Anne-Laure Dalibard
email: dalibard@ceremade.dauphine.fr

Homogenization of linear transport equations in a stationary ergodic setting

We study the homogenization of a linear kinetic equation which models the evolution of the density of charged particles submitted to a highly oscillating electric field. The electric field and the initial density are assumed to be random and stationary. We identify the asymptotic microscopic and macroscopic profiles of the density, and we derive formulas for these profiles when the space dimension is equal to one.

Introduction

This note is concerned with the homogenization of a linear transport equation in a stationary ergodic setting. The equation studied here describes the evolution of the density of charged particles in a rapidly oscillating random electric potential. This equation can be derived by passing to the semi-classical limit in the Schrödinger equation (see [START_REF] Gérard | Mesures semi-classiques et ondes de Bloch, Séminaire Équations aux dérivées partielles[END_REF], [START_REF] Lions | Sur les mesures de Wigner[END_REF], and the presentation in [START_REF] Frenod | Homogenisation of transport kinetic equations with oscillating potentials[END_REF]). Our work generalizes a result of E. Frénod and K. Hamdache (see [START_REF] Frenod | Homogenisation of transport kinetic equations with oscillating potentials[END_REF]) which was obtained in a periodic setting. The strategy of proof we have chosen here is different from the one of [START_REF] Frenod | Homogenisation of transport kinetic equations with oscillating potentials[END_REF], and allows us to retrieve some of the results in [START_REF] Frenod | Homogenisation of transport kinetic equations with oscillating potentials[END_REF] in a rather simple and explicit fashion.

Let us mention a few related works on the homogenization of linear transport equations; we emphasize that this list is by no means exhaustive. In [START_REF] Amirat | Homogénéisation d'équations hyperboliques du premier ordre et applications aux écoulements miscibles en milieu poreux[END_REF], Y. Amirat, K. Hamdache and A. Ziani study the homogenization of a linear transport equation in a periodic setting and give an application to a model describing a multidimensional miscible flow in a porous media. In [START_REF] Dumas | Homogenization of Transport Equations[END_REF] (see also [START_REF] Golse | Remarques sur l'homogénéisation des équations de transport[END_REF]), Laurent Dumas and François Golse focus on the homogenization of linear transport equations with absorption and scattering terms, in periodic and stationary ergodic settings. And in [START_REF] Weinan | Homogenization of linear and nonlinear transport equations[END_REF], Weinan E derives strong convergence results for the homogenization of linear and nonlinear transport equations with oscillatory incompressible velocity fields in a periodic setting.

Let us now present the context we will be working in : let (Ω, F , P ) be a probability space, and let (τ x ) x∈R N be a group transformation acting on Ω. We assume that τ x preserves the probability measure P for all x ∈ R N , and the group transformation is ergodic, which means ∀A ∈ F, τ x A = A ∀x ∈ R N ⇒ P (A) = 0 or 1 .

The periodic setting can be embedded the stationary ergodic setting (see [START_REF] Papanicolaou | Boundary value problems with rapidly oscillating random coefficients, Rigorous results in Statistical Mechanics and Quantum Field Theory[END_REF]). We will denote by E[•] the expectation with respect to the probability measure P ; in the periodic case, we will write f rather than E[f ] to refer to the average of f over one period. We consider a potential function u = u(y, ω) ∈ L ∞ (R N × Ω) which is assumed to be stationary, i.e.

u(y + z, ω) = u(y, τ z ω) ∀(y, z, ω) ∈ R N × R N × Ω Moreover, we assume that 0 ≤ u(y, ω) ≤ u max = sup u for all y ∈ R N , ω ∈ Ω, and u(•, ω) ∈ W 2,∞ loc (R N ) for almost every ω ∈ Ω, so that ∇ y u (y, ω) is well-defined and locally Lipschitz continuous with respect to its y variable.

Let f ε = f ε (t, x, ξ, ω), (t ≥ 0, x ∈ R N , ξ ∈ R N , ω ∈ Ω) be the solution of the transport equation

∂ t f ε (t, x, ξ, ω) + ξ • ∇ x f ε (t, x, ξ, ω) -1 ε ∇ y u x ε , ω • ∇ ξ f ε (t, x, ξ, ω) = 0, f ε (t = 0, x, ξ, ω) = f 0 x, x ε , ξ, ω . (1)
Here, we assume that the initial data f 0 = f 0 (x, y, ξ, ω) belongs to L 1 loc (R N x × R N ξ , L ∞ (R N y × Ω)) and is stationary in y, i.e. f 0 (x, y + z, ξ, ω) = f 0 (x, y, ξ, τ z ω) for all (x, y, z, ξ, ω) ∈ R 4N × Ω.

It is well-known from the classical theory of linear transport equations that for every ω ∈ Ω, there exists a unique solution f ε of (1) in L ∞ loc ((0, ∞), L 1 loc (R N x × R N ξ )). The goal of this paper is to study the asymptotic behavior of f ε as ε → 0. Thus, following [START_REF] Frenod | Homogenisation of transport kinetic equations with oscillating potentials[END_REF], we define the constraint space K :

Definition 1.1. Let ξ • ∇ y f (y, ξ, ω) -∇ y u(y, ω) • ∇ ξ f (y, ξ, ω) = 0 (2)
be the constraint equation, and let

K := {f ∈ L 1 loc (R N ξ × R N y , L 1 (Ω)); f satisfies (2) in D ′ (R N y × R N ξ ) a.s. in ω}.
We also define the projection P onto the constraint space K, characterised by P (f ) ∈ K for f ∈ L 1 loc (R N ξ × R N y , L 1 (Ω)) stationary, and

R N ×Ω
(P (f )f )(y, ξ, ω) g(y, ξ, ω) dξ dP (ω) = 0 for a.e. y ∈ R N for all stationary functions g ∈ L ∞ (R N y × R N ξ × Ω) ∩ K, with compact support in ξ. (A more precise definition of the projection P will be given in the second section). Finally, we define K ⊥ as

K ⊥ := {f ∈ L 1 loc (R N ξ × R N y , L 1 (Ω)); ∃g ∈ L 1 loc (R N ξ × R N y , L 1 (Ω)), f = P (g) -g}.
Remark 1.1. Let us indicate that the constraint equation can easily be derived thanks to a formal twoscale Ansatz : indeed, assume that

f ε (t, x, ξ, ω) ≈ f t, x, x ε , ξ, ω as ε → 0;
inserting this asymptotic expansion in equation (1), we see that f necessarily satisfies the constraint equation (2).

Remark 1.2. Let f, g ∈ L ∞ (R N y , L 2 (R N ξ × Ω))
be stationary, and assume that f ∈ K and g ∈ K ⊥ . Then for a.e. y ∈ R N ,

R N ×Ω f (y, ξ, ω)g(y, ξ, ω) dξ dP (ω) = 0. This is a characterization of K ⊥ for the class of stationary functions in L ∞ (R N y , L 2 (R N ξ × Ω)).
Here, we provide another proof for the result of E. Frénod and K. Hamdache in [START_REF] Frenod | Homogenisation of transport kinetic equations with oscillating potentials[END_REF] in the "nonperturbed case". Our proof is based on the use of the ergodic theorem, and gives a more concrete insight of the projection P and of the microscopic behavior of the sequence f ε . Moreover, it allows us to retrieve the explicit formulas of the integrable case.

The first result we prove in this paper is the following

Theorem 1. Let f 0 ∈ L 1 loc (R N x × R N ξ × R N y ; L 1 (Ω)) stationary. Let f ε = f ε (t,
x, ξ, ω) be the solution of [START_REF] Amirat | Homogénéisation d'équations hyperboliques du premier ordre et applications aux écoulements miscibles en milieu poreux[END_REF]. Then for all ε > 0, there exist f = f (t, x, y, ξ, ω) and g = g(t, x; τ, y, ξ, ω), both stationary in y, and a sequence {r ε (t, x, ξ, ω)} ε>0 such that

f ε (t, x, ξ, ω) = f t, x, x ε , ξ, ω + g t, x; t ε , x ε , ξ, ω + r ε (t, x, ξ, ω)
and :

• ||r ε || L 1 loc ((0,∞)×R N x ×R N ξ ,L 1 (Ω)) → 0 as ε → 0; • f ∈ L ∞ loc ((0, ∞); L 1 loc (R N x × R N ξ × R N y ; L 1 (Ω))), and f (t, x) ∈ K for a.e. t ≥ 0, x ∈ R N ; • For all T > 0, for all compact K ⊂ R N x × R N ξ × R N y , sup 0≤t≤T,0≤τ ≤T ||g|| L 1 (K×Ω) < ∞.
Moreover, g(t, x; τ, •) ∈ K ⊥ for a.e. (t, x, τ ) ∈ (0, ∞) × R N × (0, ∞);

• Microscopic evolution equation for g : for a.e. t, x ∈ (0, ∞) × R N , g(t, x; •) is a solution of

∂g ∂τ + ξ • ∇ y g -∇ y u • ∇ ξ g = 0. (3)
Moreover, for all T > 0

T 0 g t, x; t ε , x ε , ξ, ω dt L 1 loc (R N x ×R N ξ ,L 1 (Ω))
→ 0 as ε → 0.

