisfies B}ε

L(ϕε, ϕ) = (Ψε, ϕ)2,Π for any ϕ ∈ HεΠ, which is equivalent

to (39). Choosing ϕ = ϕ_εand (36) leads to the estimate ε2_|ϕ

ε|22,Π+ (1/2)|ϕε|21,ε ≤ M Cε2/2.

An easy consequence is ε4|ϕ_ε|2

2,Π+ (ε2/2)kϕεk21,ε≤ ε2M Cε2/2 → 0 as ε → 0.

Final step. From Lemma 6.4 below, the following formula holds ¯_Pε_xa.s.

dϕε(X ε t, Xtε) =ε −2 Lεϕε(X ε t, Xtε) dt + ε −1 σ∗(D + ε∂y)ϕε(X ε t, Xtε) dBt =ε−2[ε2ϕε− Ψε](X ε t, X ε t) dt + ε −1 σ∗(D + ε∂y)ϕε(X ε t, X ε t) dBt,

and, as a consequence, for 0 ≤ s ≤ t, Z t s Ψε(X ε r, Xrε) dr = − ε2ϕε(X ε t, Xtε) − ϕε(X ε s, Xsε) + ε2 Z t s ϕε(X ε r, Xrε) dr + ε Z t s σ∗(D + ε∂y)ϕε(X ε r, X ε r) dBr.

From Lemma 5.2, we deduce ¯ Eεx| Z t s Ψε(X ε r, X ε r) dr| 2_{ ≤ ε}4_{C(1 + t)|ϕ} ε|22,Π+ ε 2_Ctkϕ εk21,ε≤ C(1 + t)M ε 2_C2 ε/2

and the result follows.

Lemma 6.4. Keeping the notations of the proof of Theorem 6.3, the following formula holds ¯ Pεxalmost surely: dϕε(X ε t, X ε t) = ε −2 Lεϕε(X ε t, X ε t) dt + ε −1 σ∗(D + ε∂y)ϕε(X ε t, X ε t) dBt.

Proof: To simplify the notations, choose ε = 1 and omit the parameter ε (as a consequence ϕ_ε is denoted by ϕ). Once again, we follow [6, Sect 8.3]. The method is quite similar to Section 5 so that we just outline the main ideas.

As in Section 5, ek (1 ≤ k ≤ d) still stands for the k-th vector of the canonical basis of

Rd. For any ψ ∈ L2(Ω × Rd; Π) and r ∈ R∗, we define [Γrkψ](ω, y) = 1rψ(τrekω, y + rek) −

ψ(ω, y). We have again: (Γk

rφ, ψ)2,Π = −(φ, Γk−rψ)2,Π, for any φ, ψ ∈ L2(Ω × Rd; Π),

and |Γk

rφ|2,Π ≤ |(D + ∂y)φ|2,Π for any φ ∈ HΠ.

Following the proof of Proposition 5.1, we can prove that (D + ∂y)ϕ ∈ HΠ so that it

makes sense to consider (D + ∂y)2ϕ ∈ L2(Ω × Rd; Π).

Let us now consider a mollifier % : Rd → R with compact support and define, for n ≥ 1, %n = nd%(n·). Then we define ϕn(ω, y) =

R

Rdϕ(τ−x0ω, y − x 0_)%

n(x0) dx0. Again with

classical convolution technics we can establish that (ϕn)n(resp. (D + ∂y)ϕn

n, resp. (D +

∂y)2ϕn

n) converges to ϕ (resp. (D + ∂y)ϕ, resp. (D + ∂y)

2_{ϕ) in L}2_{(Ω × R}d_{; Π) as n goes}

to ∞. Moreover the function x ∈ Rd7→ ϕn_(τ

xω, x) is of class C2. Thus we can apply the Itô

formula to this function and to the process X, and then pass to the limit as n goes to ∞ with the help of Lemma 5.2 and an adapted version of Lemma 5.3, as explained in Section 5. The result then follows.

7

Homogenization Property

This section is devoted to the presentation and the proof of the homogenization property. We remind the reader that ui_λ stands for the solution of the auxiliary problems, that the solution

of the resolvent equation λui_λ(·, y) − Lyui_λ(·, y) = bi(·, y) (see Section 4). Moreover, for

i = 1, . . . , d and y ∈ Rd _{fixed, the functions (Du}i

λ(·, y))λ are convergent in L2(Ω)dtowards

ξ_i(·, y) and are bounded in L2_{(Ω)-norm independently of y and λ. It thus makes sense to}

define, for every y ∈ Rd, A(y) = lim

λ→0 M[(I + Duλ) ∗

a(I + Duλ)(., y)] = M[(I + ξ)∗a(I + ξ)(., y)],

(40a)

H(y) = lim

λ→0 M[(I + Duλ) ∗

H(I + Duλ)(., y)] = M[(I + ξ)∗H(I + ξ)(., y)],

(40b)

B(y) = (1/2)∂y[A + H](y).

