Nicolas Crouseilles

Sever Emmanuel Frenod

Adrian Hirstoaga

Emmanuel Fr Énod

Sever A Hirstoaga

Alexandre Mouton

Two-Scale Macro-Micro decomposition of the Vlasov equation with a strong magnetic field

published or not. The documents may come

Introduction

The goal of this paper is to make the first step towards the setting out of a new class of numerical methods: the Two-Scale Asymptotic-Preserving Schemes or TSAPS. We intend to use these methods in order to treat problems involving the following two numerical difficulties: first, dealing efficiently on long time scales with solutions having high frequency oscillations (the Two-Scale approach) and second, an accurate and stable managing of the transition between different regimes, using a unified model (the Asymptotic-Preserving approach). These problems naturally occur in solving a Vlasov equation implying some small parameter, for the modelling of the dynamics of charged particles in the presence of a strong magnetic field. More precisely, we are interested in developing a model whose discretization will be able to simulate both, the regime when the parameter is not small (i.e. the magnetic field is not large), and the limit regime obtained when the parameter is small. The discrete scheme will automatically shift from one regime to the other. In addition, in the limit of the small parameter, the particles mean behaviour will be efficiently described by the Two-Scale limit while the micro behaviour will be expressed by some correctors.

In the framework of a large magnetic field modelization, we refer to [START_REF] Bostan | The Vlasov-Poisson system with strong external magnetic field. Finite Larmor radius regime[END_REF][START_REF] Frénod | Two-dimensional Finite Larmor Radius approximation in canonical gyrokinetic coordinates[END_REF][START_REF] Frénod | The Finite Larmor Radius approximation[END_REF][START_REF] Golse | The Vlasov-Poisson system with strong magnetic field[END_REF][START_REF] Golse | The Vlasov-Poisson system with strong magnetic field in quasineutral regime[END_REF][START_REF] Han-Kwan | The three-dimensional finite Larmor radius approximation[END_REF] and the references therein as previous works about different regimes, obtained by taking limits of Vlasov or Vlasov-Poisson equations corresponding to small parameters. Then, we refer to [START_REF] Ailliot | Long term object drift in the ocean with tide and wind[END_REF][START_REF] Allaire | Homogenization and two-scale convergence[END_REF][START_REF] Frénod | Two-scale numerical simulation of the weakly compressible 1D isentropic Euler equations[END_REF][START_REF] Frénod | Long time simulation of a beam in a periodic focusing channel via a two-scale PIC-method[END_REF][START_REF] Mouton | Two-scale semi-lagrangian simulation of a charged particles beam in a periodic focusing channel[END_REF][START_REF] Mouton | Approximation multi-échelles de l'équation de Vlasov[END_REF][START_REF] Nguetseng | A general convergence result for a functional related to the theory of homogenization[END_REF] for a detailed description of Two-Scale numerical methods. As for the Asymptotic-Preserving schemes, although there is an abundant literature about, we first cite the classical paper [START_REF] Jin | Efficient asymptotic-preserving (AP) schemes for some multiscale kinetic equations[END_REF] and then [START_REF] Bennoune | Uniformly stable numerical schemes for the Boltzmann equation preserving the compressible Navier-Stokes asymptotics[END_REF][START_REF] Crouseilles | An asymptotic preserving scheme based on a micro-macro decomposition for Collisional Vlasov equations: diffusion and high-field scaling limits[END_REF][START_REF] Filbet | A class of asymptotic preserving schemes for kinetic equations and related problems with stiff sources[END_REF][START_REF] Klar | A Numerical Method for Kinetic Semiconductor Equations in the Drift Diffusion Limit[END_REF][START_REF] Lemou | A New Asymptotic Preserving Scheme Based on Micro-Macro Formulation for Linear Kinetic Equations in the Diffusion Limit[END_REF] which treat problems close to ours, without considering the two-scale aspect.

Next we start to detail the method, by setting a generic and formal problem

E ǫ u ǫ = 0 , (1.1) 1
in which E ǫ is a partial differential operator depending on a parameter ǫ which induces oscillations of typical size ǫ in the solution u ǫ = u ǫ (z). In the case where ǫ is uniformly small over the domain on which (1.1) is posed, a Two-Scale approach may give good results. To summarize, this approach consists in noticing that, in some sense,

u ǫ (z) ≈ U (z, z ǫ ) , (1.2)
where U = U (z, ζ) is the Two-Scale limit of u ǫ (Two-Scale convergence goes back to [START_REF] Nguetseng | A general convergence result for a functional related to the theory of homogenization[END_REF]), and that U is the solution of a well-posed problem of the form

E U = 0 , (1.3)
where E does not induce high frequency oscillations. Building a Two-Scale numerical method consists then in solving, via an usual numerical method, model (1.3) and rebuilding u ǫ using an algorithm based on (1.2).

In the case where ǫ is not a uniform parameter over the domain, in the sense that it may be small in some sub-region and of order 1 in other ones, Asymptotic-Preserving schemes are intended to be used, since they allow to simulate both sub-regions with the same method. The methodology for these numerical schemes is based on the fact that u ǫ may converge, in a strong or a weak topology, towards a limit u which is solution of E u = 0 .

