Rémi Rhodes
email: rhodes@cmi.univ-mrs.fr

On homogenization of space-time dependent degenerate random flows

We study a diffusion with time-dependent random coefficients. The diffusion coefficient is allowed to degenerate. We prove an invariance principle when this diffusion is supposed to be controlled by another one with time independent coefficients.

Introduction

We want to establish an invariance principle for a diffusive particle in a random flow described by the following Stochastic Differential Equation (SDE)

X ω t = x + t 0 b (r, X ω r , ω) dr + t 0 σ (r, X ω r , ω) dB r ,
where B is a d-dimensional Brownian motion and σ, b are stationary random fields. b is defined in such a way that the generator at time t of the diffusion coincides on smooth functions with

(1) L ω = (1/2)e 2V (x,ω) div x e -2V (x,ω) [a(t, x, ω) + H(t, x, ω)]∇ x .

Here a(t, x, ω) is equal to σσ * (t, x, ω). V and H are stationary random fields, V is bounded and H antisymmetric. We will then be in position to study the effective diffusion on a macroscopic scale of the following convection-diffusion equation

∂ t z(t, x, ω) = (1/2)Trace[a△ xx z](t, x, ω) + b • ∇ x z(t, x, ω), (2)
with certain initial condition. We will prove that, in probability with respect to ω,

lim ε→0 z(t/ε 2 , x/ε, ω) = z(t, x)
where z is the solution of a deterministic equation [START_REF] Fannjiang | Diffusion approximation for particle convection in Markovian Flows[END_REF] ∂ t z(t, x) = Trace[A△ xx z](t, x).

A is a constant matrix -the matrix of so-called effective coefficients.

Homogenization problems have been extensively studied in the case of periodic flows (cf. [START_REF] Bhattacharya | Asymptotics of solute dispersion in periodic media[END_REF], [START_REF] Pardoux | Homogenization of periodic linear degenerate PDEs[END_REF], [START_REF] Pardoux | Homogenization of linear and semilinear second order parabolic PDEs with periodic coefficients: a probabilistic approach[END_REF], and many others). The study of random flows (see [START_REF] Olla | Homogenization of diffusion processes in Random Fields[END_REF], [START_REF] Osada | Homogenization of diffusion with random stationary coefficients[END_REF], [START_REF] Sethuraman | Diffusive limit of a tagged particle in asymmetric simple exclusion processes[END_REF], and many others) spread rapidly thanks to the techniques of the environment as seen from the particle introduced by Kipnis and Varadhan in [START_REF] Kipnis | Central limit theorem for additive functionals of reversible Markov processes and application to simple exclusion[END_REF], at least in the case of time independent random flows. Recently, there have been results going beyond these techniques in the case of isotropic coefficients which are small perturbations of Brownian motion (see [START_REF] Sznitman | An invariance principle for isotropic diffusions in random environment[END_REF]). But there are only a few works in the case of space-time dependent random flows (see [START_REF] Landim | Convection-diffusion equation with space-time ergodic random flow[END_REF] or [START_REF] Komorowski | On homogenization of time-dependent random flows[END_REF] for instance in the case σ = Id). A quenched version of the invariance principle is stated in [START_REF] Fannjiang | An invariance principle for diffusion in turbulence[END_REF] provided that the diffusion coefficient satisfies a strong uniform non-degeneracy assumption. In this case, the regularizing properties of the heat kernel are widely used to face with the non-reversibility of the underlying processes. Some results stated in Markovian flows are also established in [START_REF] Fannjiang | Diffusion approximation for particle convection in Markovian Flows[END_REF] or [START_REF] Fannjiang | Invariance principle for a diffusion in a Markov field[END_REF]. The novelty of this work lies in the ergodic and regularizing properties required on the coefficients, which are not far from being minimal. The only restriction is the control of the diffusion process with an ergodic and time independent one. As a consequence, this work includes the static case where all the coefficients do not depend on time. Moreover, these assumptions allow the diffusion matrix to degenerate. Typically it can degenerate in certain directions or vanish on subsets of null measure but cannot totally reduce to zero on an open subset of R d . However, considering such strong degeneracies remains a quite open problem for random stationary coefficients (for recent advances in the static periodic case, see [START_REF] Pardoux | Homogenization of periodic linear degenerate PDEs[END_REF]).

We will outline now the main ideas of the proof. Our goal is to show that the rescaled process

εX ω t/ε 2 = ε t/ε 2 0 b (r, X ω r , ω) ds + ε t/ε 2 0 σ (r, X ω r , ω) dB s
converges in law to a Brownian motion with a certain positive covariance matrix. The general strategy (see [START_REF] Kozlov | The Method of Averaging and Walks in Inhomogeneous Environments[END_REF]) consists in finding an approximation of the first term on the right-hand side by a family of martingales and then in applying the central limit theorem for martingales. To find such an approximation, we look at the environment as seen from the particle

Y t = τ t,X ω t ω,
where {τ t,x } is a group of measure preserving transformation on a random medium Ω (see Definition 2.1). Thanks to the particular choice of the drift, an explicit invariant measure can be found for this Markov process. The ergodicity is ensured by the geometry of the diffusion coefficient σ (see Assumptions 2.3 and 2.4). The approximation that we want to find leads to study the equation (λ > 0)

(4) λu λ -(L + D t )u λ = b
where L + D t coincides with the generator of the process Y on a certain class of functions (the term D t is due to the time evolution and L is an unbounded operator on the medium Ω associated to [START_REF] Bhattacharya | Asymptotics of solute dispersion in periodic media[END_REF]). Here are arising the difficulties resulting from the time dependence. Due to the term D t , the Dirichlet form associated to L + D t does not satisfy any sector condition (even weak). However, for a suitable function b, (4) can be solved with the help of an approximating sequence of Dirichlet forms with weak sector condition. Then, usual techniques used in the static case fall short of establishing the so-called sublinear growth of the correctors u λ . To get round this difficulty, regularizing properties of the heat kernel are used in [START_REF] Fannjiang | An invariance principle for diffusion in turbulence[END_REF], [START_REF] Landim | Convection-diffusion equation with space-time ergodic random flow[END_REF] or [START_REF] Komorowski | On homogenization of time-dependent random flows[END_REF]. Here the degeneracies of the diffusion coefficient prevents us from using such arguments. The strategy here consists in separating the time and spatial evolutions (see Assumption 2.3). We introduce a new operator S whose coefficients do not depend on time. Then the spectral calculus linked to the normal operator S + D t will be determining to establish the desired estimates for the solution v λ of the equation

λv λ -( S + D t )v λ = b.
Finally, with perturbation methods, we show that these estimates remain valid for the correctors u λ .

Notations, Setup and Main Result

Let us first introduce a random medium Definition 2.1. Let (Ω, G, µ) be a probability space and τ t,x ; (t, x) ∈ R × R d a stochastically continuous group of measure preserving transformations acting ergodically on Ω:

1) ∀A ∈ G, ∀(t, x) ∈ R × R d , µ(τ t,x A) = µ(A), 2) If for any (t, x) ∈ R × R d , τ t,x A = A then µ(A) = 0 or 1, 3) For any measurable function g on (Ω, G, µ), the function (t, x, ω) → g(τ t,x ω) is measurable on (R × R d × Ω, B(R × R d ) ⊗ G).
In what follows we will use the bold type to denote a function g from Ω into R (or more generally into R n , n ≥ 1) and the unbold type g(t, x, ω) to denote the associated representation mapping (t, x, ω) → g(τ t,x ω). The space of square integrable functions on (Ω, G, µ) is denoted by L 2 (Ω), the usual norm by | • | 2 and the corresponding inner product by ( • , • ) 2 . Then, the operators on L 2 (Ω) defined by T t,x g(ω) = g(τ t,x ω) form a strongly continuous group of unitary maps in L 2 (Ω). Each function g in L 2 (Ω) defines in this way a stationary ergodic random field on R d+1 . The group possesses d + 1 generators defined for i = 1, . . . , d, by

D i f = ∂ ∂x i T 0,x f | (t,x)=0 , and
D t f = ∂ ∂t T t,0 f | (t,x)=0 ,
which are closed and densely defined. Denote by C the dense subset of L 2 (Ω) defined by

C = Span f * ϕ; f ∈ L 2 (Ω), ϕ ∈ C ∞ c (R d+1 ) , with f * ϕ(ω) = R d+1 f (τ t,x ω)ϕ(t, x) dt dx, where C ∞ c (R d+1 ) is the set of smooth functions on R d+1 with a compact support. Remark that C ⊂ Dom(D i ) and D i (f * ϕ) = -f * ∂ϕ ∂x i . This last quantity is also equal to D i f * ϕ if f ∈ Dom(D i ). Consider now the measurable functions σ : Ω → R d×d , σ : Ω → R d×d , H : Ω → R d×d and V : Ω → R and assume that H is antisymmetric. Define a = σσ * and a = σ σ * . The function V does not depend on time, that means ∀t ∈ R, T t,0 V = V .

Assumption 2.2. (Regularity of the coefficients)

• Assume that ∀i, j, k, l = 1, . . . , d, a ij , a ij , V , H ij , D l a ij and D l a ij ∈ Dom(D k ). • Define, for i = 1, . . . , d, b i (ω) = d j=1 1 2 D j a ij (ω) -a ij D j V (ω) + 1 2 D j H ij (ω) , b i (ω) = d j=1 1 2 D j a ij (ω) -a ij D j V (ω) , (5)
and assume that the applications [START_REF] Fukushima | Dirichlet Forms and Symmetric Markov Processes[END_REF]Ch. 3,Sect. 3]) which is still denoted S. Assume that each function f ∈ Dom( S) satisfying Sf = 0 must be µ almost surely equal to some function that is invariant under space translations.

(t, x) → b i (t, x, ω), (t, x) → b i (t, x, ω), (t, x) → σ(t,
Even if it means adding to V a constant (and this does not change the drift b, see ( 5)), we make the assumption that e -2V dµ = 1. Thus we can define a new probability measure on Ω by dπ(ω) = e -2V (ω) dµ(ω).

We now consider a standard d-dimensional Brownian motion defined on a probability space (Ω ′ , F, IP) (the medium and the Brownian motion are mutually independent) and the diffusions in random medium given as the solutions of the following Stochastic Differential Equations (SDE)

X ω t = x + t 0 b (r, X ω r , ω) dr + t 0 σ (r, X ω r , ω) dB r , X ω t = x + t 0 b(X ω r , ω) dr + t 0 σ(X ω r , ω) dB r . (8)
The main result of this paper is stated as follows Theorem 2.5. The law of the rescaled process εX ω t/ε 2 converges in probability (with respect to ω) to the law of a Brownian motion with a certain covariance matrix A (see (45)).

Examples

There are many ways to ensure the validity of Assumption (2.4). In particular, it is satisfied when, for almost all ω ∈ Ω, the R d -valued Markov process X ω , whose generator coincides on smooth functions with

S ω = e 2V (x,ω) 2 Div x e -2V (x,ω) a(x, ω)∇ x . ,
is irreducible in the following sense. Suppose that, starting from any point of R d , the process reaches each subset of R d of non-null Lebesgue measure in finite time. That means that there exists a measurable subset

N ⊂ Ω with µ(N ) = 0 such that ∀ω ∈ Ω \ N , for each measurable subset B of R d with λ Leb (B) > 0, ∀x ∈ R d , ∃t > 0, (9)
IP x X ω t ∈ B > 0.
This can be proved as in [START_REF] Komorowski | On homogenization of time-dependent random flows[END_REF] section 3 or in [START_REF] Olla | Homogenization of diffusion processes in Random Fields[END_REF] chapter 2 Theorem 2.1, in studying the Ω-valued Markov process Y t (ω) = τ 0, X ω t ω, whose generator coincides on C with S. As an easy consequence, if the diffusion coefficient a is uniformly elliptic or satisfies a strong Hörmander condition (see [START_REF] Kusuoka | Long time estimates for the heat kernel associated with a uniformly subelliptic symmetric second order operator[END_REF] for further details), then estimates on the transition densities of the process X ω ensure [START_REF] Kusuoka | Long time estimates for the heat kernel associated with a uniformly subelliptic symmetric second order operator[END_REF].

Let us now tackle the issue of constructing examples that do not satisfy any uniform ellipticity assumption or even strong Hörmander condition. In what follows, two examples are given. The first one deals with periodic coefficients. The second one is a random medium with a random chessboard structure and thereby does not reduce to the periodic case.