• Macroscopic evolution equation : f and g satisfy

∂ t f g + ξ ♯ (y, ξ, ω) • ∇ x f g = 0, (4)
where ξ ♯ (y, ξ, ω) := P (ξ)(y, ξ, ω);

• Initial data :

f (t = 0, x, y, ξ, ω) = P (f 0 )(x, y, ξ, ω), g(t = 0, x; τ = 0, y, ξ, ω) = [f 0 -P (f 0 )] (x, y, ξ, ω).

Before going any further, we wish to make a few comments on the above results. First, let us stress that it is not obvious that the function g is well-defined : indeed, let S(t) (t ≥ 0) denote the semi-group associated to the macroscopic evolution equation (4), and let T (τ ) (τ ≥ 0) be the semi-group associated to the microscopic evolution equation (3). Then g is well defined if and only if, for all stationary function g 0 = g 0 (x, y, ξ, ω), for all t, τ ≥ 0,

T (τ ) [S(t)g 0 ] = S(t) [T (τ )g 0 ] .
This identity follows from the fact that the speed ξ ♯ (y, ξ, ω) appearing in equation ( 4) is a stationary solution of (3) by definition of the projection P , and is thus invariant by the semi-group T (τ ).

Next, let us explain briefly the meaning of theorem 1. The idea is the following : write f 0 as

f 0 = f 0 + f 0⊥ , with f 0 (x, •) ∈ K and f 0⊥ (x, •) ∈ K ⊥ a.e. Then f ε can be written as f ε + f ε ⊥ , where f ε (resp. f ε ⊥ ) is the solution of equation (1) with initial data f 0 x, x ε , ξ, ω (resp. f 0⊥ x, x ε , ξ, ω ). Theorem 1 states that f ε -f t, x, x ε , ξ, ω → 0 strongly in L 1 loc norm.
In particular, there are no microscopic oscillations in time in this part of f ε . We wish to emphasize that this result appears to us to be new.

We now focus on the other part, namely f ε ⊥ . An easy consequence of the theorem is

T 0 f ε ⊥ (t, x, ξ, ω) dt → 0 in L ∞ loc (R N x ; L 1 loc (R N ξ ; L 1 (Ω))
) and for all T > 0. However, it would be wrong to think that

f ε ⊥ vanishes in L 1 loc ((0, ∞) × R N x × R N ξ , L 1 (Ω)), for instance. Indeed f ε ⊥ (t, x, ξ, ω) ≈ g t, x; t ε , x ε , ξ, ω in L 1 loc , and ||g(t = 0, x; τ, y)|| L 1 (R N ξ ×Ω) = ||f 0⊥ (x, y)|| L 1 (R N ξ ×Ω)
as soon as f 0⊥ (x, y) ∈ L 1 (R N ξ × Ω) for almost every x, y. Consequently, if f 0⊥ = 0, then for all T > 0 and for all compact K ⊂ R N , there exists a constant C > 0 depending only on K, ||f 0⊥ || L 1 loc , and T such that

T 0 g t, x; t ε , x ε , ξ, ω L 1 (K×R N ξ ×Ω) dt ≥ C.
Hence f ε ⊥ does not vanish strongly in general. In other words, there are fast oscillations in time, due to the ill-preparedness of the initial data (i.e. f 0 (x, •) / ∈ K), but these oscillations do not cancel out as ε vanishes.

Let us now explain briefly here how our strategy of proof differs from the one of E. Frénod and K. Hamdache. The key of our analysis lies in the study of the behavior as ε → 0 of the Hamiltonian system

   Ẏ ε (t, x, ξ, ω) = -Ξ ε (t, x, ξ, ω), t > 0 Ξε (t, y, ξ, ω) = 1 ε ∇ y u(Y ε (t, x, ξ, ω), ω), t > 0 Y ε (t = 0, x, ξ, ω) = x, Ξ ε (t = 0, x, ξ, ω) = ξ, (x, ξ, ω) ∈ R 2N × Ω. Indeed, f ε (t, x, ξ, ω) = f 0 Y ε (t, x, ξ, ω), Y ε (t, x, ξ, ω) ε , Ξ ε (t, y, ξ, ω), ω ,
so that we can deduce the asymptotic behavior of f ε from the one of (Y ε , Ξ ε ). And it is easily checked that Y ε (t, x, ξ, ω) = εY t ε , x ε , ξ, ω Ξ ε (t, y, ξ, ω) = Ξ t ε , x ε , ξ, ω , where (Y, Ξ) is the solution of the system

   Ẏ (t, y, ξ, ω) = -Ξ(t, y, ξ, ω), t > 0 Ξ(t, y, ξ, ω) = ∇ y u(Y (t, y, ξ, ω), ω), t > 0 Y (t = 0, y, ξ, ω) = y, Ξ(t = 0, y, ξ, ω) = ξ, (y, ξ, ω) ∈ R 2N × Ω. (5)
Hence, in order to study the limit of f ε as ε → 0, we have to investigate the long time behavior of the system (Y, Ξ), and this will be achieved with the help of the ergodic theorem in the second section.

In the case when N = 1, we can give explicit formulas for ξ ♯ (y, ξ, ω); the proof of this formula in the stationary ergodic case is strongly linked to methods from the Aubry-Mather theory (see [START_REF]Aubry-Mather theory and periodic solutions of the forced Burger's equation[END_REF], [START_REF] Evans | Effective Hamiltonians and averaging for Hamiltonian dynamics[END_REF], [START_REF] Lions | Correctors for the homogenization theory of Hamilton-Jacobi equations in the stationary ergodic setting[END_REF]), and thus also to the homogenization of Hamilton-Jacobi equations. Let us first recall the definition of the homogenized Hamiltonian H (see [START_REF] Lions | Homogenization of Hamilton-Jacobi equations[END_REF])

H(p) = u max + 1 2    0 if |p| < E 2(u max -u) λ if |p| ≥ E 2(u max -u) , where |p| = E 2(u max -u) + λ Proposition 1.1. Assume that N = 1. Let (y, ξ, ω) ∈ R × R × Ω such that H(y, ξ, ω) > u max .
Assume that for all Q ∈ R, for all ω ∈ Ω, there exists a Lipschitz continuous function v(•, ω), viscosity solution of

H(y, Q + ∇ y v(y, ω), ω) = H(Q) such that v(y, ω) 1 + |y| → 0 as |y| → ∞ (6)
a.s. in ω.

Let P = P (y, ξ, ω) ∈ R such that H(P ) = H(y, ξ, ω) and sgn(P ) = sgn(ξ). Then ξ ♯ (y, ξ, ω) = H′ (P )

Moreover, if L is the dual function of H, i.e.

L(y, p, ω) = sup ξ∈R (pξ -H(y, ξ ω)) = 1 2 |p| 2 -u(y, ω),
and L is the homogenized Lagrangian, then

P (L)(y, ξ, ω) = L(ξ ♯ (y, ξ, ω).)
In the periodic case, we will give another proof of the above result; the strategy chosen in that case is inspired from techniques and calculations in classical mechanics. It also allows to give a formula for ξ ♯ for low energies in the periodic setting only: Proposition 1.2. Assume that N = 1 and that the environment is periodic.

Let (y, ξ) ∈ R 2 such that H(y, ξ) < u max . Then ξ ♯ (y, ξ) = 0.

The organisation of this note is the following : in the second section, we derive some preliminary results on the long-time behavior of the system (Y, Ξ) thanks to the ergodic theorem. Those will be useful in the proof of theorem 1, to which is devoted the third section. Eventually, the fourth and last section is concerned with results in the integrable case, both in the periodic and the stationary ergodic settings.

Preliminaries

This section is largely devoted to the study of the long-time behavior of the Hamiltonian system (Y, Ξ) defined by [START_REF]Aubry-Mather theory and periodic solutions of the forced Burger's equation[END_REF]. First, notice that the Hamiltonian H(y, ξ, ω) := 1 2 |ξ| 2 + u(y, ω) is constant along the curves of the system (Y, Ξ), and if

f ∈ L ∞ (Ω, C 1 (R N y × R N ξ )) is stationary, then f ∈ K ⇐⇒ f (Y (t, y, ξ, ω), Ξ(t, y, ξ, ω), ω) = f (y, ξ, ω) ∀(y, ξ, ω) ∈ R N × R N × Ω. Indeed, for all f ∈ L ∞ (Ω, C 1 (R N y × R N ξ )), we have ∂ ∂t f (Y (t, y, ξ, ω), Ξ(t, y, ξ, ω), ω) = {H, f } (Y (t, y, ξ, ω), Ξ(t, y, ξ, ω), ω) ,
where {H, f } denotes the Poisson bracket of f and H, i.e.