(1.4)

Then, an Asymptotic-Preserving scheme consists in solving

E ǫ ∆z u ǫ ∆z = 0 , (1.5)
such that u ǫ ∆z -→ u ǫ , as ∆z → 0, (

and u ǫ ∆z -→ u ∆z , as ǫ → 0, (1.7) where u ∆z is a numerical approximation of u, meaning that u ∆z -→ u , as ∆z → 0, (1.8) and is the solution of

E ∆z u ∆z = 0 , (1.9)
where E ∆z is a numerical operator. Notice that (1.6)-(1.7)-(1.8) is called the Asymptotic-Preserving property. These convergence results may be summarized in the following diagram:

u ǫ solution of E ǫ u ǫ = 0 ǫ → 0 / / u solution of E u = 0 u ǫ ∆z solution of E ǫ ∆z u ǫ ∆z = 0 ǫ → 0 / / ∆z → 0 O O u ∆z solution of E ∆z u ∆z = 0 ∆z → 0 O O (1.10)
In the case when u ǫ contains high frequency oscillations with high amplitude when ǫ is small, solving (1.5) can be time consuming to capture the oscillations. In this case, the convergence u ǫ → u , (1.11) occurs in a weak sense only. Besides this, the Two-Scale limit U = U (z, ζ) may describe very well the oscillations of u ǫ in the sense that the convergence

U (z, z ǫ ) -u ǫ (z) → 0 , (1.12)
occurs in a stronger sense than (1.11). This is particularly the case when E ǫ generates pseudo-periodic oscillations, essentially at a sole period, of size ǫ in the solution u ǫ when ǫ becomes small. Moreover, u and U are linked by a relation of the kind

u(z) = 1 |Z| Z U (z, ζ) dζ , (1.13)
where Z is the cell such that ζ → U (z, ζ) is Z-periodic for any z, and |Z| is its measure.

In this case, we propose to add a layer in the diagram (1.10) which summarizes the ideas on which Two-Scale Asymptotic-Preserving Scheme are based:

u ǫ solution of E ǫ u ǫ = 0 ǫ → 0 , weak- * / / ǫ → 0 , Two-Scale + + X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X u solution of E u = 0 U solution of E U = 0 1 |Z| Z dζ 6 6 n n n n n n n n n n n n U ǫ solution of E ǫ U ǫ = 0 ζ = z ǫ ÀA A A A A A A A A A A A A A A A A A A A A A ǫ → 0 6 6 n n n n n n n n n n n n u ǫ ∆z solution of E ǫ ∆z u ǫ ∆z = 0 ∆z → 0 O O ǫ → 0 / / u ∆z solution of E ∆z u ∆z = 0 ∆z → 0 O O U ∆z solution of E ∆z U ∆z = 0 ∆z → 0 O O 1 |Z| NUM Z dζ 6 6 n n n n n n n n n n n n U ǫ ∆z solution of E ǫ ∆z U ǫ ∆z = 0 ζ = z ǫ ÀA A A A A A A A A A A A A A A A A A A A A A ∆z → 0 O O ǫ → 0 6 6 n n n n n n n n n n n n
(1.14) Diagram (1.14) may be looked at as a prism. At the bottom of the front edge stands the TSAPS and the back rectangle is made of the Asymptotic-Preserving scheme diagram of (1.10). The intermediate layer gathers the Two-Scale limit problem, at the top, and the Two-Scale numerical method, at the bottom.

The solution U ǫ ∆z computed with the TSAPS must be close to u ǫ when it is computed in ζ = z ǫ and when ∆z converges to 0. This is the first part of the Two-Scale Asymptotic-Preserving property and it is symbolized by the arrow going from U ǫ ∆z to u ǫ ∆z and by the one linking u ǫ ∆z and u ǫ . The second part of the Two-Scale Asymptotic-Preserving property is symbolized by the right part of the diagram (1.14) : U ǫ ∆z must be close to the numerical approximation U ∆z of the Two-Scale limit of u ǫ when ǫ → 0 and a numerical integration (symbolized by 1 |Z| NUM Z dζ) of U ∆z needs to recover an approximation u ∆z of the weak- * limit u of u ǫ . We will deduce the Two-Scale Asymptotic-Preserving model on the base of a Two-Scale Macro-Micro decomposition that yields an equation which stands at the top of the front edge. Arrows linking U ǫ and u ǫ , U ǫ and U and u may be seen as the continuous Two-Scale Asymptotic-Preserving property. They are constraints for the Macro-Micro decomposition. We refer to [START_REF] Bennoune | Uniformly stable numerical schemes for the Boltzmann equation preserving the compressible Navier-Stokes asymptotics[END_REF][START_REF] Crouseilles | An asymptotic preserving scheme based on a micro-macro decomposition for Collisional Vlasov equations: diffusion and high-field scaling limits[END_REF][START_REF] Jin | A Micro-Macro Decomposition-Based Asymptotic-Preserving Scheme for the Multispecies Boltzmann Equation[END_REF][START_REF] Lemou | A New Asymptotic Preserving Scheme Based on Micro-Macro Formulation for Linear Kinetic Equations in the Diffusion Limit[END_REF] for the use of Macro-Micro decomposition in order to design an Asymptotic-Preserving scheme.