A periodic example

Let us construct a periodic example on the torus T 3 , where the diffusion matrix reduces to zero on a certain subset with null Lebesgue measure. We define a time-independent matrix-valued function

σ(t, x, y) = (1 -cos(x))(1 -cos(y)) 1 0 0 1 .
For simplicity, we choose V = H(x, y) = 0. Thanks to the (not uniform!) ellipticity of the diffusion coefficient inside the cell C =]0, 2π

T 3 → R 2×2 satisfying α -1 Id ≤ U U * (t,
x, y) ≤ αId for some α > 0, and then in defining σ(t, x, y) = σ(t, x, y)U (t, x, y), for which Assumption 2.3 is easily checked.

An example on chessboard structures

Let us now explain how to construct a random medium with chessboard structures.

= (β t , α 1 x 1 , α 2 x 2 ) t∈R,x=(x 1 ,x 2 )∈R 2
, where the three processes α 1 , α 2 and β are mutually independent and constructed as prescribed above. Hence ω (t,x) ; (t, x) ∈ R × R 2 is an ergodic stationary process and we can consider the random medium Ω = C(R × R 2 ; R 3 ) equipped with the probability law of this process.

We define the matrix σ(ω) = 1 0 0 α 1 0 and V = 0 (or any bounded function of the random field α). We can choose any matrix-valued function U : Ω → R 2×2 such that U U * is uniformly elliptic and bounded, and then we set σ = σU . It can be proved that Assumption 2.4 is satisfied. Actually, the ergodicity property for σ is very intuitive. Indeed, the matrix σ(•, ω) degenerates only on some stripes (the white ones), and in fact only on a part of each of them (depending on the support of ϕ), and only along the y 2 -axis direction: while lying on the degenerating part of a white stripe, the diffusion associated to (1/2) 2 i,j=1 ∂ i (ã i,j ∂ j ) can only move along the y 1 -axis direction. Nevertheless, with probability 1, the process encounters a black stripe sooner or later (because the parameter p belongs to ]0, 1[): it thus manages to move up and down and to reach every subset of the space. Ergodicity follows. Rigorous arguments are however left to the reader.

We can also consider a non-null stream matrix H. For instance the matrix-valued function

H(ω) = 0 (α 1 0 ) 2 β 0 -(α 1 0 ) 2 β 0 0 , fits Assumption 2.3.

Environment as seen from the particle

We now look at the environments as seen from the particle associated to the processes X and X: they both are Ω-valued Markov processes and are defined by

(10) Y t (ω) = τ t, X ω t ω, and Y t (ω) = τ t,X ω t ω,
where the processes X ω and X ω both starts from the point 0 ∈ R d . An easy computation proves that the generators of these Markov processes respectively coincide on C with S + D t and L + D t , where L is defined on C by

(11) L = e 2V 2 d i,j=1 D i e -2V [a + H] ij D j ).
Hence π is an invariant measure for both processes (see also [START_REF] Oelschl Äger | Homogenization of a diffusion process in a divergence free random field[END_REF]). Both associated semigroups thus extend continuously to L 2 (Ω, π). We should point out that the invariant measure need not be unique.

Poisson's equation

The aim of this section is, at first, to find a solution u λ of the resolvent equation that can formally be rewritten (a rigorous definition of each term is given later), for λ > 0, as:

(12) λu λ -(L + D t )u λ = h.
Since the associated Dirichlet form satisfies no sector condition (even weak), existence and regularity of such a solution is generally a tricky work, especially in considering degeneracies both in time and in space. However, for a suitable right-hand side, this equation can be solved with the help of an approximating sequence of Dirichlet forms satisfying a weak sector condition. Thereafter we study the asymptotic behaviour of the solution u λ as λ → 0.

Setup

Let us denote by ( P t ) t the semigroup on L 2 (Ω, π) generated by the process Y and by ( P * t ) t its adjoint operator. Let us also denote by ( Pt ) t the self-adjoint semigroup on L 2 (Ω, π) generated by the process Ȳt (ω) = τ 0, X ω t ω. Its generator is S. From the time independence of the coefficients b and σ, it is readily seen that, that ∀f ∈ L 2 (Ω, π), P t f = T t,0 Pt f = Pt T t,0 f . As a consequence,

P * t = T -t,0 Pt f = Pt T -t,0 f , in such a way that P t ( P * t f ) = P * t ( P t f ).
The generator in L 2 (Ω, π) of ( P t ) t , wrongly denoted by [ S + D t ], is then normal (see Theorem 13.38 in [START_REF] Rudin | Analyse fonctionnelle[END_REF]) so that we can find a spectral resolution of the identity E on the Borelian subsets of

R + × R such that -[ S + D t ] = R + ×R (x + iy) E(dx, dy).
Actually, we have -S = R + ×R x E(dx, dy), and -D t = R + ×R iy E(dx, dy). Indeed, S and R + ×R x E(dx, dy) are both self-adjoint and coincide on C. From [5, Ch. 1, Sect. 3], they are equal. The same arguments hold for D t and R + ×R iy E(dx, dy). For any ϕ, ψ ∈ L 2 (Ω), denote by E ϕ,ψ the measure defined by E ϕ,ψ = (Eϕ, ψ) 2 . From now on, denote by (. , . ) 2 the usual inner product in L 2 (Ω, π). For any ϕ, ψ ∈ C, define [START_REF] Oelschl Äger | Homogenization of a diffusion process in a divergence free random field[END_REF] ϕ,

ψ 1 = R + ×R x E ϕ,ψ (dx, dy) = -(ϕ, Sψ) 2 and ϕ 1 = ϕ, ϕ 1 . By virtue of Assumption (6), this semi-norm is equivalent on C to the semi-norm defined by -(ϕ, Sϕ) 2 , (14) m ϕ 2 1 ≤ -(ϕ, Sϕ) 2 ≤ M ϕ 2 1 ,
where S is the Friedrich extension of the operator defined on C by (1/2)e 2V i,j D i e -2V a ij D j . Let F (respectively H) be the Hilbert space that is the closure of C in L 2 (Ω) with respect to the inner product ε (resp. κ) defined on C by 14), Φ can be extended to the whole space ID and this extension is a linear isomorphism from ID into a closed subset of (L 2 (Ω)) d . Hence, for each function u ∈ H, we define ∇ σ u = Φ((-S) 1/2 u) and this stands, in a way, for the gradient of u along the direction σ.

ε(ϕ, ψ) = (ϕ, ψ) 2 + ϕ, ψ 1 + (D t ϕ, D t ψ) 2 (resp. κ(ϕ, ψ) = (ϕ, ψ) 2 + ϕ, ψ 1 ). Define the space ID as the closure in (L 2 (Ω), | . | 2 ) of the subspace {(-S) 1/2 ϕ; ϕ ∈ C}. For any ϕ ∈ C, define Φ (-S) 1/2 ϕ = σ * D x ϕ ∈ (L 2 (Ω)) d and note that |Φ (-S) 1/2 ϕ | 2 2 = -(ϕ, Sϕ) 2 . From (
For each f ∈ L 2 (Ω) satisfying R + ×R 1 x E f,f (dx, dy) < ∞, we define (15) f 2 -1 = R + ×R 1 x E f,f (dx, dy).
We point out that f -1 < ∞ if and only if there exists

C ∈ R such that for any ϕ ∈ C, (f , ϕ) 2 ≤ C ϕ 1 . For such a function f , f -1 also matches the smallest C satisfying this inequality. Remark that f -1 < ∞ implies π(f ) = 0. Denote by H -1 the closure of L 2 (Ω) in H * (topological dual of H) with respect to the norm -1 .
Let us now focus on the antisymmetric part H. We have ( 16)

|(u, Hv)| ≤ (u, |H|u) 1/2 (v, |H|v) 1/2 ≤ C H 1 (u, au) 1/2 (v, av) 1/2 .
The second inequality follows from ( 7) and the first one is a general fact of linear algebra. We deduce ∀ϕ,

ψ ∈ C, (1/2)(H D x ϕ, D x ψ) 2 ≤ C H 1 ψ 1 ϕ 1 .

Thus there exists an antisymmetric continuous bilinear form

T H on ID × ID such that (17) ∀ϕ, ψ ∈ C, (1/2)(H D x ϕ, D x ψ) 2 = T H (-S) 1/2 ϕ, (-S) 1/2 ψ .
Likewise, with the help of Assumption 2.3, we define the continuous bilinear forms

T a , ∂ t T a , ∂ t T H , Λ s T a , Λ s T a on ID × ID ⊂ L 2 (Ω, π) × L 2 (Ω, π) as follows: ∀ϕ, ψ ∈ C, (1/2)(aD x ϕ, D x ψ) 2 = T a (-S) 1/2 ϕ, (-S) 1/2 ψ , (1/2)(D t aD x ϕ, D x ψ) 2 = ∂ t T a (-S) 1/2 ϕ, (-S) 1/2 ψ , (1/2)(D t HD x ϕ, D x ψ) 2 = ∂ t T H (-S) 1/2 ϕ, (-S) 1/2 ψ , (1/2)(Λ s aD x ϕ, D x ψ) 2 = Λ s T a (-S) 1/2 ϕ, (-S) 1/2 ψ , (1/2)(Λ s HD x ϕ, D x ψ) 2 = Λ s T H (-S) 1/2 ϕ, (-S) 1/2 ψ ,
where, for any s ∈ R * , Λ s denotes the L 2 -continuous difference operator (remind of the definition of T s,0 in section 2):

(18) ∀f ∈ L 2 (Ω), Λ s (f ) = (T s,0 f -f )/s.
From Assumption 2.3, the norms of the forms Λ s T a and Λ s T H are uniformly bounded with respect to s ∈ R * and the forms are weakly convergent respectively towards ∂ t T a and ∂ t T H . Now, denote by H the subspace of H -1 whose elements satisfy the condition: ∃C > 0, ∀s > 0 and ∀ϕ ∈ C, h, Λ s ϕ -1,1 ≤ C ϕ 1 . For any h ∈ H, the smallest C that satisfies such a condition is denoted h T . Then H is closed for the norm H = -1 + T . Finally, let us now extend the operator L defined on C by [START_REF] Komorowski | On homogenization of time-dependent random flows[END_REF]. For any λ > 0, consider the continuous bilinear form Let us additionally mention that the adjoint operator L * of L in L 2 (Ω, π) can also be described through

B λ on H × H that coincides on C × C with ∀ϕ, ψ ∈ C, B λ (ϕ, ψ) = λ(ϕ, ψ) 2 + [T a + T H ] (-S) 1/2 ϕ, (-S)
B λ . Indeed, Dom(L * ) = {ϕ ∈ H; B λ (•, ϕ) is L 2 (Ω)-continuous.}. If ϕ ∈ Dom(L * ), there exists f ∈ L 2 (Ω) such that B λ (•, ϕ) = (f , •) 2 and L * ϕ is equal to f -λϕ.
Remark 5.2. For each function ϕ ∈ C ⊂ H, the application Lϕ can be viewed as a function of

H -1 . Indeed, ∀ψ ∈ C, (Lϕ, ψ) 2 = -[T a +T H ] (-S) 1/2 ϕ, (-S) 1/2 ψ ≤ [M +C H 1 ] ϕ 1 ψ 1 .
Hence, the application ϕ → Lϕ ∈ H -1 can be extended to the whole space H so that, for each function u ∈ H, we can define Lu as an element of H -1 even if u ∈ Dom(L).

Existence of a solution:

This section is devoted to proving existence of solutions of equation ( 12) for a suitable right-hand side. The difficulty lies in the strong degeneracy of the associated Dirichlet form. It satisfies no sector condition, even weak. However, it can be approximated by a family of Dirichlet forms with weak sector condition. For any θ ∈ {0; 1}, λ > 0 and δ ≥ 0, define B θ λ,δ as the (non-symmetric) bilinear continuous form on

F × F that coincides on C × C with (19) B θ λ,δ (ϕ, ψ) = λ(ϕ, ψ) 2 + (1/2)([a + H]D x ϕ, D x ψ) 2 -θ(D t ϕ, ψ) 2 + (δ/2)(D t ϕ, D t ψ) 2 .
In what follows, the parameter θ (resp. δ) is omitted each time that it is equal to 1 (resp. 0). So the forms B 1 λ,δ , B θ λ,0 and B 1 λ,0 are respectively simply denoted by B λ,δ , B θ λ and B λ .