(40c)

Thanks to Proposition 4.3, all these coefficients are twice continuously differentiable with bounded derivatives up to order two. In particular, it makes sense to consider the strong solution of the following SDE

(41) Xt= x + Z t 0 B(Xr) dr + Z t 0 A1/2(Xr) dBr.

As observed in [14], one of the consequences of (32) is the tightness of the process Xε in C([0, T ]; Rd). We will establish that each converging subsequence of (Xε₎

ε solves the

martingale problem (41). As a consequence, Theorem 3.1 holds.

Proof of Theorem 3.1: Let us consider a converging subsequence of (Xε₎

ε, still indexed with

ε. We just have to identify the finite dimensional distributions of the limit. Applying Theorem 5.4 to the function uλand choosing λ = ε2 yields

dX_tε = − εduε2(X ε t, Xtε) + ε(∂yuε2)∗σ(X ε t, Xtε) dBt + [εuε2 + εc · ∂_yu_ε2 + (ε/2)trace(a∂_yy2 u_ε2)](X ε t, X ε t) dt + [b∂yuε2 + c · (I + Du_ε2) + trace(aD∂_yu_ε2)](X ε t, X ε t) dt + [σ + Du∗_ε2σ](X ε t, X ε t) dBt ≡dΘ1,ε_t + dΘ2,ε_t + dΘ3,ε_t + dΘ4,ε_t

From Proposition 4.3 and Lemma 5.2, it is plain to see that ¯_Eε x[(Θ

1,ε

t )2+ (Θ 2,ε

t )2] → 0 as

ε goes to 0. Let us now focus now on Θ4,ε_t . The quadratic variations exactly match R₀t[I + Duε2]∗a[I + Du_ε2](X

ε

r, Xrε) dr and satisfy ¯Eεx sup0≤s≤t| < Θ4,ε >s −

Rs

0 A(X ε

r) dr| → 0

as ε goes to 0 (Theorem 6.1). Combining this with Proposition 7.1 below, the limit is readily identified as the solution of SDE (41). So we complete the proof of Theorem 3.1.

Proposition 7.1. The following convergence holds: ¯_Eε_x|Θ3,ε_t −Rt

0 B(X ε r) dr|2  → 0 as ε tends to0.

Proof: The main difficulty actually lies in the term corresponding to b∂yuε2. The other terms

are easily treated with the help of Theorem 6.1 and the convergence of the derivatives (Duλ)λ

and (D∂yuλ)λstated in Proposition 4.3. The strategy consists in establishing that, up to

intro-ducing new correctors for b∂yuε2, this term can be divided in two parts, the former satisfying

Theorem 6.1 and the latter satisfying Theorem 6.3.

We keep the notations of the proof of Theorem 6.3 and use the convention of summa-tion over repeated indices. To find these correctors, let us consider a test funcsumma-tion ϕ ∈ CΠ.

Proceeding with successive integration by parts, we have, for 1 ≤ i ≤ d, 2(b∂yuiε2, ϕ)2,Π = (Dj(a + H)jk∂yku i ε2, ϕ)2,Π = −((a + H)jk∂yku i ε2, Djϕ)2,Π− ((a + H)jkDj∂yku i ε2, ϕ)2,Π = −((a + H)jk∂yku i ε2, (Dj + ε∂yj)ϕ)2,Π− ((a + H)jkDj∂yku i ε2, ϕ)2,Π +((a + H)jk∂yku i ε2, ε∂yjϕ)2,Π = −((a + H)jk∂yku i ε2, (D_j + ε∂_yj)ϕ)_2,Π− ((a + H)_jkD_j∂_ykui_ε2, ϕ)_2,Π −ε(∂yj(a + H)jk∂yku i ε2, ϕ)_2,Π− ε((a + H)_jk∂_y2 kyju i ε2, ϕ)_2,Π

The decomposition and the correctors now appear clearly. Define the correctors as Corri,ε= (ε/2)∂yj(a + H)jk∂yku

i

ε2 + (ε/2)(a + H)jk∂y2kyju

i ε2,

the converging part in L2(Ω × Rd_{; e}

Mt5.2)

Convi,ε = −(1/2)(a + H)jkDj∂yku

i ε2

and the exploding part that we aim to treat with Theorem 6.3 (42) Expli,ε= b∂yuiε2 + Corri,ε− Convi,ε.