The present paper implements the Two-Scale Macro-Micro decomposition in the case of the linear Vlasov equation with a strong magnetic field. This equation is the following:

       ∂f ǫ ∂t + v • ∇ x f ǫ + E ǫ + v × B ǫ + M ǫ • ∇ v f ǫ = 0 , f ǫ (t = 0, x, v) = f 0 (x, v) , (1.15)
where f ǫ = f ǫ (t, x, v) is the distribution function of an ion gaz submitted to external electric and magnetic field

E ǫ = E ǫ (t, x) and B ǫ = B ǫ (t, x), where t ∈ R + , x ∈ R 3 , v ∈ R 3
, and where M = 2π e 1 , i.e. the first vector of the canonical base (e 1 , e 2 , e 3 ) of R 3 .

The paper is organized as follows: in Section 2, we recall the results of Frénod & Sonnendrücker [START_REF] Frénod | Homogenization of the Vlasov equation and of the Vlasov-Poisson system with a strong external magnetic field[END_REF] giving the Two-Scale convergence of f ǫ . Roughly speaking they claim that, when ǫ is small,

f ǫ (t, x, v) is close to G t, x, ⊔ ⊔ ⊔( t ǫ , v) in a strong sense, (1.16
)

f ǫ (t, x, v) is close to f (t, x, v) in a weak sense, (1.17)
and the equations for G and f are known (⊔ ⊔ ⊔(τ, v) is defined by (2.9)). Then we show that the v-

dependence of f is only through v || = v • e 1 , v ⊥ = (v • e 2 ) 2 + (v • e 3 ) 2
and not through the angular velocity component α, which is such that v • e 2 = v ⊥ cos α. Based on this fact, in Section 3, we build a classical Macro-Micro decomposition of (1.15) by following the same ideas as those of Lemou & Mieussens [START_REF] Lemou | A New Asymptotic Preserving Scheme Based on Micro-Macro Formulation for Linear Kinetic Equations in the Diffusion Limit[END_REF]. It consists in decomposing f ǫ in the following way

f ǫ t, x, v e 1 + v ⊥ (cos α e 2 + sin α e 3 ) = m ǫ (t, x, v || , v ⊥ ) + n ǫ (t, x, v || , v ⊥ , α) (1.18)
for some functions m ǫ and n ǫ . Indeed, we will see in details that this decomposition is unique for every ǫ, small or not. Note that m ǫ does not depend on α, as f t, x, v e 1 +v ⊥ (cos α e 2 +sin α e 3 ) , while n ǫ does. Equations for m ǫ and n ǫ are set out, making up the Macro-Micro model. The drawback of this model is that, because of (1.17), the exact equality (1.18) implies that function n ǫ is still highly oscillating, as f ǫ . To avoid this drawback, we develop the Two-Scale Macro-Micro framework. This is done in Sections 4, 5, and 6, where we build the Two-Scale Macro-Micro approximation of f ǫ . First, based on results of Frénod, Raviart & Sonnendrücker [START_REF] Frénod | Two-scale expansion of a singularly perturbed convection equation[END_REF] we improve (1.16) by using a first order approximation, claiming that when ǫ is small

f ǫ (t, x, v) is close to G t, x, ⊔ ⊔ ⊔( t ǫ , v) + ǫ G 1 t, x, ⊔ ⊔ ⊔( t ǫ , v) + ǫ l t, t ǫ , x, v) in a strong sense, (1.19)
where G 1 is the solution of a partial differential equation and l is given by a formula (see [START_REF] Frénod | Two-scale expansion of a singularly perturbed convection equation[END_REF]). Inspired by (1.19), we construct our Two-Scale Macro-Micro decomposition in the following way

f ǫ (t, x, v) = G t, x, ⊔ ⊔ ⊔( t ǫ , v) + ǫ G ǫ 1 t, x, ⊔ ⊔ ⊔( t ǫ , v) + ǫ l(t, t ǫ , x, v) + ǫ h ǫ (t, t ǫ , x, v), (1.20)
where G ǫ 1 is intended to be close to G 1 when ǫ is small and h ǫ is the corrector to be taken into account when the order of magnitude of ǫ is 1. In other words, we use the Macro-Micro decomposition in order to make precise for every ǫ the approximation in (1.19) which is valid only for small ǫ. Then, we need to show that the decomposition in (1.20) is unique. This is done by paying attention to the properties of the operator

∂ ∂τ + (v × M) • ∇ v , (1.21)
since the expression (1.20) may also be set in the following form:

f ǫ (t, x, v) = A ǫ (t, t ǫ , x, v) + B ǫ (t, t ǫ , x, v) , (1.22)
with

A ǫ = A ǫ (t, τ, x, v) ∈ Ker ∂ ∂τ +(v×M)•∇ v and B ǫ = B ǫ (t, τ, x, v) ∈ Ker ∂ ∂τ +(v×M)•∇ v ⊥ . (1.23)
Then, we obtain the equations for the Macro part G ǫ 1 and the Micro one h ǫ . In Section 7, we study the asymptotic behaviour of the Two-Scale Macro-Micro decomposed model as ǫ → 0. We close the paper with two appendices about some properties of the operator in (1.21) leading mainly to the uniqueness of the decomposition in (1.22).