Proposition 5.4. Suppose that h ∈ L 2 (Ω) ∩ Dom(D t ) and d ∈ H. Then, for any θ ∈ {0; 1} and λ > 0, there exists a unique solution u λ ∈ F of the equation

λu λ -Lu λ -θD t u λ = h + d, in the sense that ∀ϕ ∈ F, B θ λ (u λ , ϕ) = (h, ϕ) 2 + d, ϕ -1,1 . Moreover, D t u λ ∈ H and (20a) λ|u λ | 2 2 + m u λ 2 1 ≤ |h| 2 2 /λ + d| 2 -1 /m, (20b) λ|D t u λ | 2 2 +m D t u λ 2 1 ≤ |D t h| 2 2 /λ+2 d 2 T /m+2(C a 2 +C H 2 ) 2 |h| 2 /λ+ d 2 -1 /m /m 2 . In the case d ∈ L 2 (Ω), u λ ∈ Dom(L).
Finally, u λ is the strong limit in H as δ goes to 0 of the sequence (u λ,δ ) λ,δ , where u λ,δ is the unique solution of the equation:

∀ϕ ∈ F, B θ λ,δ (u λ,δ , ϕ) = (h, ϕ) 2 + d, ϕ -1,1
, and the family

(D t u λ,δ ) δ is bounded in L 2 (Ω).
Before proving this result, we first investigate the case of time independent coefficients. On the first side, this is a good starting point for understanding the proof in the time dependent case and this will bring out the difficulties arising with the time dependency. On the other side, this result is needed in the last section of this paper in order to prove the tightness of the process X. Proposition 5.5. Suppose that h ∈ L 2 (Ω) Then, for any λ > 0, there exists a unique solution w λ ∈ H ∩ Dom(S) of the equation [START_REF] Sznitman | An invariance principle for isotropic diffusions in random environment[END_REF] λw λ -Sw λ = h.

Proof :

The main tool of this proof is the Lax-Milgram theorem. Let λ > 0 be fixed. For any ϕ, ψ ∈ C, consider the bilinear form on C × C defined by

D λ (ϕ, ψ) = λ(ϕ, ψ) 2 -(ϕ, Sψ) 2 .
Thanks to Assumption 2.3, this form is clearly coercive and continuous on C × C so that it can be extended to the whole space H × H. The extension is also coercive and continuous. Now, the application ϕ → (h, ϕ) Proof of the Proposition 5.4: Since the case θ = 0 and θ = 1 are quite similar, we only give the proof for θ = 1. The existence of a solution relies on the Lax-Milgram theorem again. However, the considered bilinear form [START_REF] Rudin | Analyse fonctionnelle[END_REF] with δ = 0 is not coercive on F because of the time differential term (D t ϕ, ψ). The strategy consists in making it coercive by adding a term (δ/2)(D t ϕ, D t ψ) (δ > 0) and then letting δ go to 0. Notice that for ϕ, ψ ∈ C, we have

[λ -L -D t -(δ/2)D 2 t ]ϕ, ψ 2 = B λ,δ (ϕ, ψ).
The continuity of B λ,δ on C × C ⊂ F × F follows from ( 6) and ( 16). As a result of the timeindependence of V , for any ϕ ∈ C, we have (ϕ, D t ϕ) 2 = 0. As a consequence, for any ϕ ∈ C,

(λ, δ/2, m)ε(ϕ, ϕ) ≤ B λ,δ (ϕ, ϕ). (22) min
Hence B λ,δ defines a continuous coercive bilinear form on F×F. The Lax-Milgram theorem applies and provides us with a solution u λ,δ of the equation

(23) ∀ϕ ∈ C, B λ,δ (u λ,δ , ϕ) = (h, ϕ) 2 + d, ϕ -1,1 .
In particular, choosing ϕ = u λ,δ in (23), we get the bound

(24) λ|u λ,δ | 2 2 + m u λ,δ 2 1 + δ|D t u λ,δ | 2 2 ≤ |h| 2 2 /λ + d 2 -1 /m.
Let us now to pass to the limit as δ goes to 0 to obtain a solution u λ ∈ F of the equation

(25) ∀ϕ ∈ C, B λ (u λ , ϕ) = (h, ϕ) 2 + d, ϕ -1,1 .
We are faced with the problem of controlling D t u λ,δ as δ goes to 0. The idea lies in differentiating equation (23) with respect to the time variable in order to establish an equation satisfied by D t u λ,δ , from which estimates will be derived. So, we define for each fixed λ, δ > 0, v s = Λ s u λ,δ (the parameters λ, δ of v s are temporarily omitted in order to simplify the notations) and we easily check that v s solves the following equation

(26) ∀ϕ ∈ F, B λ,δ (v s , ϕ) = F s (ϕ),
where F s is a continuous linear form on F defined, ∀ϕ ∈ F, by

(27) F s (ϕ) = (Λ s h, ϕ) 2 -d, Λ -s ϕ -1,1 -[Λ s T a + Λ s T H ] (-S) 1/2 T s,0 u λ,δ , (-S) 1/2 ϕ .
From Assumption 2.3, it is readily seen that

F s (ϕ) ≤ |D t h| 2 |ϕ| 2 + d T ϕ 1 + (C a 2 + C H 2 ) u λ,δ 1 ϕ 1 , for any s ∈ R * . Therefore (28) B λ,δ (v s , v s ) = F s (v s ) ≤ |D t h| 2 |v s | 2 + d T ϕ 1 + (C a 2 + C H 2 ) u λ,δ 1 v s 1 .
Using estimate (24) in (28), we have (29)

λ|v s | 2 2 + m v s 2 1 + δ|D t v s | 2 2 ≤ |D t h| 2 2 /λ + 2 d 2 T /m + 2(C a 2 + C H 2 ) 2 |h| 2 /λ + d 2 -1 /m /m 2 .
So, the family (v s ) s∈R * is bounded in F. Even if it means extracting a subsequence (still denoted by (v s ) s∈R * ), (v s ) s∈R * converges weakly in F towards some function v 0 ∈ F as s goes to 0. On the other hand, since u λ,δ ∈ F ⊂ Dom(D t ), (v s ) s∈R * also converges strongly in L 2 (Ω) towards D t u λ,δ , so that D t u λ,δ ∈ F and satisfies bound (29) instead of v s . In particular, (D t u λ,δ ) δ>0 is bounded in H independently of δ > 0 and so is (u λ,δ ) δ>0 in F. Thereby, there exists a subsequence

(u λ,δ , D t u λ,δ ) δ>0 ⊂ F × H, still indexed with δ > 0, that converges weakly in F × H towards (u λ , D t u λ ) ∈ F × H as δ → 0.
In particular, δD t u λ,δ → 0 in L 2 (Ω) as δ goes to 0. So we are in position to pass to the limit as δ goes to 0 in (23). Obviously, u λ is a solution of (25). Uniqueness of the weak limit raises no particular difficulty since two weak limits u λ and w λ satisfy ∀ϕ ∈ F, B λ (u λw λ , ϕ) = 0. It just remains to choose ϕ = u λw λ . (20a) and (20b) respectively result from ( 24) and (29

). If d ∈ L 2 (Ω), note that u λ ∈ F ⊂ H and B λ (u λ , •) = (h + d + D t u λ , •) 2 is L 2 -continuous so that u λ ∈ Dom(L).
Let us now investigate the strong convergence in F of (u λ,δ ) λ,δ towards u λ as δ goes to 0. Let us make the difference between ( 23) and ( 25) and choose ϕ = u λ,δu λ , this yields

B λ,δ (u λ,δ -u λ , u λ,δ -u λ ) = (δ/2)(D t u λ , D t u λ -D t u λ,δ ) 2 ,
and this latter quantity converges to 0 as δ goes to 0 because of the boundedness of the family (|D t u λ,δ | 2 ) λ,δ . ( 22) allows to conclude.

Control of the solution

Our goal is now to determine the asymptotic behaviour, as λ goes to 0, of the solution u i λ of the equation (in the sense of Proposition 5.4)

(30) λu i λ -Lu i λ -D t u i λ = b i .
More precisely, we aim at proving that λ|u i λ | 2 2 → 0 and that (∇ σ u i λ ) λ converges in (L 2 (Ω)) d as λ goes to 0. Our strategy consists in showing that the operator λ -L -D t is just a perturbation of the operator λ -S -D t , so that the study can be reduced to studying the solution of the equation

λv λ -Sv λ -D t v λ = b λ ,
where b λ will be defined thereafter but possesses a strong limit in H -1 . This latter equation is more convenient to study because the operators S and D t can be viewed through the same spectral decomposition. Thus, the purpose of this section is to prove the following Proposition Proposition 5.7. Let (b λ ) λ>0 be a family of functions in H -1 ∩ L 2 (Ω) which is strongly convergent in H -1 to b 0 . Suppose that there exists a constant C (which does not depend on λ) such that ∀s > 0 and ∀ϕ ∈ C,

(b λ , Λ s ϕ) 2 ≤ C ϕ 1 .
Then the solution u λ ∈ F of the equation λu λ -Lu λ -D t u λ = b λ (in the sense of Proposition 5.4) satisfies:

• there exists η ∈ ID such that (-S) 1/2 u λ → η as λ goes to 0 in ID,

• λ|u λ | 2 2 → 0 as λ goes to 0.

As for the existence of the solution, let us first investigate the time independent case by way of introduction.

Proposition 5.8. Let h be in H -1 ∩ L 2 (Ω). For any λ > 0, let w λ be defined as the unique solution in H of the equation

λw λ -Sw λ = h
Then λ|w λ | 2 2 → 0 and there exists ζ ∈ (L 2 (Ω)) d such that |∇ σ w λ -ζ| 2 → 0 as λ goes to 0.

Proof : Keeping the notations of Proposition 5.5, w λ solves the equation: ∀ϕ ∈ H, D λ (w λ , ϕ) = (h, ϕ) 2 . Choosing ϕ = w λ and using h ∈ H -1 , we have

λ|w λ | 2 2 + m w λ 2 1 ≤ h 2 -1 /m.
Thus, even if it means extracting a subsequence, we can find g ∈ L 2 (Ω) such that ((-S) 1/2 w λ ) λ converges weakly in L 2 (Ω) towards g as λ tends to 0. Moreover (λw λ ) λ clearly converges to 0 in L 2 (Ω). For any ϕ ∈ H, passing to the limit as λ goes to zero in the expression

(31) λ(w λ , ϕ) 2 + T a (-S) 1/2 w λ , (-S 1/2 ϕ) 2 = D λ (w λ , ϕ) = (h, ϕ) 2 ,
we obtain T a g, (-S) 1/2 ϕ 2 = (h, ϕ) 2 . Making the difference between the last two equalities, subtracting T a (-S) 1/2 w λg, g and then choosing (-S) 1/2 ϕ = (-S) 1/2 w λg, we obtain

λ|w λ | 2 2 + T a (-S) 1/2 w λ -g, (-S) 1/2 w λ -g = -T a (-S) 1/2 w λ -g, g .
Due to the weak convergence of ((-S) 1/2 w λ ) λ to g in ID, the right-hand side converges to 0 as λ goes to 0. So does the left-hand side. Since T a defines an inner product on ID equivalent to the canonical one (Assumption 2.3), this completes the proof of the strong convergence up to a subsequence. Uniqueness of the weak limit is clear since two weak limits g and g ′ ∈ ID satisfy: ∀ϕ ∈ C, T a (gg ′ , (-S) 1/2 ϕ) = 0. Finally, since the convergence in ID of ((-S) 1/2 w λ ) λ is equivalent to the convergence of (∇ σ w λ ) λ in (L 2 (Ω)) d , we complete the proof.

Proposition 5.9. Let (b λ ) λ>0 be a family of functions in H -1 that is strongly convergent to b 0 in H -1 . Let (v λ ) λ>0 be a family of functions in F that solves the equation (for any λ > 0)

λv λ - Sv λ -D t v λ = b λ in the following sense, (32) ∀ϕ ∈ F, λ(v λ , ϕ) 2 + v λ , ϕ 1 -(D t v λ , ϕ) 2 = (b λ , ϕ) 2 .
Then there exists η ∈ ID such that λ|v λ | 2 2 → 0 and |(-S) 1/2 v λ -η| 2 → 0 as λ goes to 0.