{H, f } (y, ξ, ω) = ξ • ∇ y f (y, ξ, ω) -∇ y u(y, ξ, ω) • ∇ ξ f (y, ξ, ω).
Let us mention an easily checked property of the trajectories (Y, Ξ) which will be used extensively in the rest of the article : for all (y, z, ξ

) ∈ R 3N , for all ω ∈ Ω, t ≥ 0, Y (t, y, ξ, τ z ω) + z = Y (t, y + z, ξ, ω), Ξ(t, y, ξ, τ z ω) = Ξ(t, y + z, ξ, ω). (7)
In the periodic case, this invariance entails that the hamiltonian system (Y, Ξ) can be considered as a dynamical system on the N dimensional torus [0, 2π) N . In this periodic setting, it is somewhat natural to introduce the semi-group of transformations (T t ) t≥0 on [0, 2π) N × R N given by

T t (y, ξ) = (Y (t, y, ξ), Ξ(t, y, ξ)), y ∈ [0, 2π) N , ξ ∈ R N .
According to Liouville's theorem, this semi-group preserves the Lebesgue measure on [0, 2π) N × R N ; moreover, we can construct a family of finite invariant measures on [0, 2π) N × R N by setting m c (y, ξ) = 1 H(y,ξ)≤c dy dξ for c > 0 (remember that the Hamiltonian is constant along the hamiltonian curves). This construction is the root of the ergodic theorem (see corollary 2.1), and thus of the study of the long-time behavior of the system (Y, Ξ).

In the stationary ergodic setting, this construction can be generalized as follows : we define the transformation

T t : R N ξ × Ω → R N ξ × Ω by T t (ξ, ω) = Ξ(t, 0, ξ, ω), τ Y (t,0,ξ,ω) ω
together with the family of measures

µ c := 1 H(ξ,ω)≤c dξ dP (ω)
where

H(ξ, ω) := 1 2 |ξ| 2 + u(0, ω). It is obvious that for all c ∈ (0, ∞), µ c is a finite measure on R N ξ × Ω.
Notice that the "good" generalization to the stationary ergodic setting of the semi-group (T t ) is a semi-group which acts on

R N ξ × Ω rather than R N y × R N ξ .
Thanks to the group of transformations (τ x ) x∈R N , the transformations in Ω can result in transformations in R N y , but the definition chosen here allows us to define a family of finite invariant measures, whereas such a construction is rather difficult if one tries to define a semi-group acting on R N y × R N ξ . This will be fundamental in the rest of the proof.

Lemma 2.1. (T t ) t≥0 is a semi-group on R N ξ × Ω and preserves the family of measures µ c .

Proof. Let us first prove the semi-group property : let t, s ∈ [0, ∞), and (ξ, ω) ∈ R N × Ω; then

T t • T s (ξ, ω) = T t Ξ(s, 0, ξ, ω), τ Y (s,0,ξ,ω) ω = Ξ(t, 0, Ξ(s, 0, ξ, ω), τ Y (s,0,ξ,ω) ω), ω ′
and using the properties [START_REF] Frenod | Homogenisation of transport kinetic equations with oscillating potentials[END_REF] we deduce

Ξ(t, 0, Ξ(s, 0, ξ, ω), τ Y (s,0,ξ,ω) ω) = Ξ(t, Y (s, 0, ξ, ω), Ξ(s, 0, ξ, ω), ω), = Ξ(t + s, 0, ξ, ω)
and

ω ′ = τ Y (t,0,Ξ(s,0,ξ,ω),τ Y (s,0,ξ,ω) ω) τ Y (s,0,ξ,ω) ω = τ Y (t,0,Ξ(s,0,ξ,ω),τ Y (s,0,ξ,ω) ω)+Y (s,0,ξ,ω) ω = τ Y (t,Y (s,0,ξ,ω),Ξ(s,0,ξ,ω),ω) ω = τ Y (t+s,0,ξ,ω) ω Thus T t • T s (ξ, ω) = Ξ(t + s, 0, ξ, ω), τ Y (t+s,0,ξ,ω) ω = T t+s (ξ, ω).
Since it is obvious that T 0 = Id, (T t ) t≥0 is a semi-group on R N × Ω. We now have to check the invariance property; let

F ∈ L 1 (R N × Ω; µ c ) arbitrary. We set f (y, ξ, ω) := F (ξ, τ y ω) for (y, ξ, ω) ∈ R N y × R N ξ
× Ω, and we compute

R N ×Ω F (T t (ξ, ω)) dµ c (ξ, ω) = E R N f (Y (t, 0, ξ, ω), Ξ(t, 0, ξ, ω), ω)1 H(Y (t,0,ξ,ω),Ξ(t,0,ξ,ω),ω)≤c dξ
Since the probability measure P is invariant by the group of transformation τ y , and

f (Y (t, y, ξ, ω), Ξ(t, y, ξ, ω), ω) = f (Y (t, 0, ξ, τ y ω), Ξ(t, 0, ξ, τ y ω), τ y ω), we have, for all y ∈ R N E f (Y (t, 0, ξ, ω), Ξ(t, 0, ξ, ω), ω)1 H(Y (t,0,ξ,ω),Ξ(t,0,ξ,ω),ω)≤c = = E f (Y (t, y, ξ, ω), Ξ(t, y, ξ, ω), ω)1 H(Y (t,y,ξ,ω),Ξ(t,y,ξ,ω),ω)≤c .
Take an arbitrary function φ ∈ L 1 (R N y ), and write

R N ×Ω F (T t (ξ, ω)) dµ c (ξ, ω) = E R 2N dy dξφ(y)f (Y (t, y, ξ, ω), Ξ(t, y, ξ, ω), ω)1 H(Y (t,y,ξ,ω),Ξ(t,y,ξ,ω),ω)≤c
We change variables in the integral in (y, ξ) by setting (x, v) = (Y (t, y, ξ, ω), Ξ(t, y, ξ, ω)); according to Liouville's theorem, the jacobian of this change of variables is equal to 1, and

(x, v) = (Y (t, y, ξ, ω), Ξ(t, y, ξ, ω)) ⇐⇒ (y, ξ) = (X(t, x, v, ω), V (t, x, v, ω)),
where (X, V ) is a solution of the Hamiltonian system

   Ẋ = V, V = -∇u(X, ω), (X, V )(t = 0, x, v) = (x, v).
Observe that in the present case, we have simply

X(t, x, v, ω) = Y (t, x, -v, ω),
so that (X, V ) satisfies relations [START_REF] Frenod | Homogenisation of transport kinetic equations with oscillating potentials[END_REF]. Hence

R 2N dy dξφ(y)f (Y (t, y, ξ, ω), Ξ(t, y, ξ, ω), ω)1 H(Y (t,y,ξ,ω),Ξ(t,y,ξ,ω),ω)≤c = R 2N dx dvφ(X(t, x, v, ω))f (x, v, ω)1 H(x,v,ω)≤c = R 2N dx dvφ(X(t, 0, v, τ x ω) + x)F (v, τ x ω)1 H(v,τxω)≤c so that R N ×Ω F (T t (ξ, ω)) dµ c (ξ, ω) = E R 2N dx dvφ(X(t, 0, v, τ x ω) + x)F (v, τ x ω)1 H(v,τxω)≤c = E R 2N dx dvφ(X(t, 0, v, ω) + x)F (v, ω)1 H(v,ω)≤c = E R N dv R N φ(X(t, 0, v, ω) + x) dx F (v, ω)1 H(v,ω)≤c = E R N dvF (v, ω)1 H(v,ω)≤c = R N ×Ω F dµ c since the integral of φ is equal to 1.
The following corollary is an immediate consequence of Birkhoff's ergodic theorem:

Corollary 2.1. Let F ∈ L 1 (R N × Ω; µ c ). There exists a function F ∈ L 1 (R N × Ω; µ c ) such that as T → ∞, 1 T T 0 F (T t (ξ, ω)) dt → F (ξ, ω)
a.e. on R N × Ω and in L 1 (µ c ). Moreover, F is invariant by T t for all t > 0, and

R N ×Ω F dµ c = R N ×Ω F dµ c . (8)
Additionally, if f = f (y, ξ, ω) is the stationary function associated to F , that is, f (y, ξ, ω) = F (ξ, τ y ω), then f is invariant by the hamiltonian flow (Y, Ξ); precisely, for all (y, ξ, ω) ∈ R 2N × Ω, t > 0 f (Y (t, y, ξ, ω), Ξ(t, y, ξ, ω), ω) = f (y, ξ, ω).

Proof. We only have to prove the invariance of f by the Hamiltonian flow; first, for y = 0, we have

f (Y (t, 0, ξ, ω), Ξ(t, 0, ξ, ω), ω) = F (Ξ(t, 0, ξ, ω), τ Y (t,0,ξ,ω) ω) = F (T t (ξ, ω)) = F (ξ, ω) = f (0, ξ, ω)
and the property is proved when y = 0.

For

y ∈ R N arbitrary, f (Y (t, y, ξ, ω), Ξ(t, y, ξ, ω), ω) = f (Y (t, 0, ξ, τ y ω) + y, Ξ(t, 0, ξ, τ y ω), ω) = f (Y (t, 0, ξ, τ y ω), Ξ(t, 0, ξ, τ y ω), τ y ω) = f (0, ξ, τ y ω) = f (y, ξ, ω)
according to the result in the case y = 0.