Two-Scale convergence of f ǫ

In this section, we briefly recall the results of Frénod & Sonnendrücker [START_REF] Frénod | Homogenization of the Vlasov equation and of the Vlasov-Poisson system with a strong external magnetic field[END_REF] where the asymptotic behaviour of (1.15) is studied. First, under the following hypotheses

f 0 ≥ 0 , f 0 ∈ L 2 (R 6 ) , (2.1)
and

E ǫ → E strong in L ∞ 0, T ; L 2 loc (R 3 ) , B ǫ → B strong in L ∞ 0, T ; L 2 loc (R 3 ) , (2.2)
we have that, for any T > 0, (f ǫ ) ǫ > 0 is bounded in L ∞ 0, T ; L 2 (R 6 ) . Second, as ǫ → 0, up to a subsequence, (f ǫ ) ǫ > 0 Two-Scale converges towards (see [START_REF] Allaire | Homogenization and two-scale convergence[END_REF][START_REF] Nguetseng | A general convergence result for a functional related to the theory of homogenization[END_REF])

F = F (t, τ, x, v) ∈ L ∞ 0, T ; L ∞ #1 (R + ; L 2 (R 6 )) , (2.3)
where the subscript # 1 indicates that τ → F (t, τ, x, v) is 1-periodic for any (t, x, v). Introduced in [START_REF] Nguetseng | A general convergence result for a functional related to the theory of homogenization[END_REF] and further analyzed and used in several homogenization problems in [START_REF] Allaire | Homogenization and two-scale convergence[END_REF], the Two-Scale convergence means that for any regular function ψ = ψ(t, τ, x, v) such that (t, x, v) → ψ(t, τ, x, v) is compactly supported in Ω = [0, T ) × R 6 for any τ , and τ → ψ(t, τ, x, v) is 1-periodic for any (t, x, v), we have lim

ǫ → 0 Ω f ǫ (ψ) ǫ dt dx dv = Ω 1 0 F ψ dτ dt dx dv , (2.4)
where (ψ

) ǫ = (ψ) ǫ (t, x, v) = ψ(t, t ǫ , x, v) . (2.5)
It was shown in [START_REF] Frénod | Homogenization of the Vlasov equation and of the Vlasov-Poisson system with a strong external magnetic field[END_REF] that the Two-Scale limit F has the following property:

F ∈ Ker ∂ ∂τ + (v × M) • ∇ v , (2.6)
where ∂ ∂τ +(v×M)•∇ v is seen as anti-symmetric, non-bounded, and with open range from L ∞ 0, T ; L ∞ #1 (R + ; L 2 (R 6 )) on itself. The direct consequence of (2.6) is that, introducing the rotation matrix r as

r(τ ) =   1 0 0 0 cos (2πτ ) -sin (2πτ ) 0 sin (2πτ ) cos (2πτ )   , (2.7)
the Two-Scale limit F expresses itself in terms of a function G = G(t, x, u) ∈ L ∞ 0, T ; L 2 (R 3 ) by

F (t, τ, x, v) = G t, x, ⊔ ⊔ ⊔(τ, v) , (2.8)
where

⊔ ⊔ ⊔(τ, v) = r(τ ) v .
(2.9)

Using oscillating test functions defined by (2.5) for regular functions ψ which are moreover in Ker

∂ ∂τ + (v × M) • ∇ v , it may be brought out that G is the solution of    ∂G ∂t + u || • ∇ x G + (E || + u × B || ) • ∇ u G = 0 , G(t = 0, x, u) = f 0 (x, u) , (2.10)
where we have used the notation u || = (u • e 1 ) e 1 .

Having in mind diagram (1.14), we can say at this point that equation (1.15) plays the role of E ǫ u ǫ = 0, and (2.8)-(2.10) plays the role of E U = 0. So, if we build a numerical method to approximate (2.10), this numerical method, coupled with (2.8) would play the role of E ∆z U ∆z = 0.

In [START_REF] Frénod | Homogenization of the Vlasov equation and of the Vlasov-Poisson system with a strong external magnetic field[END_REF], it was also proved that f , the weak- * limit of (f ǫ ) ǫ>0 , is the solution to

       ∂f ∂t + v || • ∇ x f + (E || + v × B || ) • ∇ v f = 0 , f (t = 0, x, v) = 1 0 f 0 x, ⊔ ⊔ ⊔(τ, v) dτ . (2.11)
This equation is obtained from (2.8)-(2.10) by performing an integration in τ and by using the link between the weak- * limit f and the Two-Scale limit F given by

f (t, x, v) = 1 0 F (t, τ, x, v) dτ . (2.12)
Then, equation (2.11) plays the role of E u = 0 in diagram (1.14) and any of its numerical approximations would play the role of E ∆z u ∆z = 0.

3 Classical Macro-Micro decomposition

Rewriting the weak- * limit model

Before going further, we will rewrite model (2.11) in a form involving cylindrical variables in velocity. In this coordinate system, the weak- * limit is independent of the angle variable, and consequently it is easily gotten that the v × B || piece in the force term in (2.11) is removed.