Proof: From Lemma 5.10 and Lemma 5.11 below, we can assume that, for any λ > 0, b λ ∈ L 2 (Ω) ∩ Dom(D t ) ∩ H -1 and converges to b 0 ∈ H -1 . Then v λ ∈ Dom( S) (see Proposition 5.4).

Remind that -S = R + ×R x E(dx, dy) and -D t = R + ×R iy E(dx, dy). Choosing ϕ = v λ in (32), we have

(33) λ|v λ | 2 2 + v λ 2 1 = (b λ , v λ ) 2 ≤ C v λ 1 ≤ C 2 ,
where C = sup λ>0 b λ -1 . Thus we can find h ∈ ID and a subsequence, still denoted by (v λ ) λ , such that (-S) 1/2 v λ λ converges weakly in L 2 (Ω) to h.

Now we claim sup

λ>0 λv λ -1 < ∞ and sup λ>0 D t v λ -1 < ∞. |(λv λ , ϕ) 2 | = R + ×R λ(λ + x + iy) -1 dE b λ ,ϕ ≤ R + ×R λ 2 x[(λ + x) 2 + y 2 ] dE b λ ,b λ 1/2 R + ×R x dE ϕ,ϕ 1/2 ≤ sup λ>0 R + ×R x -1 dE b λ ,b λ 1/2 ϕ 1 = sup λ>0 b λ -1 ϕ 1 . Since D t v λ = λv λ -Sv λ -b λ and Sv λ -1 ≤ v λ 1 , D t v λ ∈ H -1 and sup λ>0 D t v λ -1 < ∞.
Then there exists a bounded family (F λ ) λ≥0 of continuous linear forms on

ID ⊂ L 2 (Ω) such that ∀λ > 0, ∀ϕ ∈ C, F λ ((-S) 1/2 ϕ) = (D t v λ , ϕ) 2 . Moreover, from (33), (λv λ ) λ converges to 0 in L 2 (Ω) so that, ∀ϕ ∈ C F λ ((-S) 1/2 ϕ) = (λv λ , ϕ) 2 + ((-S) 1/2 v λ , (-S) 1/2 ϕ) 2 -b λ , ϕ -1,1 → (h, (-S) 1/2 ϕ) 2 -b 0 , ϕ -1,1
as λ goes to 0. Hence, (F λ ) λ≥0 is weakly convergent in ID * (topological dual of ID) to a limit denoted by F 0 .

We now aim at proving F 0 (h) = 0. Using the antisymmetry of the operator D t

F λ ((-S) 1/2 v µ ) = (D t v λ , v µ ) 2 = -(D t v µ v λ ) 2 = -F µ ((-S) 1/2 v λ ),
we pass to the limit as λ goes to 0 and obtain F 0 ((-S) 1/2 v µ ) = -F µ (h). It just remains to pass to the limit as µ goes to 0, it yields F 0 (h) = -F 0 (h) = 0. Let us investigate now the limit equation, which connects F 0 , h and b 0 . First remind of (33), which states λ|v λ | 2 2 ≤ C 2 and as a consequence λv λ → 0 as λ goes to 0. Then, we are in a position to pass to the limit as λ tends to 0 in (32), and this yields, for any ϕ ∈ F,

(34) (h, (-S) 1/2 ϕ) 2 -F 0 ((-S) 1/2 ϕ) = b 0 , ϕ -1,1 .
Let us now establish the uniqueness of the weak limit. Let h and h ′ be two possible weak limits of two subsequences of (v λ ) λ , and F 0 ,F ′ 0 the corresponding linear forms defined as described above. Then (34) provides us with he following equality:

(35) ∀ϕ ∈ F, (h -h ′ , (-S) 1/2 ϕ) = [F 0 -F ′ 0 ]((-S) 1/2 ϕ).
Using the antisymmetry of the operator D t again, we obtain

F λ ((-S) 1/2 v µ ) = (D t v λ , v µ ) 2 = -(D t v µ , v λ ) 2 = -F µ ((-S) 1/2 v λ ).
Let us first pass to the limit as λ goes to 0 along the first subsequence, and then pass to the limit as µ goes to 0 along the second subsequence, we obtain

F 0 (h ′ ) = -F ′ 0 (h).
Now, it just remains to choose (-S) 1/2 ϕ = hh ′ in (35) and this yields

|h -h ′ | 2 2 = -F 0 (h ′ ) -F ′ 0 (h) = 0.
Hence the weak convergence holds for the whole family. Let us now tackle the strong convergence of (v λ ) λ . Choosing ϕ = v λ in (34), using F 0 (h) = 0 and passing to the limit a λ goes to 0, this yields

(36) (h, h) 2 = lim λ→0 b 0 , v λ -1,1 = lim λ→0 b λ , v λ -1,1 = lim λ→0 λ|v λ | 2 2 + v λ 2 1 .
In particular,

|h| 2 = lim λ→0 |(-S) 1/2 v λ | 2 .
Thus, the convergence of the norms implies the strong convergence of the sequence ((-S)

1/2 v λ ) λ to h in L 2 (Ω).
(λw λ , ϕ) 2 = R + ×R λ(λ + x) -1 dE b,ϕ (dx, dy) ≤ R + ×R λ 2 x -1 (λ + x) -2 dE b,b (dx, dy) 1/2 ϕ 1 . Since b ∈ H -1 , we have R + ×R x -1 dE b,b (dx, dy) < ∞.
Thus the Lebesgue theorem ensures that the above integral converges to 0 as λ goes to 0. Hence, λw λ -1 converges to 0 as λ goes to 0. We can now choose a family (ϕ λ ) λ in C such that w λϕ λ 1 → 0 as λ goes to 0. Finally,

b -Sϕ λ -1 ≤ b -Sw λ -1 + Sw λ -Sϕ λ -1 ≤ λw λ -1 + w λ -ϕ λ 1
also converges to 0 as λ tends to 0 and, clearly, Sϕ λ ∈ L 2 (Ω) ∩ Dom(D t ).

λ -v ′ λ | 2 2 + v λ -v ′ λ 2
1 → 0 as λ goes to 0. Proof: Making the difference between the two equations corresponding to v λ and v ′ λ , this yields for any ϕ ∈ F,

λ(v λ -v ′ λ , ϕ) 2 + v λ -v ′ λ , ϕ 1 -(D t v λ -D t v ′ λ , ϕ) 2 = b λ -b ′ λ , ϕ -1,1 . Choosing ϕ = v λ -v ′ λ , we easily deduce λ|v λ -v ′ λ | 2 2 + v λ -v ′ λ 1 ≤ b λ -b ′ λ -1 . The result follows.
Let us now investigate the general case, that means that we aim at replacing S by L in Proposition 5.9. We first set out the main ideas of the proof. Let us formally write

λ -L -D t = λ -S -D t -(L -S) = I -L -S (λ -S -D t ) -1 (λ -S -D t )
If we can prove that L-S (λ-S -D t ) -1 defines a strictly contractive operator, then we will be in position to inverse it. It turns out that it is actually bounded but not strictly contractive. To overcome this difficulty, we introduce a small parameter δ to make the operator δ L -S (λ -S -D t ) -1 strictly contractive. Then, an iteration procedure proves that δ can be chosen equal to 1. Proposition 5.12. Let (b λ ) λ>0 be a family of functions in H -1 that is strongly convergent in H -1 to some b 0 ∈ H -1 and bounded in H. Then there exists δ 0 > 0 such that, for any 0 ≤ δ ≤ δ 0 , for any λ > 0, the solution (in the sense of Proposition 5.4)

u λ ∈ F (with D t u λ ∈ H) of the equation λu λ -δLu λ -(1 -δ) Su λ -D t u λ = b λ , satisfies: ∃η ∈ L 2 (Ω) such that λ|u λ | 2 2 + |(-S)
> 0 such that δ 0 P λ H→H < 1 (actually δ 0 < 2(2 + M + C H 1 )(1 + C a 2 + C H 2 ) -1
). For 0 < δ < δ 0 , we can then define the operator

[I -δP λ ] -1 : H -→ H. Note that (λ -δL -(1 -δ) S -D t ) -1 = (λ -S -D t ) -1 I -δP λ -1
. Thanks to Proposition 5.9, it is sufficient to prove that

I -δP λ -1 (b λ ) is convergent in H -1 . But I -δP λ -1 (b λ ) = ∞ n=0 (δP λ ) n (b λ ). Lemma 5.
13 ensures that the sum converges uniformly with respect to λ > 0. It just remains to prove that, for each fixed n ≥ 0, ((δP λ ) n (b λ )) λ converges in H -1 . This can be proved by induction on n ∈ N. For n = 0, (b λ ) λ>0 is convergent by assumption. Then, if the family ((δP λ ) n (b λ )) λ is convergent in H -1 , we can apply Proposition 5.9 to ensure that the family

(-S) 1/2 (λ -S -D t ) -1 [(δP λ ) n (b λ )] λ converges in L 2 (Ω). This implies the convergence of ((δP λ ) n+1 (b λ )) λ in H -1 .
∀ϕ ∈ F, λ(u λ , ϕ) 2 + u λ , ϕ 1 -(D t u λ , ϕ) 2 = b, ϕ -1,1 . It derives from (20a) that λ|u λ | 2 2 + u λ 2 1 ≤ b 2 -1
, in such a way that

P λ (b) -1 = (L -S)u λ -1 ≤ (1 + M + C H 1 ) u λ 1 ≤ (1 + M + C H 1 ) b -1 .
This proves the first point. Consider now u ∈ F with D t u ∈ H. An easy computation proves that, for any s ∈ R * and ϕ ∈ C,

T a (-S) 1/2 u, (-S) 1/2 Λ s ϕ = -Λ -s T a (-S) 1/2 u, (-S) 1/2 ϕ -T a (-S) 1/2 Λ s u, (-S) 1/2 T s,0 ϕ ≤ C a 2 u 1 ϕ 1 + M D t u 1 ϕ 1 . (37)
In the above inequalities, we use u 1 = T s,0 u 1 and Λ s u 1 ≤ D t u 1 . This latter point can be proved for u ∈ C as follows

Λ s u 2 1 = -(Λ s u, SΛ s u) 2 = - 1 0 1 0 (D t T r,0 u, SD t T u,0 u) 2 dr du ≤ -(D t u, SD t u) 2 .
The general case is treated by density arguments.

As in (37), we have

T H (-S) 1/2 u, (-S) 1/2 Λ s ϕ ≤ C H 2 u 1 ϕ 1 + C H 1 D t u 1 ϕ 1 . Hence, (L -S)(u) T ≤ (C H 2 + C a 2 ) u 1 + (C H 1 + M + 1) D t u 1 .
Then, Proposition 5.4 ensures that D t u λ ∈ H and

D t u λ 1 ≤ 2 b T + 2(C H 2 + C a 2 ) b -1 (see ( 20b
)) so that we finally obtain

P λ (b) T ≤ (C H 2 + C a 2 ) b -1 + 2(C H 1 + M + 1) b T + (C H 2 + C a 2 ) b -1 . (38)
The result follows. Proof of Proposition 5.7: The last step before proving Proposition 5.7 consists in lifting the restriction of the smallness of δ 0 . The previous construction provides us with δ 0 strictly less than 1.

We perform an induction to get round this restriction whose initialization is the construction of δ 0 . The second step consists in iterating our arguments to the operator

λ -(δ 0 + δ 1 )L -(1 -δ 0 -δ 1 ) S -D t = I -δ 1 (L -S)[λ -δ 0 L -(1 -δ 0 ) S -D t ] -1 (λ -δ 0 L -(1 -δ 0 ) S -D t ).
We exactly repeat the proof of Proposition 5.12 except that the operator λ -(1 -δ 0 -δ 1 ) S -(δ 0 + δ 1 )L-D t plays the role of the operator λ-(1-δ 0 ) S-δ 0 L-D t and we apply Proposition 5.12 with the operator λ-(1-δ 1 ) S-δ 1 L-D t instead of applying Proposition 5.9 with λ-S-D t . Of course, a restriction about the smallness of δ 1 is imposed by this procedure. Even if it means substituting a with m a, we assume, without loss of generality, that m = 1. Thus Lemma 5.13 remains valid for the operator P 1 λ : H → H defined by

P 1 λ (b) = (L -S)(λ -(1 -δ 0 ) S -δ 0 L -D t ) -1 (b)
. This is of the utmost importance because that means that we can choose δ 1 = δ 0 . Thus we can iterate these arguments until we find δ N such that δ 0 + δ 1 + • • • + δ N > 1 and such that Proposition 5.12 still holds except that δ < δ 0 is everywhere replaced by δ < δ 0 + δ 1 + • • • + δ N . Proposition 5.7 follows.