Remark 2.1. We mention here an important but easy consequence of the relations (7) and the invariance of the measure P w.r.t. τ y , y ∈ R N : for any stationary function

f = f (y, ξ, ω) = F (ξ, τ y ω), F ∈ L ∞ (R N × Ω), we have E[f (Y (t, y, ξ, •), Ξ(t, y, ξ, •), •)] = E[F (T t (ξ, •))]
for all t > 0, y, ξ ∈ R N ; in particular, the left-hand side of the above equality does not depend on y. This property was used in the proof of lemma 2.1

Remark 2.2. Let us precise a little what happens when the function

F ∈ L 1 loc (R N ξ , L 1 (Ω)). In that case, F ∈ L 1 (R N ξ × Ω; µ c
) for all c > 0. Consequently, for any c > 0, we can define the function Fc associated to F by corollary 2.1.

It is then easily proved that for any 0 < c < c ′ , Fc = Fc ′ , µ c -almost everywhere. Setting A n = {(ξ, ω) ∈ Suppµ n ; Fn (y, ξ) = Fn+1 (y, ξ)}, and A = ∪ ∞ n=0 A n , we see that µ c (A) = 0 for all c > 0. Moreover, for all (ξ, ω) ∈ R N × Ω \ A, for all integers k, l such that (ξ, ω) ∈ Suppµ k ∩ Suppµ l , we have Fk (ξ, ω) = Fl (ξ, ω). We can thus define a function

F (ξ, ω) on R N × Ω \ A by F (ξ, ω) = Fn (ξ, ω) for any n ∈ N such that (ξ, ω) ∈ Suppµ n We then now that 1 T T 0 F (T t (ξ, ω)) dt → F (ξ, ω) (9)
as T → ∞, and the convergence holds in L 1 (µ c ) for all c > 0, and µ n almost everywhere for n ∈ N.

Eventually, setting

B := {(ξ, ω) ∈ R N × Ω \ A; 1 T T 0 F (T t (ξ, ω)) dt does not converge towards F (ξ, ω)}
it is easily proved that µ c (B) = 0 for all c > 0 (the equality is true for c ∈ N, and is then deduced for c > 0 arbitrary because the family of measures (µ c ) is increasing in c).

Eventually, we have found a function F ∈ L 1 loc (R N , L 1 (Ω)), independent of c, such that (9) holds in L 1 (µ c ) and µ c -almost everywhere for all c > 0.

Remark 2.3. The construction above allows us to make more precise what we mean by projection P :

let f = f (y, ξ, ω) be a stationary function, f ∈ L ∞ (R N y , L 1 loc (R N ξ , L 1 (Ω))), and set F (ξ, ω) = f (0, ξ, ω) ∈ L 1 loc (R N ξ , L 1 (Ω)). We can then associate to F a function F ∈ L 1 loc (R N ξ , L 1 (Ω)
) such that (9) holds in L 1 (µ c ) for all c (see remark 2.2). We set

P (f )(y, ξ, ω) := F (ξ, τ y ω).
It follows from corollary 2.1 that P (f ) is invariant by the hamiltonian flow (5), and thus satisfies the constraint equation. From now on, we take this definition for the projection P , instead of the one given in the introduction. Notice that, for all y ∈ R N and µ c -almost everywhere,

P (f )(y, ξ, ω) = lim T →∞ 1 T T 0 F (T t (ξ, τ y ω)) dt = lim T →∞ 1 T T 0 f (Y (t, 0, ξ, τ y ω), Ξ(t, 0, ξ, τ y ω), τ y ω) dt = lim T →∞ 1 T T 0 f (Y (t, y, ξ, ω), Ξ(t, y, ξ, ω), ω) dt
And we also give a more precise definition of ξ ♯ (y, ξ, ω) : let

ξ : R N × Ω → R N (ξ, ω) → ξ .
Then

ξ ♯ (y, ξ, ω) = P ( ξ)(y, ξ, ω) = lim T →∞ 1 T T 0 Ξ(t, y, ξ, ω) dt
a.e. and in L 1 (µ c ) for all 0 < c < ∞.

Eventually, we mention here a property that will be used in the proof of the theorem; with the same notations as above, let φ(τ, y, ξ, ω) = F (T τ (ξ, τ y ω)) .

Then φ is a solution of the evolution equation

∂ τ φ + ξ • ∇ y φ -∇ y u • ∇ ξ φ = 0,
with initial data φ(τ = 0, y, ξ, ω) = f (y, ξ, ω) = F (ξ, τ y ω).

The general N -dimensional case

This section is devoted to the proof of theorem 1. The proof is divided in three steps : first, we study the case of an initial data which does not depend on x, then the case when the initial data only depends on x (and not on y, ξ, ω), and eventually, we treat the general case.

3.1 First case : f 0 does not depend on x

Here, we assume that

f 0 = f 0 (y, ξ, ω) ∈ L 1 loc (R N ξ ; L ∞ (R N y × Ω).
Recall that f 0 is stationary, i.e. f 0 (y + z, ξ, ω) = f 0 (y, ξ, τ z ω) a.s. in ω, for all (y, z, ξ) ∈ R 3N . In the rest of the subsection, we set

F 0 (ξ, ω) := f 0 (0, ξ, ω) and F0 (ξ, ω) := lim T →∞ 1 T T 0 F (T t (ξ, ω)) dt, f0 (y, ξ, ω) = F0 (ξ, τ y ω). Notice that F ∈ L 1 loc (R N ξ ; L ∞ (Ω)), and thus F ∈ L 1 (R N × Ω; µ c ) for all c > 0.
In that case,

f ε (t, x, ξ, ω) = f 0 Y t ε , x ε , ξ, ω , Ξ t ε , x ε , ξ, ω , ω = f 0 Y t ε , 0, ξ, τ x ε ω , Ξ t ε , 0, ξ, τ x ε ω , τ x ε ω = F 0 T t ε ξ, τ x ε ω = f0 x ε , ξ, ω + F 0 T t ε ξ, τ x ε ω -F0 ξ, τ x ε ω In accordance with theorem 1, we set g(τ, y, ξ, ω) = F 0 -F0 (T τ (ξ, τ y ω)) ,
and r ε = 0. Then g satisfies the microscopic evolution equation (3) thanks to the remark at the end of the preceding section. Moreover, g(τ ) ∈ K ⊥ by definition of K ⊥ and because P (F 0 (T τ (ξ, τ y ω))) = F0 (ξ, τ y ω).

It only remains to check that

T 0 g t ε , x ε , ξ, ω dt → 0 as ε → 0 in L 1 loc (R N x , L 1 (R N × Ω, µ c
)) for all T > 0 and c > 0. The invariance of the measure P with respect to the group of transformations (τ x ) x∈R N (see remark 2.1) entails that

Ω×R N ξ 1 T ε T ε 0 f 0 Y t, x ε , ξ, ω , Ξ t, x ε , ξ, ω , ω dt -f0 x ε , ξ, ω dµ c (ξ, ω) = Ω×R N ξ 1 T ε T ε 0 F 0 (T t (ξ, ω)) dt -F0 (ξ, ω) dµ c (ξ, ω)
and the term above goes to 0 as ε → 0 according to corollary 2.1 and is independent of x ∈ R N . There remains to check that f0 = P (f 0 ). This follows directly from remark 2.3. Thus theorem 1 is proved in the case when f 0 does not depend on the macroscopic variable x.

The following remark will prove to be useful when treating the general case :

Remark 3.1. If f 0 ∈ L ∞ , then for any function a ∈ L ∞ ((0, ∞) × R N y × R N ξ × Ω), stationary in y, we have T 0 a t, x ε , ξ, ω g t ε , x ε , ξ, ω dt → 0 as ε → 0 in L 1 loc (R N x , L 1 (R N × Ω, µ c
)) for all T > 0 and c > 0. Indeed, prove the property first for a = a 1 (t)a 2 (y, ξ, ω), with a 1 , a 2 ∈ L ∞ . For a arbitrary, take a sequence a δ with δ > 0, converging to a in L 1 loc , and such that

a δ = n δ k=0 a δ 1 (t)a δ 2 (y, ξ, ω).
with a δ 1 , a δ 2 in L ∞ . The property is known for a δ , and it is thus easily deduced for a.