Because of the average over τ and of the definition of ⊔ ⊔ ⊔(τ, v), it is obvious to see that the initial data in (2.11)

only depends on x, v || and v ⊥ = v 2 2 + v 2 3 .
In other words, writing the velocity in the following cylindrical variables

   v || ∈ R such that v || = v • e 1 , v ⊥ ∈ R + such that v ⊥ = v 2 2 + v 2 3 , α ∈ [0, 2π] such that v 2 = v ⊥ cos α and v 3 = v ⊥ sin α, (3.1) there exists a function m 0 = m 0 (x, v || , v ⊥ ), independent of α, such that m 0 (x, v || , v ⊥ ) = 1 2π 2π 0 f 0 x, ⊔ ⊔ ⊔(τ, v || e 1 + v ⊥ cos α e 2 + v ⊥ sin α e 3 ) dα . (3.2)
The quickest way to get this is to notice that

⊔ ⊔ ⊔(τ, v || e 1 + v ⊥ cos α e 2 + v ⊥ sin α e 3 ) =   v || v ⊥ cos(α + 2πτ ) v ⊥ sin(α + 2πτ )   , (3.3)
and that, consequently, integrating in τ suppresses the α-dependency. Hence, (3.2) writes also

m 0 (x, v || , v ⊥ ) = 1 2π 2π 0 f 0 (x, v || e 1 + v ⊥ cos α e 2 + v ⊥ sin α e 3 ) dα . (3.4) Concerning the function f itself, we introduce the function m = m(t, x, v || , v ⊥ , α) linked with f by m(t, x, v || , v ⊥ , α) = f (t, x, v || e 1 + v ⊥ cos α e 2 + v ⊥ sin α e 3 ) . (3.5)
Because of (2.12) and (2.8), and using the same argument as just above, it is easily seen that m does not depend on α and that

m(t, x, v || , v ⊥ ) = 1 0 G(t, x, v || e 1 + v ⊥ cos(α + 2πτ ) e 2 + v ⊥ sin(α + 2πτ ) e 3 ) dτ = 1 2π 2π 0 G(t, x, v || e 1 + v ⊥ cos α e 2 + v ⊥ sin α e 3 ) dα . (3.6)
Noticing that

∂m ∂v ⊥ (t, x, v || , v ⊥ ) = cos α ∂f ∂v 2 (t, x, v || e 1 + v ⊥ cos α e 2 + v ⊥ sin α e 3 ) + sin α ∂f ∂v 3 (t, x, v || e 1 + v ⊥ cos α e 2 + v ⊥ sin α e 3 ) , ∂m ∂α (t, x, v || , v ⊥ ) = -v ⊥ sin α ∂f ∂v 2 (t, x, v || e 1 + v ⊥ cos α e 2 + v ⊥ sin α e 3 ) + v ⊥ cos α ∂f ∂v 3 (t, x, v || e 1 + v ⊥ cos α e 2 + v ⊥ sin α e 3 ) , (3.7)
the equation for m can be deduced from (2.11). For this purpose, we develop v × B || in the new variables:

v ⊥ (cos α e 2 + sin α e 3 ) × B || = |B || | v ⊥ (sin α e 2 -cos α e 3 ) . (3.8)
As a consequence, we have

v ⊥ (cos α e 2 + sin α e 3 ) × B || • ∇ v f (t, x, v || e 1 + v ⊥ cos α e 2 + v ⊥ sin α e 3 ) = -|B || | ∂m ∂α (t, x, v || , v ⊥ ) = 0 , (3.9)
meaning that the force term in (2.11) may be reduced to

E || . Then, m is the solution of    ∂m ∂t + v || ∂m ∂x || + E || ∂m ∂v || = 0 , m(t = 0, x, v || , v ⊥ ) = m 0 (x, v || , v ⊥ ) .
(3.10)

Macro-Micro decomposition

Inspired by (2.11) or more precisely by (3.10), we now write a classical Macro-Micro decomposition of the solution f ǫ of (1.15). The aim of this decomposition is (see [START_REF] Bennoune | Uniformly stable numerical schemes for the Boltzmann equation preserving the compressible Navier-Stokes asymptotics[END_REF][START_REF] Lemou | A New Asymptotic Preserving Scheme Based on Micro-Macro Formulation for Linear Kinetic Equations in the Diffusion Limit[END_REF]) to build a new model which is an equivalent reformulation of (1.15) for all ǫ > 0 and which straightforwardly gives the weak- * limit model (3.10) when ǫ → 0. Thus, this new formulation of problem (1.15) by using the Macro-Micro decomposition is expected to be well adapted to the building of an Asymptotic-Preserving scheme.

The main idea of the Macro-Micro decomposition is to write f ǫ as a sum of a function being in Ker (v × M) • ∇ v and a function being in Ker (v × M) • ∇ v ⊥ . Adapting this idea to our context and having in mind (3.10), we seek

m ǫ 1 = m ǫ 1 (t, x, v || , v ⊥ ) ∈ L ∞ 0, T ; L 2 (R 3 × R × R + ; v ⊥ dx dv || dv ⊥ ) and p ǫ = p ǫ (t, x, v || , v ⊥ , α) ∈ L ∞ 0, T ; L 2 #2π (R; L 2 (R 3 × R × R + ; v ⊥ dx dv || dv ⊥ )) such that n ǫ (t, x, v || , v ⊥ , α) = ∂p ǫ ∂α (t, x, v || , v ⊥ , α) , (3.11)
and

f ǫ (t, x, v || e 1 + v ⊥ cos α e 2 + v ⊥ sin α e 3 ) = m(t, x, v || , v ⊥ ) + m ǫ 1 (t, x, v || , v ⊥ ) + n ǫ (t, x, v || , v ⊥ , α). (3.12) The subscript # 2π indicates that α → p ǫ (t, x, v || , v ⊥ , α) is 2π-periodic for any (t, x, v || , v ⊥ ).
We have proved in the previous section that m ∈ Ker ∂ ∂α . In addition, considering the antisymmetric, non-bounded operator with closed range