Now let us prove that the drift b of the diffusion process X fulfills the assumptions of Proposition 5.7. To this purpose, let us establish Lemma 5.14. For each i ∈ {1, . . . , d}, b i belongs to

H -1 and ∀s ∈ R, ∀ϕ ∈ C, b i , Λ s ϕ -1,1 ≤ (C a 2 + C H 2 )|( aE i , E i ) 2 | 1/2 ϕ 1 .
Proof: Let (E 1 , . . . , E d ) be the canonical basis of R d . Then we have

(b i , ϕ) 2 = 1/2 j e 2V D j (e -2V [a + H] ij ), ϕ 2 = -1/2 [a -H]Dϕ, E i 2 ≤ 1/2 aDϕ, E i 2 + 1/2 HDϕ, E i 2 Cauchy-Schwarz ≤ M ϕ 1 |( aE i , E i ) 2 | 1/2 + C H 1 ϕ 1 |( aE i , E i ) 2 | 1/2
and this proves the first point. Then, ∀s > 0, ∀ϕ ∈ C, we have

b i , Λ s ϕ -1,1 = -(1/2) [a + H]E i , Λ s Dϕ 2 = (1/2) Λ -s [a + H]E i , Dϕ 2 Assumption 2.3 ≤ (C a 2 + C H 2 )|( aE i , E i ) 2 | 1/2 ϕ 1 6

Itô's formula

We are not in a lucky situation of working on an explicit Dirichlet form connected with the generator in L 2 (Ω, π) of Y , wrongly denoted by [L + D t ]. This raises the following issue: given a function f ∈ L 2 (Ω) and the function u λ that weakly solves (see Proposition 5.4) λu λ -(L + D t )u λ = f , does the "Ito formula" apply to u λ and to the process Y . Indeed, it is not clear that the construction of u λ in Proposition 5.4 belongs to the domain of the generator of Y . The key tool is the regular approximation (u λ,δ ) δ provided by Proposition 5.4 for a suitable function f . Let us consider a standard 1-dimensional Brownian motion {B ′ t ; t ≥ 0} that is independent of {B t ; t ≥ 0} in such a way that {(B ′ t , B t ); t ≥ 0} is a standard d + 1-dimensional Brownian motion. Define then the d + 1-dimensional diffusion process X ω,δ , starting from 0, as the solution of the SDE:

(39) X ω,δ t = t 0 1 b(X ω,δ r , ω) dr + t 0 √ δ 0 0 σ(X ω,δ r , ω) d(B ′ , B) r .
The associated diffusion in random medium Y δ defined by

Y δ t (ω) = τ X ω,δ t
ω is a Ω-valued Markov process, which admits π as invariant measure (similar to section 4). It also defines a continuous semi-group on L 2 (Ω). The associated (non-symmetric) Dirichlet form is given by ( 19) (with θ = 1) with domain F × F and satisfies a weak sector condition (see [12, Ch. 1, Sect 2.] for the definition). The generator L δ is defined on

Dom(L δ ) = {u ∈ F; B λ,δ (u, •) is L 2 (Ω)-continuous} (see [12, Ch. 1, Sect 2.] for further details). It coincides on C with L + D t + (δ/2)D 2
t . Since b and σ are globally Lipschitz (Assumption 2.2), classical tools of SDE theory ensures that (40)

Ω IE sup 0≤t≤T |(t, X ω t ) -X ω,δ t | 2 dπ → 0 as δ goes to 0,
where both diffusions start from 0. Proposition 6.1. Let f ∈ L 2 (Ω) and a family (u λ ) λ>0 in F such that:

1) ∀ϕ ∈ F, B λ (u λ , ϕ) = (f , ϕ) 2 ,
2) for each λ > 0, there exists a sequence (u λ,δ ) δ>0 in F that converges in H towards u λ . Moreover (u λ,δ ) δ>0 ∈ Dom(L δ ) and satisfies λu λ,δ -L δ u λ,δ = f . 3) for each fixed λ > 0, (D t u λ,δ ) δ is bounded in L 2 (Ω). 4) each function u λ,δ has continuous trajectories, that is, for µ almost every ω ∈ Ω, the function

(t, x) ∈ R d+1 → u λ,δ (τ t,x ω) is continuous.
Then, IP π a.s., the following formula holds

u λ (Y t ) = u λ (Y 0 ) + t 0 (λu λ -f )(Y r ) dr + t 0 ∇ σ u * λ (Y r ) dB r
where IP π is the law of the process Y starting with initial distribution π on Ω.

Proof: Since u λ,δ ∈ Dom(L δ ) and λu λ,δ -L δ u λ,δ = f , we can write (see Lemma 6.2 below)

u λ,δ (Y δ t ) -u λ,δ (Y δ 0 ) = t 0 L δ u λ,δ (Y δ r ) dr + δ 1/2 t 0 D t u λ,δ (Y δ r ) dB ′ r + t 0 ∇ σ u * λ,δ (Y δ r ) dB r = t 0 [λu λ,δ -f ](Y δ r ) dr + δ 1/2 t 0 D t u λ,δ (Y δ r ) dB ′ r + t 0 ∇ σ u * λ,δ (Y δ r ) dB r . (41)
Thanks to (40), the convergence, as δ → 0, of (u λ,δ ) λ,δ towards u λ in H and the boundedness of (D t u λ,δ ) δ in L 2 (Ω), we can pass to the limit in (41) and complete the proof. Lemma 6.2. Keeping the notations of Proposition 6.1, the following formula holds, IP π a.s.,

u λ,δ (Y δ t ) -u λ,δ (Y δ 0 ) = t 0 L δ u λ,δ (Y δ r ) dr + δ 1/2 t 0 D t u λ,δ (Y δ r ) dB ′ r + t 0 ∇ σ u * λ,δ (Y δ r ) dB r . Proof: Since u λ,δ ∈ Dom(L δ ) , the difference u λ,δ (Y δ t ) -u λ,δ (Y δ 0 ) - t 0 L δ u λ,δ (Y δ r ) dr is a square-integrable continuous IP π -martingale, denoted by M δ t . Moreover, for a function ϕ ∈ C, the classical Ito formula yields ϕ(Y δ t ) -ϕ(Y δ 0 ) = t 0 L δ ϕ(Y δ r ) dr + δ 1/2 t 0 D t ϕ(Y δ r ) dB ′ r + t 0 ∇ σ ϕ * (Y δ r ) dB r . Then the process t → u λ,δ (Y δ t ) -ϕ(Y δ t
) is a continuous semimartingale and Theorem 32 in [18, Ch. 2, Sect. 7] (applied with the function x ∈ R → x 2 ) yields IP π a.s.,

(u λ,δ (Y δ t )-ϕ(Y δ t )) 2 = (u λ,δ (Y δ t ) -ϕ(Y δ 0 )) 2 + 2 t 0 (u λ,δ -ϕ)L δ (u λ,δ -ϕ)(Y δ r ) dr + 2 t 0 (u λ,δ -ϕ)(Y δ r ) dM δ r -δ 1/2 D t ϕ(Y δ r ) dB ′ r -∇ σ ϕ * (Y δ r ) dB r + 2 M - • 0 δ 1/2 D t ϕ(Y δ r ) dB ′ r - • 0 ∇ σ ϕ * (Y δ r ) dB r t , (42)
where [X] stands for the quadratic variations of the martingale X. Integrating with respect to the measure π, the martingale term vanishes and we deduce

(43) IE π 2 M - • 0 δ 1/2 D t ϕ(Y δ r ) dB ′ r - • 0 ∇ σ ϕ * (Y δ r ) dB r t ≤ 2B λ,δ (u λ,δ -ϕ, u λ,δ -ϕ).
Choosing a sequence (ϕ n ) n in C that converges in F towards u λ,δ , we easily complete the proof with the help of (43). Note that the time reversed process t → Y δ T -t is a Markov process with respect to the backward filtration (G δ t ) 0≤t≤T , where G δ s is the σ-algebra on Ω generated by Y δ r ; t ≤ r ≤ T , and admits the adjoint operator (L δ ) * of L δ in L 2 (Ω, π) as generator, which coincides on C with L * -D t + (δ/2)D 2 t . From (40), t → Y δ T -t approximates the process t → Y T -t as δ tends to 0. It is then readily seen that we can slightly modify the proof of Proposition 6.1 and prove the Proposition 6.3. Let f ∈ L 2 (Ω) and a family (u λ ) λ>0 in F such that:

1) ∀ϕ ∈ F, B λ (ϕ, u λ ) = (f , ϕ) 2 ,
2) for each λ > 0, there exists a sequence (u λ,δ ) δ>0 in F that converges in H towards u λ . Moreover (u λ,δ ) δ>0 ∈ Dom(L δ ) * and satisfies λu λ,δ -(L δ ) * u λ,δ = f . 3) for each fixed λ > 0, (D t u λ,δ ) δ is bounded in L 2 (Ω). 4) each function u λ,δ has continuous trajectories, that is, for µ almost every ω ∈ Ω, the function (t, x) ∈ R d+1 → u λ,δ (τ t,x ω) is continuous. Then, IP π a.s., the following formula holds

u λ (Y T -t) = u λ (Y T ) + t 0 (λu λ -f )(Y T -r ) dr + (M t -M 0 )
where M is a martingale with respect to the backward filtration (G t ) 0≤t≤T , and G s is the σalgebra on Ω generated by {Y r ; t ≤ r ≤ T }. Moreover, the quadratic variations of M exactly match

t 0 ∇ σ u * λ • ∇ σ u λ (Y T -r ) dr.

Ergodic Theorem

Let us now exploit the ergodic properties of the operator S stated in Assumption 2.4 and prove Theorem 7.1. Let f ∈ L 1 (Ω). Then

IE π 1 t t 0 f (Y r ) dr -π(f ) → 0 as t goes to ∞.
Proof: We suppose at first that f ∈ C. Even if it means considering f -π(f ) instead of f , we assume that π(f ) = 0. Clearly, f ∈ Dom(D t ) and Proposition 5.4 applies. For each λ > 0, it provides us with a function u λ ∈ F such that

(44) ∀ϕ ∈ F, B λ (u λ , ϕ) = (f , ϕ) 2 .
Moreover, (20a) and (20b) ensures that the families (λu λ ) λ , (λD t u λ ) λ and (λ 1/2 (-S)

λ(f , ϕ) 2 = λB λ (u λ , ϕ) = λ 2 (u λ , ϕ) 2 -(λu λ , L * ϕ) 2 -(λD t u λ , ϕ) 2 .
Passing to the limit as λ goes to 0, we deduce (g, L * ϕ) 2 = -(D t g, ϕ) 2 . Hence g ∈ Dom(L * * ) = Dom(L) ⊂ H and Lg = -D t g. In particular m g 2 1 ≤ -(g, Lg) 2 = (D t g, g) 2 = 0 so that g ∈ Dom((-S) 1/2 ) and (-S) 1/2 g = 0. As a consequence, g ∈ Dom( S) and Sg = 0. From Assumption 2.4, g is invariant under space translations in such a way that D t g = -Lg = 0 and g is also invariant under time translations. Thus the ergodicity of the measure µ implies that g is constant (µ a.s.). Choosing ϕ equal to the constant function 1 in (44), we deduce g = 0. We now aim at proving that the convergence of (λu λ ) λ towards 0 holds in the strong sense. In what follows, we make no distinction between 0 ∈ R and the constant function that matches 0 over Ω. We just have to write

0 = (0, f ) 2 = lim λ→0 (λu λ , f ) 2 = lim λ→0 B λ (λu λ , λu λ ) 2 ≥ lim sup λ→0 |λu λ | 2 2 .
Note now that the approximating family (u λ,δ ) δ provided by Proposition 5.4 is given by u λ,δ (ω) = ∞ 0 e -λr IE 0 [f (X ω,δ r , ω)] dr. For each (t, x) ∈ R d+1 , the law of the process (t, x) + X τt,xω,δ , X τt,xω,δ starting from 0 ∈ R d+1 , is the same as the law of the process X ω,δ starting from (t, x) ∈ R d+1 (see the proof at the end of Section 8). Hence u λ,δ (τ t,x ω) = ∞ 0 e -λr IE t,x [f (X ω,δ r , ω)] dr. Since f is smooth and X ω,δ is a Feller process, u λ,δ has continuous trajectories. Thus Proposition 6.1 applies and it yields

t 0 f (Y r ) dr = (u λ (Y δ 0 ) -u λ (Y t )) + t 0 λu λ (Y r ) dr + t 0 ∇ σ u * λ (Y r ) dB r .
Thanks to (20a) and the invariance of the measure π for the process Y , we can find a constant C, which depends neither on λ nor on t, such that

IE π 1 t t 0 f (Y r ) dr 2 ≤ C/(tλ) 2 + C|λu λ | 2 2 + C/(tλ 1/2 ).
It just remains to choose λ small enough and then t large enough to complete the proof in the case f ∈ C. The general case is treated with the density of C in L 1 (Ω) and the invariance of the measure π. Since it raises no particular difficulty, details are left to the reader.