Second case :

f 0 = f 0 (x)
Unlike the preceding subsection, we now focus on the case when f 0 only depends on the macroscopic variable x. In order to simplify the analysis, we assume that f 0 ∈ W 1,∞ (R N x ) (the case when f 0 is not smooth in x will be treated in the next subsection). In that case,

f ε (t, x, ξ, ω) = f 0 εY t ε , x ε , ξ, ω .
Hence we have to investigate the behavior as ε → 0 of

εY t ε , x ε , ξ, ω .
We prove the following Lemma 3.1. Let T > 0 arbitrary. As ε vanishes,

εY t ε , x ε , ξ, ω -x + tξ ♯ x ε , ξ, ω → 0 in L ∞ ((0, T ) × R N x ; L 1 (R N ξ × Ω, µ c )). Proof. Let us write, for t > 0 εY t ε , x ε , ξ, ω -x + tξ ♯ x ε , ξ, ω = ε t ε 0 Ẏ s, x ε , ξ, ω ds + tξ ♯ x ε , ξ, ω = -t ε t t ε 0 Ξ s, x ε , ξ, ω ds + tξ ♯ x ε , ξ, ω = -t ε t t ε 0 ξ T s (ξ, τ x ε ω) ds -ξ ♯ x ε , ξ, ω
Let 0 < α < T arbitrary. For α ≤ t ≤ T , we have

R N ξ ×Ω εY t ε , x ε , ξ, ω -x + tξ ♯ x ε , ξ, ω dµ c (ξ, ω) = t R N ξ ×Ω ε t t ε 0 ξ (T s (ξ, ω)) ds -ξ ♯ (0, ξ, ω) dµ c (ξ, ω) ≤ T sup τ ≥ α ε 1 τ τ 0 ξ (T s (ξ, ω)) ds -ξ ♯ (0, ξ, ω) L 1 (R N ×Ω,µc)
and the upper-bound vanishes as ε → 0 for any α > 0 thanks to corollary 2.1. Notice that the upperbound does not depend on x, hence the convergence holds in L ∞ (R N x ; L 1 (µ c )). We now have to investigate what happens when t is close to 0; notice that sup

x∈R N ||ξ ♯ x ε , ξ, ω || L 1 (R N ×Ω,µc) ≤ C 0
where the constant C 0 only depends on N and c. Similarly, for all t ≥ 0, sup

x∈R N ξ T s (ξ, τ x ε ) L 1 (R N ×Ω,µc) ≤ C 0 .
Hence, if 0 ≤ t ≤ α, we have sup

x∈R N R N ξ ×Ω εY t ε , x ε , ξ, ω -x + ξ ♯ x ε , ξ, ω dµ c (ξ, ω) ≤ 2αC 0 .
Eventually,

εY t ε , x ε , ξ, ω -x + tξ ♯ x ε , ξ, ω L ∞ ((0,T )×R N ;L 1 (µc)) ≤ ≤ inf 0<α<T 2C 0 α + T sup τ ≥ α ε 1 τ τ 0 ξ (T s (ξ, ω)) ds -ξ ♯ (0, ξ, ω) L 1 (µc)
and the lemma is proved.

We easily deduce that theorem 1 is true when

f 0 ∈ W 1,∞ (R N ) with f (t, x, y, ξ, ω) := f 0 (x -tξ ♯ (y, ξ, ω)), g = 0, r ε (t, x, ξ, ω) := f ε (t, x, ξ, ω) -f t, x, x ε , ξ, ω
and it is easily checked that f satisfies P (f ) = f , f (t = 0) = P (f 0 ) = f 0 (since f 0 is independent of y and ξ), and that f is a solution of the evolution equation (4).

Third case : f 0 arbitrary

We now tackle the case of an arbitrary stationary function

f 0 ∈ L 1 loc (R N x × R N ξ , L ∞ (R N y × Ω)).
We begin with the case when f 0 (x, y, ξ, ω) = a(x)b(y, ξ, ω),

with a ∈ W 1,∞ (R N ) and b ∈ L 1 loc (R N ξ , L ∞ (R N y × Ω)) ∩ L ∞ (R N y × R N ξ × Ω
), b stationary. This case follows directly from the two first subsections. Indeed, let f (t, x, y, ξ, ω) = a(xtξ ♯ (y, ξ, ω)) P (b)(y, ξ, ω), and g(t, x; τ, y, ξ, ω) = a(xtξ ♯ (y, ξ, ω)) (b -P (b)) (T τ (y, ξ, ω)).

It is already known that f and g satisfy ( 4), that f (t, x, •) ∈ K, and that g satisfies (3) thanks to the preceding subsections and the fact that ξ ♯ (y, ξ, ω) is invariant by the Hamiltonian flow (Y, Ξ). Notice that it is capital here that the coefficient ξ ♯ (y, ξ, ω) in the transport equation ( 4) belongs to K.

There remains to check that g(t, x; τ, •) ∈ K ⊥ , that the remainder r ε goes to 0 strongly in L 1 loc and that g(t, x; t/ε, x/ε, ξ, ω) goes weakly to 0 in the sense of theorem 1. First, notice that a(xtξ ♯ (y, ξ, ω)) ∈ K and (b -P (b)) (T τ (y, ξ, ω) ∈ K ⊥ . Thus, a(xtξ ♯ )P (b) = P (a(xtξ ♯ )b) almost everywhere (because a(xtξ ♯ (0, ξ, ω)) is invariant by the semi-group T τ ), and consequently

g(t, x; τ, y, ξ, ω) = a(x -tξ ♯ )b -P a(x -tξ ♯ )b (T τ (ξ, τ y ω)).
Hence g(t, x; τ ) ∈ K ⊥ a.e.

Then, setting

r ε (t, x, ξ, ω) = f ε (t, x, ξ, ω) -f t, x, x ε , ξ, ω -g t, x; t ε , x ε , ξ, ω ,
we have to prove that r ε goes to 0 strongly in L 1 loc . We compute the difference

f ε (t, x, ξ, ω) -f t, x, x ε , ξ, ω -g t, x; t ε , x ε , ξ, ω = a εY t ε , x ε , ξ, ω b Y t ε , x ε , ξ, ω , Ξ t ε , x ε , ξ, ω , ω -a x -tξ ♯ x ε , ξ, ω P (b) x ε , ξ, ω -a x -tξ ♯ x ε , ξ, ω [b -P (b)] Y t ε , x ε , ξ, ω , Ξ t ε , x ε , ξ, ω , ω = a εY t ε , x ε , ξ, ω -a x -tξ ♯ x ε , ξ, ω b Y t ε , x ε , ξ, ω , Ξ t ε , x ε , ξ, ω , ω
The right-hand side of the above equality is bounded by

||a|| W 1,∞ ||b|| L ∞ εY t ε , x ε , ξ, ω -x + tξ ♯ x ε , ξ, ω
and thus converges to 0 as ε → 0 in L ∞ ((0, T )×R N x ; L 1 (R N ξ ×Ω, µ c ))) according to the second subsection.

Moreover, it is easily proved that as ε → 0,

T 0 g t, x; t ε , x ε , ξ, ω dt → 0 strongly in L 1 loc (R N x × R N ξ , L 1 (Ω)) thanks to remark 3.1. Hence theorem 1 is proved in that case. Now, let f 0 ∈ L 1 loc (R N x × R N ξ , L ∞ (R N y × Ω))
arbitrary, and set F 0 (x, ξ, ω) := f 0 (x, 0, ξ, ω). Take a sequence of functions

F n ∈ L 1 (R N x × R N ξ × Ω) such that • F n → F 0 as n → ∞ in L 1 loc (R N x × R N ξ × Ω);
• For all n ∈ N, there exist functions

a n k ∈ L 1 ∩ W 1,∞ (R N ), b n k ∈ L 1 ∩ L ∞ (R N ξ × Ω), 1 ≤ k ≤ n such that F n (x, ξ, ω) = n k=1 a n k (x) b n k (ξ, ω) a.e.
Let f ε n be the solution of ( 1) with initial data F n x, ξ, τ x ε ω , and let f n = f n (t, x, y, ξ, ω), g n = g n (t, x; τ, y, ξ, ω) be the functions associated to f ε n by theorem 1 for all n. Let f (t, x, y, ξ, ω), g(t, x; τ, y, ξ, ω) be the solutions of the system

P (f ) = f, P (g) = 0, ∂ t f g + ξ ♯ (y, ξ, ω) • ∇ x f g = 0, ∂ τ g + ξ • ∇ y g -y u(y, ω) • ∇ ξ g = 0, f (t = 0) = P (f 0 ), g(t = 0, x; τ = 0, y, ξ, ω) = [f 0 -P (f 0 )] (x, y, ξ, ω).
We have already proved that f n , g n satisfy the above system. We denote by F0 , Fn , the functions associated to F 0 , F n respectively by corollary 2.1, so that P (f 0 )(x, y, ξ, ω) = F0 (x, ξ, τ y ω), and f n (t = 0, x, y, ξ, ω) = Fn (x, ξ, τ y ω), g n (t = 0, x, τ = 0, y, ξ, ω) = (F n -Fn )(x, ξ, τ y ω).