1 ∂ ∂α : L ∞ 0, T ; L 2 #2π (R; L 2 (R 3 × R × R + ; v ⊥ dx dv || dv ⊥ )) -→ L ∞ 0, T ; L 2 #2π (R; L 2 (R 3 × R × R + ; v ⊥ dx dv || dv ⊥ )) , (3.13)
we clearly have that

m ǫ 1 ∈ Ker ∂ ∂α and n ǫ ∈ Ker ∂ ∂α ⊥ = Im ∂ ∂α (3.14)
exist and are unique. Then, if we inject the decomposition (3.12) in Vlasov equation (1.15), we obtain

∂m ∂t + ∂m ǫ 1 ∂t + ∂n ǫ ∂t +   v || v ⊥ cos α v ⊥ sin α   • ∇ x (m + m ǫ 1 + n ǫ ) +     E ǫ || + v ⊥ (cos α B ǫ 3 -sin α B ǫ 2 ) cos α E ǫ 2 + sin α E ǫ 3 -v || (cos α B ǫ 3 -sin α B ǫ 2 ) 1 v ⊥ -sin α E ǫ 2 + cos α E ǫ 3 + v || (sin α B ǫ 3 + cos α B ǫ 2 ) -B ǫ || - 1 ǫ     •         ∂m ∂v || + ∂m ǫ 1 ∂v || + ∂n ǫ ∂v || ∂m ∂v ⊥ + ∂m ǫ 1 ∂v ⊥ + ∂n ǫ ∂v ⊥ ∂n ǫ ∂α         = 0 . (3.15) 1 See Appendix A.
Combining this equation with (3.10), we finally obtain

∂m ǫ 1 ∂t + ∂n ǫ ∂t +   0 v ⊥ cos α v ⊥ sin α   • ∇ x m +   v || v ⊥ cos α v ⊥ sin α   • ∇ x (m ǫ 1 + n ǫ ) +     E ǫ || -E || + v ⊥ (cos α B ǫ 3 -sin α B ǫ 2 ) cos α E ǫ 2 + sin α E ǫ 3 -v || (cos α B ǫ 3 -sin α B ǫ 2 ) 1 v ⊥ -sin α E ǫ 2 + cos α E ǫ 3 + v || (sin α B ǫ 3 + cos α B ǫ 2 ) -B ǫ || - 1 ǫ     •       ∂m ∂v || ∂m ∂v ⊥ 0       +     E ǫ || + v ⊥ (cos α B ǫ 3 -sin α B ǫ 2 ) cos α E ǫ 2 + sin α E ǫ 3 -v || (cos α B ǫ 3 -sin α B ǫ 2 ) 1 v ⊥ -sin α E ǫ 2 + cos α E ǫ 3 + v || (sin α B ǫ 3 + cos α B ǫ 2 ) -B ǫ || - 1 ǫ     •         ∂m ǫ 1 ∂v || + ∂n ǫ ∂v || ∂m ǫ 1 ∂v ⊥ + ∂n ǫ ∂v ⊥ ∂n ǫ ∂α         = 0 . (3.16)
By projecting (3.15) onto Ker( ∂ ∂α ) and Ker( ∂ ∂α ) ⊥ = Im( ∂ ∂α ), we will obtain the Macro-Micro decomposition of f ǫ . In order to do this, we use the fact that (see Appendix A and [START_REF] Bostan | Transport equations with disparate advection fields. Application to the gyrokinetic models in plasma physics[END_REF]) projecting a function onto Ker( ∂ ∂α ) consists in computing its average in α and projecting it onto Im( ∂ ∂α ) consists in substracting from it its average value. This is what we do in the next lines.

Since m, m ǫ 1 , E, E ǫ , B and B ǫ do not depend on α, and recalling the definition of n ǫ given in (3.11), the projection of (3.15) onto Ker( ∂ ∂α ) gives

∂m ǫ 1 ∂t + v || ∂m ǫ 1 ∂x || + E ǫ || ∂m ǫ 1 ∂v || = -(E ǫ || -E || ) ∂m ∂v || - v ⊥ 2π   0 1 1   •       0 2π 0 sin α ∂p ǫ ∂x 2 dα - 2π 0 cos α ∂p ǫ ∂x 3 dα       - 1 2π E ǫ •       0 2π 0 sin α ∂p ǫ ∂v ⊥ + p ǫ v ⊥ dα - 2π 0 cos α ∂p ǫ ∂v ⊥ + p ǫ v ⊥ dα       + 1 2π   -v ⊥ v B ǫ 3 v B ǫ 2   •         2π 0 (cos α B ǫ 2 + sin α B ǫ 3 ) ∂p ǫ ∂v || dα 2π 0 sin α ∂p ǫ ∂v ⊥ + p ǫ v ⊥ dα 2π 0 cos α ∂p ǫ ∂v ⊥ + p ǫ v ⊥ dα         . (3.17)
In the second, third and last terms of the right hand side of (3.17), we use integrations by parts to tackle α-integrations. Finally, we perform the projection onto Im( ∂ ∂α ) by substracting (3.17) from (3.15), we obtain a second order equation for p ǫ in which m and m ǫ 1 appear as right hand side members:

∂ 2 p ǫ ∂t ∂α +   v || v ⊥ v ⊥   •         ∂ 2 p ǫ ∂x || ∂α cos α ∂ 2 p ǫ ∂x 2 ∂α - 1 2π 2π 0 cos α ∂ 2 p ǫ ∂x 2 ∂α dα sin α ∂ 2 p ǫ ∂x 3 ∂α - 1 2π 2π 0 sin α ∂ 2 p ǫ ∂x 3 ∂α dα         + E ǫ •         ∂ 2 p ǫ ∂v || ∂α cos α ∂ 2 p ǫ ∂v ⊥ ∂α - 1 2π 2π 0 cos α ∂ 2 p ǫ ∂v ⊥ ∂α dα - 1 v ⊥ sin α ∂ 2 p ǫ ∂α 2 + 1 2π v ⊥ 2π 0 sin α ∂ 2 p ǫ ∂α 2 dα sin α ∂ 2 p ǫ ∂v ⊥ ∂α - 1 2π 2π 0 sin α ∂ 2 p ǫ ∂v ⊥ ∂α dα + 1 v ⊥ cos α ∂ 2 p ǫ ∂α 2 - 1 2π v ⊥ 2π 0 cos α ∂ 2 p ǫ ∂α 2 dα         +   v ⊥ -v || B ǫ 3 v || B ǫ 2   •         (cos α B ǫ 3 -sin α B ǫ 2 ) ∂ 2 p ǫ ∂v || ∂α - 1 2π 2π 0 (cos α B ǫ 3 -sin α B ǫ 2 ) ∂ 2 p ǫ ∂v || ∂α dα cos α ∂ 2 p ǫ ∂v ⊥ ∂α - 1 v ⊥ sin α ∂ 2 p ǫ ∂α 2 - 1 2π 2π 0 cos α ∂ 2 p ǫ ∂v ⊥ ∂α - 1 v ⊥ sin α ∂ 2 p ǫ ∂α 2 dα sin α ∂ 2 p ǫ ∂v ⊥ ∂α + 1 v ⊥ cos α ∂ 2 p ǫ ∂α 2 - 1 2π 2π 0 sin α ∂ 2 p ǫ ∂v ⊥ ∂α + 1 v ⊥ cos α ∂ 2 p ǫ ∂α 2 dα         -(B ǫ || + 1 ǫ ) ∂ 2 p ǫ ∂α 2 = -   0 v ⊥ cos α v ⊥ sin α   • ∇ x m -E ǫ •       0 cos α ∂m ∂v ⊥ sin α ∂m ∂v ⊥       -   v ⊥ (cos α B ǫ 3 -sin α B ǫ 2 ) -v || B ǫ 3 v || B ǫ 2   •         ∂m ∂v || cos α ∂m ∂v ⊥ sin α ∂m ∂v ⊥         -   0 v ⊥ cos α v ⊥ sin α   • ∇ x m ǫ 1 -E ǫ •       0 cos α ∂m ǫ 1 ∂v ⊥ sin α ∂m ǫ 1 ∂v ⊥       -   v ⊥ (cos α B ǫ 3 -sin α B ǫ 2 ) -v || B ǫ 3 v || B ǫ 2   •         ∂m ǫ 1 ∂v || cos α ∂m ǫ 1 ∂v ⊥ sin α ∂m ǫ 1 ∂v ⊥         . (3.18)
Concerning the initial conditions for m ǫ 1 and p ǫ , we invoke the initial distribution f 0 as follows:

f 0 (x, v || e 1 + v ⊥ cos α e 2 + v ⊥ sin α e 3 ) = 1 2π 2π 0 f 0 (x, v || e 1 + v ⊥ cos θ e 2 + v ⊥ sin θ e 3 ) dθ + f 0 (x, v || e 1 + v ⊥ cos α e 2 + v ⊥ sin α e 3 ) - 1 2π 2π 0 f 0 (x, v || e 1 + v ⊥ cos θ e 2 + v ⊥ sin θ e 3 ) dθ = m 0 (x, v || , v ⊥ ) + f 0 (x, v || e 1 + v ⊥ cos α e 2 + v ⊥ sin α e 3 ) -m 0 (x, v || , v ⊥ ) . (3.19)
Since m |t = 0 = m 0 , we have

m ǫ 1 (t = 0, x, v || ,v ⊥ ) + ∂p ǫ ∂α (t = 0, x, v || , v ⊥ , α) = f 0 (x, v || e 1 + v ⊥ cos α e 2 + v ⊥ sin α e 3 ) -m 0 (x, v || , v ⊥ ).
(3.20)