Invariance principle

Notation :

Up to the end of this paper, for i ∈ {1, . . . , d} we denote by u i λ the solution of the equation (in the weak sense of Proposition 5.4)

λu i λ -Lu i λ -D t u i λ = b i .
From Proposition 5.7, there exists

ξ i ∈ (L(Ω)) d such that λ|u i λ | 2 2 + |ξ i -∇ σ u i λ | 2 → 0 as λ goes to 0.
Applying the Ito formula (see Proposition 6.1) to the function u ε 2 yields

εX ω t/ε 2 = H ε,ω t + ε t/ε 2 0 (σ + ∇ σ u * λ )(r, X ω r , ω) dB r ,
where

H ε,ω t = ε 3 t/ε 2 0 u ε 2 (r, X ω r , ω) dr -εu ε 2 (t/ε 2 , X ω t/ε 2 , ω) + εu ε 2 (0, 0, ω).
For the reader's convenience, it is worth recalling that Y t = τ t,X ω t and IP π is the law of the process Y with initial distribution π. We want to show that the finite dimensional distributions of the process H ε,ω converges in IP π -probability to 0. Using the Cauchy-Scharz inequality and the invariance of the measure π, we get the estimate

IE π [(H ε,ω t ) 2 ] ≤ 3(2 + t 2 )ε 2 |u ε 2 | 2 2
and this latter quantity converges to 0 as ε goes to 0.

Let us now investigate the convergence of the process t → ε t/ε 2 0 (σ + ∇ σ u * λ )(Y r ) dB r whose quadratic variations are given by

ε 2 t/ε 2 0 (σ + ∇ σ u * ε 2 )(σ + ∇ σ u * ε 2 ) * (Y r ) dr = ε 2 t/ε 2 0 (σ + ξ * )(σ + ξ * ) * (Y r ) dr + ε 2 t/ε 2 0 (σ + ∇ σ u * ε 2 )(σ + ∇ σ u * ε 2 ) * (Y r ) dr -ε 2 t/ε 2 0 (σ + ξ * )(σ + ξ * ) * (Y r ) dr .
With the help of Theorem 7.1, the finite dimensional distributions of the former term in the righthand side converge in L 1 (IP π ) to the ones of the process t → At, where the matrix A is given by

(45) A = Ω (σ + ξ * )(σ + ξ * ) * dπ.
The finite dimensional distributions of the latter term in the right-hand side converge in L 1 (IP π ) to 0. Indeed, after integrating with respect to the probability measure IP π , it is bounded by Ct|∇ σ u ε 2 -ξ| 2 2 . Hence we conclude by applying the central limit theorem for martingales that the finite dimensional distributions of the process εX ω t/ε 2 converge in law to the ones of the process A 1/2 B t . Proposition 8.1. The process εX ω t/ε 2 is tight in the space C([0, T ]; R d ). Hence it converges in law in the space C([0, T ]; R d ) towards the process A 1/2 B t .

Proof :

The next section is devoted to the proof of the tightness Let us now to determine the limit when the starting point is not 0 but x/ε.

IE x/ε f (εX ω t/ε 2 ) = IE 0 f (x + εX τ (0,x/ε) ω t/ε 2 ) in law w.r.t. µ = IE 0 f (x + εX ω t/ε 2 ) π prob ---→ ε→0 IE f (x + A 1/2 B t )
For the first above equality we used the following fact. If

X t = x + t 0 b (r, X r , ω) dr + t 0 σ (r, X r , ω) dB r and Z t ∆ = X t -x then Z t solves the SDE Z t = t 0 b r, Z r , τ (0,x) ω dr + t 0 σ r, Z r , τ (0,x) ω dB r ,
so that the law of the process X ω starting from x ∈ R d is equal to the law of the process x + X τxω where X τxω is starting from 0. We sum up: Theorem 8.2. Let f be a continuous, bounded function on R d . Then the solution z(x, t, ω) of the partial differential equation (2) with initial condition z(0, x, ω) = f (x) satisfies the following convergence: z(x/ε, t/ε 2 , ω) converges in π-probability as ε → 0 to IE f (x + A 1/2 B t ) , which is the viscosity solution of the deterministic equation (3) with the same initial condition. The matrix A is given by

A = Ω (σ + ξ * )(σ + ξ * ) * dπ.

Tightness

Let us now investigate the tightness in C([0, T ]; R d ) of the process

εX ω t/ε 2 = ε t/ε 2 0 b(r, X ω r , ω) dr + ε t/ε 2 0 σ(r, X ω r , ω) dB r .
The tightness of the first term in the above right-hand side is readily derived from the Burkholder-Davis-Gundy inequality and the boundedness of the diffusion coefficient σ. Concerning the second term, we are going to exploit ideas of [START_REF] Sethuraman | Diffusive limit of a tagged particle in asymmetric simple exclusion processes[END_REF] or [START_REF] Wu | Forward-Backward martingale decomposition and compactness results for additive functionals of stationary ergodic Markov processes[END_REF].

For any i ∈ {1, . . . , d} and λ > 0, we put w λ = (λ -S) -1 b i ∈ H ∩ Dom(S) (see Proposition 5.5). Proposition 5.4 (with θ = 0 and H = 0) also ensures that w λ ∈ F, D t w λ ∈ H. For each fixed λ > 0, we can find a sequence (ψ n λ ) n in C such that ψ n λw λ 1 + D t ψ n λ -D t w λ 1 converges to 0 as n goes to ∞. Define where M n,λ is a martingale with respect to the forward filtration (F t ) 0≤t≤T , and F t is the σ-algebra

on Ω generated by {Y r ; 0 ≤ r ≤ t}. From Proposition 6.3, we also have

v n λ (Y 0 ) -v n λ (Y t ) = t 0 [L * v n λ -D t v n λ ](Y r ) dr + M n,λ 0 -M n,λ t = t 0 [λv n λ -b i -A n λ -D t v n λ ](Y r ) dr + M n,λ 0 -M n,λ t ,
where M n,λ is a martingale with respect to the backward filtration (G t ) 0≤t≤T , and G s is the σalgebra on Ω generated by {Y r ; t ≤ r ≤ T }. Adding up these equalities, we obtain, for any 0 ≤ t ≤ T ,

2 t 0 b i (Y r ) dr =[v n λ -v n λ ](Y t ) + [v n λ -v n λ ](Y 0 ) + t 0 [λ(v n λ + v n λ ) + D t (v n λ -v n λ )](Y r ) dr + M n,λ t -M n,λ 0 + M n,λ 0 -M n,λ t
Fix R > 0 and choose λ = ε 2 . Integrating with respect to the measure IP π , we have (the sup below is taken over 0 ≤ t, s ≤ T )

IE π sup |t-s|≤α 2ε t/ε 2 s/ε 2 b i (Y r ) dr ≥ R ≤20R -2 (1 + T )ε 2 |v n ε 2 | 2 2 + |v n ε 2 | 2 2 + 10R -2 T /ε 2 |D t v n ε 2 -D t v n ε 2 | 2 2 +5ε 2 IE π sup |t-s|≤α |M n,ε 2 t/ε 2 -M n,ε 2 s/ε 2 | 2 ≥ R 2 + 5ε 2 IE π sup |t-s|≤α |M n,ε 2 s/ε 2 -M n,ε 2 t/ε 2 | 2 ≥ R 2 . ( 46
)
We are now going to explain how to choose n to make each term of the above right-hand side go to 0 as ε goes to 0. Since (λ -S)w λ = b i and (λ -L)v n λ = b i -A n λ , we can subtract these equalities and obtain, for each ϕ ∈ F, B 0 λ (w λv n λ , ϕ) = T H w λψ n λ , ϕ (remind of the definition of B 0 λ and T H in ( 19) and ( 17 46), all the terms in the right-hand side except the martingale terms converge to 0 as ε goes to 0. Let us now focus on the martingale terms. In order to prove the tightness of the two martingales, it is sufficient to prove the tightness of their brackets (see [START_REF] Jacod | Limit Theorems for Stochastic Processes[END_REF] Theorem 4.13), which respectively match 2 dπ in L 1 under the measure IP π . Then Theorem 3.37 in [START_REF] Jacod | Limit Theorems for Stochastic Processes[END_REF] says that the brackets are tight in C([0, T ]; R). The same arguments remain valid for the brackets of M n ε 2 ,ε 2 . Hence, the right-hand side in (46) converges to 0 as ε goes to 0 and the tightness of t → εX ω t/ε 2 follows.

ε 2 t/ε 2 0 |∇ σ v n ε 2 ε 2 (Y r )|

  As a bypass, (36) also implies the convergence of λ|v λ | 2 2 λ to 0. Lemma 5.10. For each function b ∈ H -1 , there exists a family (b λ ) λ of functions in L 2 (Ω) ∩ Dom(D t ) ∩ H -1 such that bb λ -1 converges to 0 as λ goes to 0. Proof: Let us consider the solution w λ ∈ H of the equation λw λ -Sw λ = b (see Proposition 5.5). Then, for any ϕ ∈ C,

Lemma 5 . 11 .

 511 Let (b λ ) λ and (b ′ λ ) λ be two families in H -1 such that b λb ′ λ -1 → 0 as λ goes to 0. Let (v λ ) λ and (v ′ λ ) λ two families in F solving equation (32) with respectively b λ and b ′ λ as right-hand side. Then λ|v

Lemma 5 . 13 .

 513 The norms of P λ : (H, • -1 ) → (H -1 , • -1 ) and P λ : (H, • H ) → (H, • H ) are both bounded from above by 2(2 + M + C H 1 )(1 + C a 2 + C H 2 ). Proof : Fix b ∈ H. Let u λ ∈ F (with D t u λ ∈ H) be the solution of the equation (apply Proposition 5.4 with a = a, H = 0, h = 0 and m = 1)

0 [

 0 A n λ = (1/2) k,l D l H kl D k ψ n λ . From Proposition 5.4, we can find two sequences (v n λ ) n ⊂ F ∩ Dom(L) and (v n λ ) n ⊂ F ∩ Dom(L * ) that respectively solve the equations (λ -L)v n λ = b i -A n λ and (λ -L * )v n λ = b i + A n λ .Moreover, the functions v n λ and v n λ possess a corresponding approximation sequence (v n λ,δ ) δ>0 and (v n λ,δ ) δ>0 (see Proposition 5.4), which both have continuous trajectories since b i ± A n λ have. We are then in position to apply Proposition 6.1. For any 0 ≤ t ≤ T and λ > 0v n λ (Y t )v n λ (Y 0 ) = t Lv n λ + D t v n λ ](Y r ) dr + M λb i + A n λ + D t v n λ ](Y r ) dr + M

2 dr and ε 2 t/ε 2 0σ v n ε 2 ε 2 ( 2 ε 2 - 2 0σ v n ε 2 ε 2 (

 22222222 |∇ Y r )| 2 dr. Note that |∇ σ v n ε ζ| 2 → 0 as ε tends to 0 so that the process t → ε 2 t/ε |∇ Y r )| 2 dr has the same limit in C([0, T ]; R) as the process t → ε 2 t/ε 2 0 |ζ(Y r )| 2 dr.Finally, for each fixed t, Theorem 7.1 proves that ε 2 t/ε 2 0 |ζ(Y r )| 2 dr converges to the deterministic non-decreasing process t Ω |ζ| 2

  Given d ≥ 1, consider a sequence (ε (k 1 ,...,k d ) ) (k 1 ,...,k d )∈Z d of independant Bernouilli random variables with parameter p ∈]0, 1[ and define a process η as follows: for each x ∈ R d , there exists a unique(k 1 , . . . , k d ) ∈ Z d such that x belongs to the cube [k 1 , k 1 + 1[× • • • × [k d , k d + 1[. Then define the process η : R d → R by: ∀x ∈ R d , η x = ε (k 1 ,...,k d ) .The law of this process is invariant and ergodic with respect to Z d translations. Roughly speaking, we are drawing a d-dimensional chessboard on R d , for which we are coloring each cube of the chessboard either in black with probability p or in white with probability 1 -p. It remains to make the process invariant under R d translations. To this purpose, choose a uniform variable U on the cube [0, 1[ d independent of the sequence (ε (k 1 ,...,k d ) ) (k 1 ,...,k d )∈Z d and define for x ∈ R d , η x = η x+U .