We use the following lemma, of which we postpone the proof : Lemma 3.2. Let g ε be a solution of (1) with initial data g 0 x, x ε , ξ, ω , and

g 0 ∈ L 1 loc (R N x ×R N ξ , L ∞ (R N y × Ω)) stationary. Then for all R, R ′ , T > 0, for all t ∈ [0, T ], x∈BR, ξ∈B R ′ |g ε (t, x, ξ, ω)| dx dξ ≤ ||g 0 || L 1 (K T ,R,R ′ ,L ∞ (R N y ×Ω))
where

K T,R,R ′ = (x, ξ) ∈ R N × R N , |x| ≤ R + T R ′ 2 + 2u max , |ξ| ≤ R ′ 2 + 2u max .
Similarly, if g is a solution of (4) with initial data

g 0 ∈ L 1 loc (R N x × R N ξ , L ∞ (R N y × Ω)), then x≤R |g(t, x, y, ξ, ω)| dx ≤ x≤R+T √ ξ 2 +2umax |g 0 (x, y, ξ, ω)| dx Consequently, with C R,T,ξ := {x ∈ R N , |x| ≤ R + T ξ 2 + 2u max }, we have ||f (t, •, y, ξ, ω) -f n (t, •, y, ξ, ω)|| L 1 (BR) ≤ F0 (•, ξ, τ y ω) -Fn (•, ξ, τ y ω) L 1 (C R,T ,ξ ) ≤ |F 0 -F n |(•, ξ, τ y ω) L 1 (C R,T ,ξ ) ||f (t, x, y, ξ, ω) -f n (t, x, y, ξ, ω)|| L 1 (BR×R N ξ ×Ω,dx dµc(ξ,ω)) ≤ |F 0 -F n |(x, ξ, ω) L 1 (C R,T , √ 2c ×R N ξ ×Ω,dxdµc(ξ,ω)) ≤ ||F 0 -F n (x, ξ, ω)|| L 1 (C R,T , √ 2c ×R N ξ ×Ω,dxdµc(ξ,ω)) .
In the last inequality, we have used property [START_REF] Golse | Remarques sur l'homogénéisation des équations de transport[END_REF]. And similarly,

||f ε (t, x, ξ, ω) -f ε n (t, x, ξ, ω)|| L 1 (BR×B R ′ ×Ω) ≤ ||F 0 -F n || L 1 (K T ,R,R ′ ,L ∞ (Ω)) , ||g(t, x; τ, y, ξ, ω) -g n (t, x; τ, y, ξ, ω)|| L 1 (BR×R N ξ ×Ω,dx dµc(ξ,ω)) ≤ 2 ||F 0 -F n (x, ξ, ω)|| L 1 (C R,T , √ 2c ×R N ξ ×Ω,dxdµc(ξ,ω))
. The above inequalities are true for all t ∈ [0, T ] and for all τ ≥ 0. Set

r ε (t, x, ξ, ω) := f ε (t, x, ξ, ω) -f t, x, x ε , ξ, ω -g t, x; t ε , x ε , ξ, ω .
Then for all t ∈ [0, T ], for all n ∈ N,

setting c = 1 2 R ′ 2 + u max , ||r ε (t)|| L 1 (BR×B R ′ ×Ω) ≤ ||f ε (t) -f ε n (t)|| L 1 (BR×B R ′ ×Ω) +||f (t) f n (t)|| L ∞ (R N y ;L 1 (BR x ×R N ξ ×Ω,dx dµc(ξ,ω))) +||g(t) -g n (t)|| L ∞ ((0,∞)τ ×R N y ;)L 1 (BR×R N ξ ×Ω,dx dµc(ξ,ω)) +||r ε n (t)|| L 1 (BR×B R ′ ×Ω) ≤ 4 ||F 0 -F n || L 1 (C R,T , √ 2c ×R N ξ ×Ω,dxdµc(ξ,ω)) + ||r ε n (t)|| L 1 (BR×B R ′ ×Ω) Thus r ε → 0 as ε → 0 in L ∞ ([0, ∞); L 1 loc (R N x × R N ξ ; L 1 (Ω)
). There only remains to check that T 0 g(t, x; t/ε, x/e, ξ, ω) dt goes strongly to 0 in L 1 loc norm as ε vanishes; this result follows immediately from the same property for g n and the above inequalities. Therefore, we skip its proof.

Proof of Lemma 3.2. First, let us recall that

f ε (t, x, ξ, ω) = f 0 εY t ε , x ε , ξ, ω , Y t ε , x ε , ξ, ω , Ξ t ε , x ε , ξ, ω , ω ,
and the Jacobian of the change of variables

(x, ξ) → εY t ε , x ε , ξ, ω , Ξ t ε , x ε , ξ, ω , ω is equal to 1.
On the other hand, since

1 2 |Ξ(t, y, ξ, ω)| 2 + u (Y (t, y, ξ, ω)) = 1 2 |ξ| 2 + u(y, ω) we have |Ξ(t, y, ξ, ω)| ≤ |ξ| 2 + 2u(y, ω) ≤ |ξ| 2 + 2u max and εY t ε , x ε , ξ, ω -x ≤ t |ξ| 2 + 2u max . Thus x∈BR, ξ∈B R ′ |f ε (t, x, ξ, ω)| dx dξ = x∈BR, ξ∈B R ′ f 0 εY t ε , x ε , ξ, ω , Y t ε , x ε , ξ, ω , Ξ t ε , x ε , ξ, ω , ω dx dξ ≤ x∈BR, ξ∈B R ′ sup y∈R N f 0 εY t ε , x ε , ξ, ω , y, Ξ t ε , x ε , ξ, ω , ω dx dξ ≤ R N ×R N 1 (εY ( t ε , x ε ,ξ,ω),Ξ( t ε , x ε ,ξ,ω))∈K T ,R,R ′ sup y∈R N f 0 εY t ε , x ε , ξ, ω , y, Ξ t ε , x ε , ξ, ω , ω dx dξ = K T ,R,R ′ sup y∈R N |f 0 (x, y, ξ, ω)| dx dξ
The proof of the other inequality goes along the same lines.

The integrable case

In this section, we treat independently the periodic and the stationary ergodic case. Indeed, some results of the periodic case are no longer true in the stationary ergodic setting, and the results which do remain valid are not proved with the same tools.

Let us make precise what we mean about "integrable case" : in the periodic case, we take a function u(y) which has the form

u(y) = N i=1 u i (y i ), (10)
where each function u is periodic with period 1 (1 ≤ i ≤ N ). The Hamiltonian H(y, ξ) can be written

H(y, ξ) = 1 2 |ξ| 2 + u(y) = N i=1 H i (y i , ξ)
where

H i (y i , ξ) = 1 2 |ξ i | 2 + u i (y i ) (1 ≤ i ≤ N )
. And the Hamiltonian system (5) becomes

   Ẏi = -Ξ i , Ξi = u ′ i (Y i ), Y i (t = 0) = y i , Ξ i (t = 0) = ξ i . ( 11
)
Thus it is enough to investigate the behavior of each one-dimensional Hamiltonian system (11) individually, and for most calculations, we can assume without loss of generality that N = 1, and we drop all indices i. However, for the calculation of the projection P , a more thorough discussion will be needed, and we will come back to the case when N > 1 in the corresponding paragraph.

In the stationary ergodic setting, expression (10) can be transposed in the following way : assume that Ω = Π N i=1 Ω i , where each Ω i is a probability space, and assume that for 1 ≤ i ≤ N , an ergodic group transformation, denoted by (τ i,y ) y∈R , acts on each Ω i .

Then for ω = (ω 1 , • • • , ω N ) ∈ Ω, and y = (y 1 , • • • , y N ) ∈ R N , we set τ y ω := (τ 1,y1 ω 1 , • • • , τ N,yN ω N ). And we assume that the function u can be written

u(y, ω) = N i=1 U i (τ i,yi ω i ) ,
where U i ∈ L ∞ (Ω i ) for all 1 ≤ i ≤ N . The same remarks as in the periodic case can be made, and thus we will only consider the case N = 1; note that in the stationary ergodic case, we are unable to compute the projection P when N > 1.

Periodic setting

The goal of this subsection is to give another proof of the results of K. Hamdache and E. Frénod in [START_REF] Frenod | Homogenisation of transport kinetic equations with oscillating potentials[END_REF], based on the study of the system

   Ẏ = -Ξ, Ξ = u ′ (Y ), Y (t = 0) = y, Ξ = ξ, y ∈ R, ξ ∈ R. (12)
The Hamiltonian H(y, ξ) = 1 2 |ξ| 2 + u(y) is constant along the trajectories of the system (12), so that

1 2 |Ξ(t, yξ)| 2 + u(Y (t, y, ξ)) = H(y, ξ).
We now fix y, ξ ∈ R N . Without any loss of generality, we assume y ∈ [-1 2 , 1 2 ), and we set E := H(y, ξ). The above equation describes the movement of a single particle in a periodic potential u, with 0 ≤ u ≤ u max . It is well-known that there are two kinds of behavior, depending on the value of the energy E : if E < u max , the particle is "trapped" in a well of potential around y, and Y (t) remains bounded as t → ∞. In that case, the trajectories in the phase space are closed curves. If E > u max , the trajectory of the particle is unconstrained and |Y (t)| → ∞ as t → ∞. We study more precisely these two cases and their consequences on the expression of the projection P in the following; we refer for instance to [START_REF] Arnold | Ordinary differential equations[END_REF] for further calculations and results about Hamiltonian dynamics and ordinary differential equations in general. We begin with the case when H(y, ξ) < u max . In that case, u(y) ≤ H(y, ξ) < u max . By continuity of the potential u, there exists y -< y and y + > y such that H(y, ξ) < u(y ± ) < u max , and the periodicity of u allows us to choose y ± such that |y +y -| < 1. Then y -< Y (t, y, ξ) < y + for all t ≥ 0. Indeed, assume that there exists t > 0 such that Y (t, y, ξ) ≥ y + > y = Y (t = 0, y, ξ). Since the trajectory Y is continuous in time, there exists 0 < t 0 ≤ t such that Y (t = t 0 , y, ξ) = y + , which is absurd since

H(Y (t = t 0 , y, ξ), Ξ(t = t 0 , y, ξ)) = H(y, ξ) ≥ u(Y (t = t 0 , y, ξ)) > H(y, ξ). Thus Y (t, y, ξ) is bounded. Since ξ ♯ (y, ξ) = lim T →∞ 1 T T 0 Ξ(t, y, ξ) dt = -lim T →∞ 1 T T 0 Ẏ (t, y, ξ) dt = lim T →∞ y -Y (T, y, ξ) T
we deduce that ξ ♯ (y, ξ) = 0 for all y, ξ such that H(y, ξ) < u max .