Clearly, from Assumptions 2.2 and 2.3 and Proposition 4.3, the family (Corri,ε)εconverges to

0 in L2_{(Ω × R}d; e_Mt_5.2) to 0. Then Lemma 5.2 ensures that ¯

Eεx|

Z t

0

Corri,ε(Xε_r, X_rε) dr| → 0, as ε → 0.

From Proposition 4.3 and Theorem 6.1, we can prove that ¯_Eε_x| R₀tConvi,ε(Xε_r, X_rε) dr − Rt

0 K(X ε

r) dr| → 0 as ε tends to 0, where K(y) = M[−(1/2)(a + H)jk∂ykξi(·, y)].

It remains to treat the part Expli,ε, for which we have already proved that

(Expli,ε, ϕ)2,Π = −(1/2)((a + H)jk∂yku

i

ε2, (Dj+ ε∂yj)ϕ)2,Π.

We aim at applying Theorem 6.3. The only technical obstacle is that the function Expli,εneed not satisfyR

Rd|Expl

i,ε_{(·, y)|}2

2dy < +∞. However, it can be easily overcome. Indeed, it seems

natural to introduce a nonnegative smooth function % : Rd _{→ R with compact support and to}

define

(44) Expli,ε_% (ω, y) = Expli,ε(, y)%(y) + (ε/2)(a + H)jk∂yku

i ε2∂yj%.

Obviously,R

Rd|Expl i,ε

ρ (·, y)|22dy < +∞. Furthermore, from Lemma 5.2 and Assumption 2.2,

¯ Eεx|

Z t

0

[Expli,ε− Expli,ε ρ ](X ε r, X ε r) dr| ≤2Λ sup y∈Rd |∂y%(y)| eMt5.2[|ε∂yku i ε2|] + sup y∈Rd

|Expli,ε_{(·, y)|}

2Met_5.2[|1 − %(y)|]. (45)

From (42) and (26a), sup_y∈Rd|Expli,ε(·, y)|₂ is finite and bounded by a constant that depends

neither on ε nor on y ∈ Rd and the family Met_5.2[|ε∂_ykui_ε2|]

ε converges to 0 as ε tends to 0.

Moreover, from (43) and (44), for a test function ϕ ∈ CΠ, it is readily seen that

(Expli,ε_ρ , ϕ)2,Π = (Expli,ε, ϕ%)2,Π+ (ε/2) (a + H)jk∂yku

i ε2∂yj%, ϕ
2,Π = −(1/2) (a + H)jk∂yku i ε2, (D_j + ε∂_yj)(ϕ%)
2,Π +(ε/2) (a + H)jk∂yku i ε2∂yj%, ϕ
2,Π = −(1/2) %(a + H)jk∂yku i ε2, (D_j + ε∂_yj)ϕ
2,Π ≤ Cε  M Z Rd |(D + ε∂y)ϕ(., y)|2dy 1/2 , where Cε= (1/2)(M +Λ) R Rd|%(y)∂yu i

ε2(·, y)|2dy. From Proposition 4.3, εCε → 0 as ε → 0

so that Theorem 6.3 can now be applied. Consequently, ¯_Eε x| R t 0 Expl i,ε ρ (X ε r, Xrε) dr| → 0 as

ε tends to 0, for each fixed function %. Hence, we just have to fix a function % in order for the second term in the right-hand side of (45) to be small and then to choose ε small enough to make the first term andR₀tExpli,ε_ρ (Xε_r, Xε

r) dr small. The result follows.

It just remain to check that the limit can be expressed in terms of ¯B, as stated in Proposition 7.1. All we know at present is ¯Eεx|Θ

3,ε t −

Rt

0 F (X ε

r) dr|2 → 0 as ε tends to 0, where the entries

Fi, 1 ≤ i ≤ d, of F are given by (δ stands for the Kroenecker symbol)

Hence Fi = M(1/2)∂yk(a + H)kj(δij + ξij) + akj∂ykξij − (1/2)(a − H)kj∂ykξij)(·, y)  = M(1/2)∂yk(a + H)kj(δij + ξij) + (1/2)(a + H)kj∂ykξij)(·, y)  = (1/2)∂ykM(a + H)kj(δij + ξij)(·, y).

The proof is over if we can prove that (A + H)(y) = M(a + H)(I + ξ)(·, y). Choosing ϕ = uj_λ(·, y) in (25) and passing to the limit as λ goes to 0 yields (a+H)(ei+ ξi), ξj

2 = 0,

that is Mξ∗(a + H)(I + ξ)(·, y) = 0. We now complete the proof.

Acknowledgments. The author is thankful to Stefano Olla for fruitful discussions about this topic and to an anonymous referee for a careful reading of the paper.

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