	In a way, this corresponds to
	a random change of the origin of the chessboard. It can be checked that we get a stationary ergodic
	random field on R d . Let us now tackle the issue of the regularity of the trajectories. Consider
	a C ∞ (R d ) function ϕ with a compact and very small support (for instance, included in the ball
	B(0, 1/4)) and define a new process η x = R d η y ϕ(x -y) dy = η * ϕ(x), which is a stationary ergodic random process with smooth trajectories. That is enough for a general framework.
	Let us now consider the process ω (t,x)

  Thanks to Assumption 2.3 and the antisymmetry of H, this form is clearly coercive. Thus it defines a strongly continuous resolvent operator and consequently, the generator L associated to this resolvent operator. More precisely, ϕ ∈ H belongs to Dom(L) if and only if B λ (ϕ, •) is L 2 -continuous. In this case, there exists f ∈ L 2 (Ω) such that B λ (ϕ, •) = (f , •) 2 and Lϕ is equal to f -λϕ. It can be proved that this definition is independent of λ > 0 (see[START_REF] Ma | Introduction to the Theory of (Non-Symmetric) Dirichlet Forms[END_REF] Ch. 1, Sect. 2] for further details).

1/2 ψ .

  2 is obviously continuous on H so that the Lax-Milgram theorem applies. It allows to construct a strongly continuous resolvent associated to λ -S by way of classical tools (see [5, Ch. 1, Sect. 3] or [12, Ch. 1, Sect. 2] for further details).

  1/2 u λ -η| 2 → 0 as λ goes to 0. Consider the operator P λ : H → H defined by P λ (b) = (L -S)(λ -S -D t ) -1 (b). Note that Proposition 5.4 applies for all coefficients a and H satisfying Assumption 2.3. In particular, it works for a = a and H = 0, so that P λ is well defined. Lemma 5.13 below proves that P λ H→H is bounded with a norm that only depends on the constants M, C H 1 , C a 2 and C H 2 (see Assumption 2.3). Therefore, we can choose δ 0

Proof:

  1/2 u λ ) λ are bounded in L 2 (Ω). Even if it means considering a subsequence, we assume that (λu λ ) λ , (λD t u λ ) λ and (λ 1/2 (-S) 1/2 u λ ) λ weakly converge respectively to g, g ′ and G in L 2 (Ω). Since the operator D t is closed, it turns out that g ′ = D t g. Let us now prove now that g ∈ Dom(L). Consider ϕ ∈ Dom(L * ) . Then we derive from (44) that

  )). Choosing ϕ = w λψ n λ , we obtain a first estimate Following Proposition 5.4, we can differentiate the equationB 0 λ (w λv n λ , ϕ) = T H w λψ n λ , ϕ with respect to the time variable. So we have, for each ϕ ∈ H, B 0 λ (D t w λ -D t v n λ , ϕ) = T H D t w λ -D t ψ n λ , ϕ + ∂ t T H w λψ n λ , ϕ -[∂ t T a + ∂ t T H ] w λv n λ , ϕ . Choosing ϕ = D t w λ -D t ψ nλ , we obtain a second estimateλ|D t w λ -D t v n λ | 2 2 + (m/2) D t w λ -D t v nLikewise, (47) and (48) remain valid for v n λ instead of v n λ . For each fixed λ > 0, we can then choosen λ ∈ N large enough to ensure that |w λv n λ λ | 2 2 + w λv n λ λ 2 1 + λ -1 |D t w λ -D t v n λ λ | 2 2 ≤ λ and |w λv n λ λ | 2 2 + w λv n λ λ 2 1 + λ -1 |D t w λ -D t v n λ λ | 2 2 ≤ λ.From Proposition 5.8, there exists ζ ∈ (L 2 (Ω)) d such that λ|w λ | 2 2 + |∇ σ w λ -ζ| 2 → 0 as λ goes to 0. From (47) (with n = n λ ), λ|v n λ λ | 2 2 + λ|v n λ λ | 2 2 → 0 as λ goes to 0. Hence, choosing n = n ε 2 in (

	(47)	λ|w λ -v n λ | 2 2 + (m/2) w λ -v n λ	2 1 ≤ (2m) -1 (C H 1 ) 2 w λ -ψ n λ	2 1 .
	(48)		λ	2 1	λ 1	2 .

≤ (2m) -1 C H 1 D t w λ -D t ψ n λ 1 + C H 2 w λψ n λ 1 + (C a 2 + C H 2 ) w λv n


Sepideh Ghanavati
email: sepideh.ghanavati@tudor.lu

Silvia Ingolfo
email: silvia.ingolfo@disi.unitn.it

Alberto Siena
email: siena@fbk.eu

Exploring Legal Business Process Paths

Nowadays, enterprises are very complex systems, often comprised of a large number of business processes run by actors working together to achieve business objectives. Ensuring compliance with applicable laws is mandatory to avoid heavy penalties or even business failure. To this purpose, an increasingly important challenge consists of finding and resolving discrepancies between strategic goals, business processes and laws. In this paper, we envisage a formal approach that uses two modeling languages, User Requirements Notation (URN) and Nòmos, to represent enterprise goals, processes and applicable laws. Automated reasoning techniques allow us to analyze models for compliance checking and detecting conditions of unwanted concurrent executions.

Introduction

Modern enterprises are complex systems comprised of actors, as well as software and hardware components working together in interleaved processes to support high-level enterprise objectives. Well-designed processes are essential to ensure efficiency and effectiveness in the production of goods and services. Enterprises are also subject to laws and regulations which can be costly to comply with. Being non-compliant may introduce more cost to the enterprise, such as heavy fines, bad reputation or business disruption. Since business processes can be very large and articulated, they generate a potentially large number of alternative execution paths. Similarly, laws are generally comprised of a large set of conditional elements, such as conditions, exceptions and so on, which create an even potentially larger number of admissible paths to comply. This raises the need to ensure that (i) every possible process or process path respect complies with applicable laws; and (ii) for every law or law fragment, one (or more) process or process branch exists, which fulfills the necessary legal accomplishments. While doing (i) and (ii), the achievement of strategic objectives must also be ensured.

Several work has been done to analyze the compliance of business processes with laws and regulations. Legal-URN [START_REF] Ghanavati | Legal-URN framework for legal compliance of business processes[END_REF], [START_REF] Sepideh Ghanavati | Legal goal-oriented requirement language (Legal-GRL) for modeling regulations[END_REF], based on User Requirements Notation (URN) [START_REF]User Requirements Notation (URN) -Language definition[END_REF], is one of the main frameworks which aims to model regulations, organizational goals and business processes in the same notation and provides analysis for compliance. However, this approach does not explore the alternatives for compliance in detail. In [START_REF] Shamsaei | Indicator-based policy compliance of business processes[END_REF] obligations are decomposed into operational level rules, and Key Performance Indicators are used to measure compliance objectives and balance them with business objectives.

Although few approaches exist for business process compliance, in a recent systematic literature review [START_REF] Ghanavati | A systematic review of goal-oriented requirements management frameworks for business process compliance[END_REF], the authors identify the needs for having a concrete framework with guidelines on how to map legal prescriptions to business processes and how to analyze the compliance.

In [START_REF]The Practice of Enterprise Modeling -6th IFIP WG 8.1 Working Conference, PoEM 2013[END_REF] a modeling approach is presented, which focuses on modeling the organizational regulatory space (ORS), comprised by both, the internals of the enterprise, such as business processes and goals, and applicable regulations. In [START_REF] Governatori | Ict support for regulatory compliance of business processes[END_REF] a new approach is proposed for assessing compliance of business processes using the compliance checker Regorous. The used language defines a more fine-grained concept of obligation and compliance. Using FCL, a rule-based logic which combines defeasible and deontic logic, the authors formalize regulations and validate the business process model.

In this position paper we envision an approach for the exploration of business process paths to detect potential violations of the law. We define an legal business process path a path in the process such that the supported requirements are satisfied, while the applicable norms are not violated. We model the process in Use Case Maps (UCM), which is the scenario notation part of URN, and the requirements in Goal-oriented Requirements Language (GRL), which is the goal modeling notation of URN. Other business process modeling such as BPMN or UML activity diagrams can be used instead of UCM, however, the advantage of UCM over these approaches is that it has links to GRL for goal models and includes capabilities for analyzing scenarios, conflict detection and propagating the result of the analysis from UCM to GRL and vice-versa.

The applicable norms are represented in Nòmos [START_REF] Siena | Capturing variability of law with Nòmos 2[END_REF], a modeling language for representing laws and regulations. The novelty of our approach stays in the capability to model alternatives in the combination of processes, law prescriptions and goals. Also in our approach, we support automated reasoning to explore exhaustively the models, ensuring compliance with laws and allowing at the same time, to achieve enterprise goals that the processes have to support.

The paper is structured as follows: Section 2 introduces the underlying modeling techniques; Section 3 presents the envisioned modeling approach and shows how reasoning is supported; finally, Section 4 concludes the paper and outlines some future work.

Baseline

User Requirements Notation (URN)

The User Requirements Notation (URN) [START_REF]User Requirements Notation (URN) -Language definition[END_REF] is an International Telecommunications Union (ITU-T) standard which helps requirements engineers and business analysts documenting requirements and/or analyzing these requirements for correctness and completeness. URN combines two modeling notations, Goaloriented Requirement Language (GRL) and the Use Case Maps (UCM) [START_REF] Weiss | Designing and evolving business models with URN[END_REF]. GRL is used to model goals and Non-Functional Requirements (NFR) with goals and softgoals, and provides means to reason about alternatives. UCM aims at modeling scenario concepts of operational and functional requirements, and help reasoning about performance and architectural decisions [START_REF] Amyot | User Requirements Notation: The first ten years, the next ten years (invited paper)[END_REF].

Goal-oriented Requirements Language (GRL) -The Goal-oriented Requirement Language (GRL) is a goal modeling notation based on i* language and the NFR Framework's concepts and syntax. High-level business goals, NFRs and the alternatives are modeled with intentional elements in GRL. GRL also models beliefs to capture the rationales behinds stakeholders' decisions, as well as stakeholder dependencies.

GRL intentional elements are: softgoals ( ), goals ( ), tasks ( ), beliefs or resources (

). Softgoals are abstract and have no clear measure of satisfaction while goals are quantifiable and can be fully satisfied. Softgoals usually model NFRs and quality requirements, whereas goals deal with and model functional requirements. Tasks capture solutions or alternative means to achive goals or softgoals. Resources are sometimes utilized to achieve tasks, goals and softgoals. Actors ( ), represent stakeholders of the system who can have certain goals to achieves and set of tasks to perform.

GRL includes a set of link to connect intentional elements to each other. These To analyze the satisfaction value of each intentional element as well as the overall satisfaction of actors in the system, GRL provides both bottom-up and top-down evaluation mechanisms. These evaluation mechanisms can also be quantitative, qualitative, hybrid or constraint-based. The detail of these evaluations mechanism are explained in [START_REF] Amyot | Evaluating goal models within the goal-oriented requirement language[END_REF].