We now study the case H(y, ξ) > u max . Since

| Ẏ (t, y, ξ)| 2 = 2 (H(y, ξ) -u(Y (t, y, ξ))) ≥ 2(H(y, ξ) -u max ) > 0
Ẏ does not vanish for t ≥ 0. Consequently, Ξ(t, y, ξ) = -Ẏ (t, y, ξ) = sgn(ξ) 2 (H(y, ξ)u(Y (t, y, ξ))),

and since |Y (t, y, ξ) -y| ≥ 2 (H(y, ξ)u max )t, |Y (t)| → ∞ as t → ∞. We immediately deduce that Ξ(t, y, ξ) is periodic in time: indeed, there exists t 0 > 0 such that Y (t 0 , y, ξ) = ysgn(ξ).

And Ξ(t = t 0 , y, ξ) = sgn(ξ) 2 (H(y, ξ)u(Y (t 0 , y, ξ))) = sgn(ξ) 2 (H(y, ξ)u(y)) = ξ = Ξ(t = 0, y, ξ), so that for s ≥ 0,

Y (t 0 + s, y, ξ) = Y (s, y, ξ) -sgn(ξ), Ξ(t 0 + s, y, ξ) = Ξ(s, yξ),
and Ξ is periodic with period t 0 . Consequently, We close this paragraph with a calculation which allows us to express ξ ♯ in terms of the homogenized Hamiltonian H. The result we will obtain will be justified in more abstract and theoretical terms in the last subsection, using arguments similar to those of the theory of Aubry-Mather.

ξ ♯ (y, ξ) = lim T →∞ 1 T T 0 Ξ(t, y, ξ) dt = 1 t 0 t0 0 Ξ(t, y, ξ) dt.
First, let us recall the expression of the homogenized Hamiltonian H (see [START_REF] Lions | Homogenization of Hamilton-Jacobi equations[END_REF]) : we have

H(y, ξ) = 1 2 |ξ| 2 + u(y), with inf u = 0, sup u = u max ,
and thus

H(p) = u max + 1 2    0 if p < 2(u max -u) λ if |p| ≥ 2(u max -u) , where |p| = 2(u max -u) + λ
In other words, setting

θ : [0, ∞) → [0, ∞) λ → 2(u max -u) + λ we have H(p) = u max + 1 2 1 |p|≥θ(0) θ -1 (|p|).
Hence, H′ (p) = sgn(p) 1 2

1 |p|≥θ(0) 1 θ ′ (θ -1 (|p|)) ; and θ ′ (λ) = 1 2 1 2(u max -u) + λ , θ -1 (|p|) = 2 H(p) -u max ∀|p| ≥ θ(0), |p| > θ(0) ⇐⇒ H(p) > u max ∀p.
Gathering all the terms, we are led to

H′ (p) = sgn(p) √ 21 H(p)>umax 1 1 √ H(p)-u = sgn(p)ϕ H(p)
Thus, the final expression is

ξ ♯ (y, ξ) = H′ (p),
where p is such that H(p) = H(y, ξ) ∨ u max , sgn(p) = sgn(ξ).

Expression of the projection P

We also mention here how to find a general expression of the projection P in the special case N = 1, and we explain how to generalize this expression in some particular cases when

N > 1. Recall that if f = f (y, ξ) ∈ L 1 loc (R N y × R N ξ ) is periodic in y, then P (f )(y, ξ) = lim T →∞ 1 T T 0 f (Y (t, y, ξ), Ξ(t, y, ξ)) dt
and the limit holds almost everywhere and in L 1 ([0, 1) × R N , m c ), with dm c (y, ξ) = 1 H(y,ξ)≤c dy dξ.

We begin with the case H(y, ξ) > u max . We have seen in the previous paragraph that there exists t 0 > 0, which depends only on H(y, ξ) such that for all t > 0, for all k ∈ N Y (t + kt 0 , y, ξ) = Y (t, y, ξ)ksgn(ξ), Ξ(t + kt 0 , y, ξ) = Ξ(t, y, ξ).

Thus f (Y (t), Ξ(t)) is periodic in time with period t 0 , and

P (f )(y, ξ) = 1 t 0 t0 0 f (Y (t, y, ξ), Ξ(t, y, ξ)) dt.
We use once again the change of variables s = Y (t), so that

t0 0 f (Y (t, y, ξ), Ξ(t, y, ξ)) dt = y-sgn(ξ) y f (s, sgn(ξ) 2 (H(y, ξ) -u(s))) 1 -sgn(ξ) 2 (H(y, ξ) -u(s)) ds = f •, sgn(ξ) 2 (H(y, ξ) -u(•)) 1
2 (H(y, ξ)u(•)) .

And eventually, P (f )(y, ξ) = f (sgn(ξ), H(y, ξ))

with

f (η, E) := f •, η 2 (E -u(•)) 1 √ (E-u(•)) 1 √ (E-u) η = ±1, E > u max
We now focus on the case 0 < E < u max . In order to simplify the analysis we assume that E / ∈ {u(y) ; u has a local extremum at y} (this set is finite or countable), and that ∀y ∈ R, u ′ (y) = 0 ⇒ u has a local extremum at y.

In that case, it can be easily proved that Y (t, y, ξ) is periodic in t; this follows directly from the fact that the trajectory in the phase space is closed (see [START_REF] Arnold | Ordinary differential equations[END_REF]). Indeed, pushing a little further the analysis of the previous paragraph, we construct z ± such that

|z + -z -| < 2π, z -< z + , u(z ± ) = E, z -≤ y ≤ z + , u(z) < E ∀z ∈ (z -, z + ).
Sketch of proof. By density, it is enough to prove the lemma for f 1 , •, f N ∈ C ∞ (R). Write f i as a Fourier series (the series converges thanks to the regularity assumption), and use the fact that for all α = 0, 1 T T 0 e iαt dt → 0 as T → ∞.

In the present setting, we deduce the following result :

Proposition 4.1. Let ϕ : (y, ξ) → ϕ 1 (y 1 , ξ 1 ) • • • ϕ N (y N , ξ N ), where ϕ i ∈ L ∞ per (R y × R ξ ). Let (y, ξ) ∈ [0, 1) N × R N , and let θ i = θ i (y i , ξ i ) = T i (H i (y i , ξ i )) for 1 ≤ i ≤ N . Assume that (θ 1 , • • • , θ N ) satisfy condition (15). Then P (ϕ)(y, ξ) = P 1 (ϕ 1 )(y 1 , ξ 1 ) • • • P N (ϕ N )(y N , ξ N ) ( 16
)
where each P i is the projection in dimension 1 with potential u i , given by expressions (13) and (14).

In particular, when the set

{(y, ξ) ∈ [0, 1] N × R N ; (θ 1 (y 1 , ξ 1 ), • • • , θ N (y N , ξ N )) satisfy condition (15)}
has zero Lebesgue measure, equality (16) holds almost everywhere. It can then be generalized to arbitrary functions ϕ ∈ L ∞ per (R N × R N ) (always by linearity and density). The correct expression of the projection P is then

P = P 1 • P 2 • • • • • P N , (17)
where each projection P i acts on the variables (y i , ξ i ) only. Notice that all projections P i thus commute with one another; hence the order in which they are taken is unimportant.

We wish to emphasize that on the open set 17) is true. Indeed, the function T i is strictly decreasing on (max u i , +∞), and thus the set

{(y, ξ) ∈ R 2N , ∀i ∈ {1, • • • , N } H i (y i , ξ i ) > max u i }, the expression (
{(E 1 , • • • , E N ) ∈ R N ; E i > max u i and ∃k ∈ Z N \ {0}, k 1 /T 1 (E 1 ) + • • • + k N /T N (E N ) = 0}
is countable. As a consequence, the set {(y, ξ) ∈ R 2N , H i (y i , ξ i ) > max u i ∀i and (θ 1 (y 1 , ξ 1 ), • • • , θ N (y N , ξ N )) satisfy condition (15)} has zero Lebesgue measure.

However, let us mention here that in general, condition (15) cannot be relaxed : indeed, assume for instance that u i = u j := u for i = j and assume that the function u is such that

∃y 0 > 0, u(y) = y 2 for |y| < y 0 , and u(y) > y 2 0 if y ∈ [-1 2 , 1 2 ] \ [-y 0 , y 0 ]. Then if |E| ≤ √ y 0 , we have T (E) = √ E - √ E 1 E -y 2 dy = 2 1 0 1 √ 1 -z 2 dz =: T 0
Thus, if H i (y i , ξ i ) ≤ √ y 0 , then (Y i , Ξ i )(t, y i , ξ i ) is periodic with period T 0 . Notice that T 0 does not depend on the energy H i (y i , ξ i )

In that case, the function ϕ(Y (t), Ξ(t)) is also periodic with period T 0 . Thus we have to compute the limit of 1

T T 0 f 1 (t) • • • f N (t) dt
as T → ∞, where the f i are arbitrary functions with period T 0 . It is then easily proved that

1 T T 0 f 1 (t) • • • f N (t) dt → k ∈ Z N , k 1 + • • • + k N = 0 a 1,k1 • • • a N,kN (18)
where

a j,l = 1 T 0 T0 0 f j (t)e -2ilπt
T 0 dt, 1 ≤ j ≤ N, l ∈ Z.

In general, the right-hand side of (18) differs from a 1,0 • • • a N,0 , and thus

P = P 1 • • • • • P N
for (y, ξ) in a neighbourhood of the origin.

Stationary ergodic setting

In the stationary ergodic setting, some of the expressions or properties above are no longer true. The most significant difference occurs when the energy H(y, ξ) < u max ; indeed, in that case the particle is not necessarily trapped, depending on the profile of the potential u. Hence, in the rest of the subsection, we focus on the case H(y, ξ) > u max . In that case, the movement of the particle is unbounded and has many similarities with the periodic case. In particular, the particle sees "all the potential" during its evolution, and this will be fundamental in the use of the ergodic theorem.

Expression of ξ ♯ (y, ξ, ω)

This paragraph is devoted to the proof of proposition 1.1 in the stationary ergodic setting. We refer for instance to [START_REF] Lions | Correctors for the homogenization theory of Hamilton-Jacobi equations in the stationary ergodic setting[END_REF] for conditions on the existence of correctors for all P ∈ R in the case of a general coercive hamiltonian. In the present case, there exist correctors if {y ∈ R; u(y, ω) = u max } = ∅ a.s. in ω ∈ Ω.

Remark 4.1. We wish to point out that the expressions in the periodic and in the stationary ergodic case when H(y, ξ, ω) > u max are exactly the same (compare proposition 1.1 and the end of paragraph 4.1.1). This expression, and more precisely, the equality ξ ♯ = H′ (P ) for some P , is in fact strongly linked to Aubry-Mather theory. Indeed, and ξ ♯ (y, ξ, ω) is thus (up to a multiplication by -1) the rotation number associated to the Hamiltonian flow starting at (y, ξ). The interested reader should compare our proposition 1.1 to lemma 2.8 in [START_REF]Aubry-Mather theory and periodic solutions of the forced Burger's equation[END_REF] or theorem 4.1 in [START_REF] Evans | Effective Hamiltonians and averaging for Hamiltonian dynamics[END_REF], and our proof to the ones in these articles. We refer to [START_REF]Aubry-Mather theory and periodic solutions of the forced Burger's equation[END_REF][START_REF] Evans | Effective Hamiltonians and averaging for Hamiltonian dynamics[END_REF] for further references to Aubry-Mather theory and its applications to partial differential equations.

Proof of proposition 1.1. In all the proof, we fix y, ξ, ω such that H(y, ξ, ω) > u max , and we set P = P (y, ξ, ω). Let Q ∈ R be arbitrary, and let v be a corrector, i.e.

H(y, Q + ∇ y v(y, ω), ω) = H(Q).

Then according to the theory of viscosity solutions, for all (y, ξ, ω) ∈ R × R × Ω, for all T > 0, v(y, ω) ≤ v(Y (T, y, ξ, ω)) +

Expression of the projection P

The same method as in the periodic case can be used in order to find the expression of the projection P when H(y, ξ, ω) =: E > u max ; indeed, in that case, remember that P (f )(y, ξ, ω) = lim

4. 1 . 1

 11 Expression of ξ ♯ (y, ξ)

  y, ξ) dt = -(Y (t 0 , y, ξ)y) = sgn(ξ).Thus we only have to compute t 0 . With this aim in view, we use the change of variables s = Y (t), with Jacobian ds = Ẏ dt (recall that Ẏ (t, y, ξ) = -sgn(ξ) 2 (H(y, ξ)u(Y (t, y, ξ))) ), in the formula ξ) 2 (H(y, ξ)u(s)) ds = -sgn(ξ) y, ξ)u(s)) dsEventually, we deduce ξ ♯ (y, ξ) = sgn(ξ)ϕ(H(y, ξ)),

ξ 1 T T 0 Ξ

 10 ♯ (y, ξ, ω) = lim T →∞ (t, y, ξ, ω) dt =lim T →∞ Y (T, y, ξ, ω)y T ,

T 0 L

 0 (Y (t, y, ξ, ω), Ξ(t, y, ξ, ω), ω) dt + Q [Y (T, y, ξ, ω)y] + H(Q)T. (19)

1 η 2 (

 12 t, y, ξ, ω), Ξ(t, y, ξ, ω), ω) dt and we can use the change of variablesdt = 1 Ẏ dY = 1 -sgn(ξ) 2(Eu(Y, ω)) dY in order to obtain T 0 f (Y (t, y, ξ, ω), Ξ(t, y, ξ, ω), ω) dt = Y (T,y,ξ,ω) y f z, sgn(ξ) 2(Eu(z, ω)), ω 1 -sgn(ξ) 2(Eu(z, ω))dz.Since the group transformation (τ x ) is ergodic, andY (T ) → ∞ as T → ∞, for all E > u max , 1 Y (T )y Y (T,y,ξ,ω) y f z, sgn(ξ) 2(Eu(z, ω)), ω 1 -sgn(ξ) 2(Eu(z, ω)) dz → → E F sgn(ξ) 2(Eu(0, ω)), ω 1 -sgn(ξ) 2(Eu(0, ω))Thus, we obtainP (f )(y, ξ, ω) = ξ ♯ (y, ξ, ω) f (sgn(ξ), H(y, ξ, ω)),where f (η, E) = E F η 2(Eu(0, ω)), ω Eu(0, ω)) .

Then the particle starting from y with initial speed -ξ reaches either z + or z -in finite time; the speed of the particle is 0 at that moment since

but its acceleration is -u ′ (z ± ) = 0, so the particle turns around and goes back in the reverse direction. It then reaches the other extremity of the interval (z -, z + ) in finite time, and the same phenomena occurs. Hence after a finite time t 0 , the particle is back at its starting point y with the same speed -ξ. Consequently, the movement of the particle is periodic in time with period t 0 . Thus, we have

where t 1 ≥ 0 is arbitrary. It is convenient to choose for t 1 the first time when the particle hits z -. In that case, it is easily seen that t 0 is twice the time it takes to the particle to go from z -to z + , so that

and

Gathering all the terms, we are led to

Expressions ( 13) and ( 14) are compatible with the ones in [START_REF] Frenod | Homogenisation of transport kinetic equations with oscillating potentials[END_REF].

Let us now come back to the case when N > 1, and take a function ϕ(y, ξ)

, where each ϕ i is periodic with period 1. We want to compute the limit

In general, knowing the behavior of each trajectory (Y i , Ξ i ) independently is not enough to compute such a product. However, here, we recall that each function

The period depends only on H i (y i , ξ i ) and on the function u i . More precisely, setting

We can thus use the following result :

such that f i is periodic with period θ i , and set

Then as T → ∞,

Thus there exist constants α, β > 0 depending only on H(y, ξ, ω) and u max , such that

(remember ( 6)).

On the other hand, as T → ∞,

Hence, passing to the limit in (20), we derive

Thus

In order to prove the proposition, we have to find a special Q 0 ∈ R such that equality holds in (19). This will entail that

and the sup is obtained for ξ ♯ (y, ξ, ω) = H′ (Q 0 ).

Let us thus prove that with Q = P = P (y, ξ, ω), equality holds in (19). 2( H(P )u(z, ω)) dz -P y → 0 as y → ∞, a.s. in ω. Thus v satisfies (6), and v ∈ L ∞ (Ω; C 1 (R N )). Thus the method of characteristics, for instance, can be used to prove that equality holds in (19), with (y, ξ) replaced by any couple (y ′ , ξ ′ ) = (y ′ , P + ∇ y v(y ′ , ω)). We have to prove that we can take (y ′ , ξ ′ ) = (y, ξ). First, notice that