Use Case Maps (UCM) -UCM aims at modeling scenarios and use cases. It depicts the causal sequences of tasks and activities allocated to components ( ). Components can be actors, agents, roles, software modules, sub-systems, etc. and they can be decomposed into sub-components. Scenario paths connect start points ( ) to end points ( ). Paths contain responsibilities ( ) which are the actions and activities that need to be done. They can be performed in sequence, concurrently ( ), or as alternatives (

). UCM has the capability to decompose complex scenario maps into several sub-maps (i.e. plug-in maps) via stubs ( ). Stubs can be static or dynamic. Input and output of the stubs are connected to the start points and end points in the plug-in to ensure scenario continuity across various levels of details. Dynamic stubs are used to specify alternative maps in the same location.

To define scenarios in UCM, preconditions and postconditions are captured. Scenarios are triggered and started when a set of preconditions are satisfied. Based on the set of the preconditions, one alternative path is taken at any point in time. Resposibilities which are usually linked to tasks or other intentional elements in GRL can get the satisfaction value of the tasks or intentional elements linked to them. When the preconditions, postconditions and other values in a scenario are defined, a a path traversal mechanism helps simulating different scenarios and the traversed paths. Path traversal mechanism can be used in regression testing and it provides operational semantics to UCM models.

Nòmos 2

Nòmos 2 is a modeling language that aims at capturing the variability of compliance alternatives for norms [START_REF] Siena | Capturing variability of law with Nòmos 2[END_REF]. Nòmos 2 models allow us to represent fragments of laws or regulations by representing the different conditions and rules described by a law or regulation, and the alternative ways to comply with it. The conditions on a norm's applicability and satisfiability are represented through the concept of situations denoting states-of-affairs (partial states of the world), such as "Christmas season" or "Driving on the highway". Situations are partial states of the world that we may know to hold (meaning the situation is satisfied), not hold, or neither (when we can't conclude satisfaction or denial). If some situations are satisfied, the norm will apply, and when other situations are satisfied, the norm will be satisfied. In our model, situations are linked to norms in terms of four basic relations acting as a label-propagation mechanism to identify when situations make a norm applicable/not-applicable (activate, block), or satisfied/not-satisfied (satisfy, break). When a norm (duty or right) is applicable, it is complied with when it is satisfied and violated if it is not satisfied. If the norm is not applicable or it is unknown whether the norm applies or not, it is either tolerated or undefined (see [START_REF] Siena | Capturing variability of law with Nòmos 2[END_REF] for more details). Relations between norms are used to represent exceptions and special other cases where norms may make other norms applicable, not applicable, or complied with. Figure 1 represents an example of a Nòmos 2 of a hypothetical tax law. When a product is bought from a seller the duty to pay taxes applies (S 1 activate -----→ D 1 ). In order to satisfy the norm -and therefore comply with it -it is possible to either pay the taxes (S 2 satisfy ----→ D 1 ), or by filling in the VAT-claim tax form (S 3 satisfy ----→ D 1 ). For example, since VAT-free product are untaxed by definition, if the product is VAT-free (and S 4 holds), the duty is no longer applicable (S 4 block ---→ D 1 ). For the purpose of this example, we consider that the stores allow a return policy when a valid receipt of the purchase is shown. The two situations -of 1) having bought a product and 2) having a valid receipt -activate the right to return the product to the store, which in turn is complied with when the product is indeed returned (S 6 satisfy ----→ R 1 ). However, if the product is damaged, then the right is no longer applicable (S 7 break ---→ R 1 ). The main idea behind Nòmos 2 is that leveraging on the applicability and satisfiability of different norms and situations holding, it is possible to identify how to comply with a law in different ways (e.g., by paying the VAT tax, by buying a VAT-free product, . . . ).

Legal Business Process Paths

A business process consists of a set of activities to be executed in sequence. Gateways, such as decision points and parallelisms, split the activity sequence into different possible flows, potentially executed in parallel by multiple actors (lanes). The presence of parallelism, in particular, generates a sort of indeterminism in process execution, which in turn causes multiple execution instances to be possible, depending on the concurrency conditions. For example, given two activities, a and b, a parallelism between them implies that three executions exist: a is executed before b, b is executed before a, and a and b simultaneously. If there is a legal constraint on the output of a, b, or both, it becomes necessary to identify these alternative executions and explore them to detect potential violation conditions of a business process, in order to change the process structure.

Modeling

To design a process that satisfies strategic goals while complying with applicable laws, we need a modeling language capable to model business processes, goals and laws. There are mature, standardized languages to model business processes, such as BPMN; also, there are modeling languages to model goals and some to model legal prescriptions. However, to model the three aspect of the problem and be able to support automated reasoning, we need to combine two or more existing languages. In this paper, we evaluate the combination of URN and Nòmos since the integration of goals and business process models has already been made in URN standard. Figure 2 depicts an example of our approach. The picture represents an electronic commerce scenario, in which tax laws have to be taken into account. Two actors are represented, the Buyer and the Seller, and two business processes are shown: one process involves both the Buyer and the Seller, while the other concerns only the Buyer. Different activities have been introduced in both processes to comply with the legal provisions already depicted in Figure 1. The processes run in parallel, so there is no way to ensure that a certain flow of activities is executed. While we can ensure that when each activity is performed, its corresponding legal requirement is satisfied, it is possible that performing a combination of activities results in a violation.

To represent a satisfiability conditions between the business process and the law, we use traceability links from the path branch of the business process to the corresponding situation in Nòmos model. For example, in the business process related to the tax law, two conditional verifications must happen. First, it is necessary to check if the product is tax free. If it is, situation S4 is satisfied. Next, if the product is not tax-free, the second check has to be done to verify whether the buyer filled up the tax exemption form or not. The two paths created from this satisfy S2 and S3 in Nòmos.

In the process of 'returning the product', the seller only accepts the product as returned if three conditions (i.e. having bought the product, having valid receipt, and returning the purchased product with no damage) are satisfied. These three conditions satisfy the situations, S5, S6 and S7 in Nòmos respectively. If the buyer does not satisfy all of the three conditions, then the main goal, return product will not be satisfied, as shown with red X in Figure 2.

In previous works [START_REF] Siena | Capturing variability of law with Nòmos 2[END_REF], we have used Nòmos to model pieces of real laws and studied the scalability of this modeling approach for legal document. Therefore -despite using a preliminary small example for this paper -we are confident in the ability of this approach in capturing all of the possible combinations that may emerge from very complex processes.The combinations emerge in particular from the design of the business processes and law, and their execution. This is achieved by using formal reasoning on the models, as described in the following.

Formal reasoning

Once a process starts, at any given time t k , a set of activities, from 0 to n, are executed. The trace of the process at time t k consists of all the activities that have been executed from time t 0 to time t k . Figure 3 illustrates the exhaustive generation of multiple paths within a given business process. In the left part, the figure depicts a simple business process with 5 activities and two branches executed in parallel. Due to the parallelism, many different execution traces can be defined: in the right part, 9 different partial traces defined by the process are shown. In the first trace, the process has been started (activity 's' executed); in the second, 'x1' has been executed; and so on. At a certain point, when the activities can be executed in parallel, the number of possible traces grows rapidly because of the concurrency conditions described above. We hypothesize that, given a process and a law, each activity of the process can bring about an arbitrary number of conditions defined in the law to be compliant with, that are the legal prescriptions. The set of activities in a trace form an execution assignment to that prescriptions: the activities that belong to the trace are considered to actually fulfill the legal prescriptions; while the activities that do not belong to the trace leave the corresponding prescriptions not fulfilled. Depending on how the law is structured, failing in fulfilling one or more prescriptions can generate a violation to that law. To generalize, given a process P and a set of norms L, we define a legal business process path as a path in the process, such that for each possible trace in the path L is not violated. The intuition behind the present work is that we can use URN and Nómos to enforce modeling business processes that satisfy stakeholder requirements and at the same time comply with applicable laws. In particular, URN offers basics to model requirements with GRL, and business processes with UCM, and to establish traceability links between them. Nòmos allows modeling of the legal prescriptions and provides support for automated reasoning. Specifically, Nòmos enables for exhaustive exploration of legal alternatives within a given model of law. Adopting similar traceability links between business processes and laws, we can perform exhaustive search in the space of traces generated by a given business process.

Table 1 illustrates how our approach can be formalized using Disjunctive Logic Programming (DLP) [START_REF] Minker | Overview of disjunctive logic programming[END_REF]. DLP is a declarative, first-order logic language and a deductive system, where facts and deduction rules are expressed as predicates of the logic language. Disjunctions may appear in the rule heads to allow multiple alternative consequences to be drawn from a rule. Solvers, such as DLV [START_REF] Alviano | The disjunctive datalog system dlv[END_REF] allows exhaustively exploring the alternative models defined on a set of predicates, searching for desired properties.

We use DLP to introduce alternative traces, as in [START_REF] Siena | Automated reasoning for regulatory compliance[END_REF]. Each activity in a given process is represented as a variable. If an activity x is executed in a process p, the predicate done(x,p) is triggered. If the activity that precedes x has been executed, but x still has not been executed, the predicate pending(x,p) is triggered. After x has been executed, it is possible that either the next one, say y is executed, or that the transition has not accomplished, so we have that done(y,p) v pending(y,p) :-done(x,p). If an activity is far from being executed, nothing is deducted. In both cases, whether the activity is pending or discarded, not done(y,p) holds. This mechanism allows us to provide a Nòmos model with the true values coming from the process trace. In the table, a certain situation s1 is considered satisfied (i.e., id holds) when the activity x1 is executed in the process p.

Discussion and Conclusion

In this position paper, we have proposed an approach to support modeling enterprises in their strategic dimension (goals) as well as operational dimension (business processes), while ensuring compliance to applicable laws. Our approach relies on two existing modeling languages, URN and Nòmos, and extends them to allow linking activities of the business processes to the situations belonging to the legal model. Models are translated into disjunctive logic programs and exhaustively checked by means of an automated reasoning tool. Our approach allows exhaustively generating all possible execution traces on a given process, and contrast them to the linked situations to check for compliance.

While aligning business processes to the goals is a consolidated method in both research and practice, to the best of our knowledge there is a lack in analysis capabilities to what concerns the alignment of the internal structure of the enterprise to the legal environment, where the enterprise operates in. The main drawback of this approach is the need to explicitly model the links between business process and law models. However, supporting this kind of analysis through automated reasoning, is of particular importance when considering large enterprises, with many processes distributed over several legislations.

In our current work in progress, we are planning to apply our approach to a larger example for a more thorough evaluation of our proposal. This will allow us to evaluate its feasibility in a real-size case study, and to evaluate the scalability of the approach in this larger settings by means of artificial models (see for example [START_REF] Siena | Automated reasoning for regulatory compliance[END_REF]). An important limitation that we plan to investigate in our future work is for sure the creations of our models (currently done manually), where we envision a systematic process to help the analyst in the compliance evaluation of the model, as well as in the amendment of such model to reach compliance.

  links are: decomposition links ( ), contribution links ( ), correlation links ( ) or dependency links ( ). Decomposition links are used to decompose an intentional element into sub-elements and can be a type of AND, IOR, or XOR. XOR and IOR decomposition links. Contribution links which can have qualitative or quantitative values illustrate the impact of one intentional element's satisfaction on another intentional element's satisfaction value. Correlation links are similar to contribution links, but indicate side-effects of one intentional element on the other. Dependency links model relationships and the dependencies between actors.

Fig. 1 .

 1 Fig. 1. An example of a Nòmos 2 model.

Fig. 2 .

 2 Fig. 2. Example of URN-Nòmos Reasoning

Table 1 .

 1 An excerpt of the formalisation in DLP done(s,p) :-start(p). done(x1,p) v pending(x1,p) :-done(s,p). %parallelism done(x21,p) v pending(x21,p) :-done(x1). done(x22,p) v pending(x22,p) :-done(x1). done(x23,p) v pending(x23,p) :-done(x22). done(x3,p) v pending(x3) :-done(x21,p), done(x23,p). done(e,p) v pending(e,p) :-done(x3,p). end(p) :-done(e,p). % link to situations st(s1) :-done(x1,p). sf(s1) :-not done(x1,p). start(p) v not_start(p).

Fig. 3 .

 3 Fig. 3. Set of possible traces defined on a given process.

Acknowledgments. This work has been partially funded by AFR -PDR grant #5810263 and by the ERC advanced grant 267856 "Lucretius: Foundations for Software Evolution".