By integrating this equation in α over [0, 2π], we find

m ǫ 1 (t = 0, x, v || , v ⊥ ) = 0. (3.21)
Then we have

p ǫ (t = 0, x, v || , v ⊥ , α) = α 0 f 0 (x, v || e 1 + v ⊥ cos θ e 2 + v ⊥ sin θ e 3 ) dθ -α m 0 (x, v || , v ⊥ ) . (3.22)
Remark 3.1. On the one hand, we mention that we have existence and uniqueness for solutions of the classical Macro-Micro model (3.10), (3.17), (3.18), (3.21), (3.22). Indeed, the existence was proved by construction in the previous sections; more precisely, function m exists as the weak- * limit of (f ǫ ) ǫ and the existence of functions m ǫ 1 and n ǫ is guaranteed by Appendix A. Then, the linear transport problem (3.10) clearly has a unique solution. As for the linear system (3.17), (3.18), by classical results in linear partial differential equations, we can prove that the solution of (3.17 A time discretization of this reformulation can be derived following [START_REF] Bennoune | Uniformly stable numerical schemes for the Boltzmann equation preserving the compressible Navier-Stokes asymptotics[END_REF][START_REF] Crouseilles | An asymptotic preserving scheme based on a micro-macro decomposition for Collisional Vlasov equations: diffusion and high-field scaling limits[END_REF][START_REF] Lemou | A New Asymptotic Preserving Scheme Based on Micro-Macro Formulation for Linear Kinetic Equations in the Diffusion Limit[END_REF]. It is based on a semiimplicit discretization in which the stiffest term in equation (3.18) is considered implicit to ensure stability as ǫ goes to 0. The so-obtained numerical scheme enjoys the Asymptotic-Preserving property since it is consistent with (1.15) when ǫ > 0 and consistent with its weak- * limit (2.11) when ǫ goes to 0. This scheme would play the role of problem E ǫ ∆z u ǫ ∆z = 0 in the diagrams shown in (1.10) and (1.14).

Two-Scale Macro-Micro decomposition: preliminaries

The aim exposed in the first paragraph of Section 3.2 will now be changed in order to obtain the Two-Scale limit model (2.10) when ǫ → 0. Eventually, we will see in Section 7.2, that the Two-Scale Macro-Micro decomposition described in the following sections, will lead by integration in τ to the classical Macro-Micro decomposition.

From now on, we assume that the input electric field and the input magnetic field do not depend on ǫ, i.e. we take

E ǫ = E , B ǫ = B . (4.1)

Complements on convergence results

Before entering in the core of the Two-Scale Macro-Micro decomposition, we integrate some knowledge which can be deduced from Frénod, Raviart & Sonnendrücker [START_REF] Frénod | Two-scale expansion of a singularly perturbed convection equation[END_REF] where the asymptotic expansion of f ǫ is presented. Using hypotheses (2.1) and (2.2) and adding some regularity assumptions for E and B, we can claim that

1 ǫ f ǫ (t, x, v) -F (t, t ǫ , x, v) Two-Scale converges to F (t, τ, x, v) as ǫ → 0, (4.2)
and that F expresses in terms of other functions G and l in the following way:

F (t, τ, x, v) = G t, x, ⊔ ⊔ ⊔(τ, v) + l(t, τ, x, v) , (4.3)
where

l(t, τ, x, v) = r(τ + 1 4 ) -r( 1 4 ) v • ∇ x ⊥ G t, x, ⊔ ⊔ ⊔(τ, v) + r(τ + 1 4 ) -r( 1 4
)

E(t, x) + ⊔ ⊔ ⊔(τ, v) × r(τ + 1 4 ) -r( 1 4
) B(t, x)

• ∇ u ⊥ G t, x, ⊔ ⊔ ⊔(τ, v) , (4.4)
and where G satisfies an equation of the form

∂ G ∂t + u || ∂ G ∂x || + E || + u • B || • ∇ v G = RHS(t, x, u, E, B, G) , (4.5)
where the right hand side can be explicitly computed as in Theorem 4.2 of [START_REF] Frénod | Two-scale expansion of a singularly perturbed convection equation[END_REF]. In these equations, G is the solution of (2.10), r is defined in (2.7), and the gradients ∇ x ⊥ and ∇ u ⊥ are defined by

∇ x ⊥ =       0 ∂ ∂x 2 ∂ ∂x 3       , ∇ u ⊥ =       0 ∂ ∂u 2 ∂ ∂u 3       . (4.6)

Sought shape of f ǫ

As announced in the introduction, we want to use a shape given by (1.22)-(1.23) for f ǫ . We clearly have that operator

∂ ∂τ + (v × M) • ∇ v : L ∞ 0, T ; L ∞ #1 (R + ; L 2 (R 6 )) → L ∞ 0, T ; L ∞ #1 (R + ; L 2 (R 6 )) , (4.7)
is anti-symmetric, non bounded and with closed range (see Appendix B) and satisfies

Ker ∂ ∂τ + (v × M) • ∇ v ⊕ Im ∂ ∂τ + (v × M) • ∇ v = L ∞ 0, T ; L ∞ #1 (R + ; L 2 (R 6 )) . (4.8)
On another hand, we have

F ∈ Ker ∂ ∂τ + (v × M) • ∇ v . (4.9)
Furthermore, we can easily prove that the function l defined in (4.4) is such that

l ∈ Im ∂ ∂τ + (v × M) • ∇ v . (4.10)
Indeed, any regular function in Ker

∂ ∂τ +(v×M)•∇ v reads γ(t, x, ⊔ ⊔ ⊔(τ, v)) for some γ ∈ L ∞ 0, T ; L 2 (R 6 ) . Multiplying l(t, τ, x, v) by γ(t, x, ⊔ ⊔ ⊔(τ, v)), integrating over R 6 × [0, 1] and performing the change of vari- ables v → u = ⊔ ⊔ ⊔(τ, v) , (4.11)
we obtain

R 6 γ(t, x, u) 1 0 r(τ - 1 4 ) -r( 1 4
)

dτ u • ∇ x ⊥ G(t, x, u) - 1 0 r(τ - 1 4 ) -r( 1 4
) dτ E(t, x)

+ u × 1 0 r(τ - 1 4 ) -r( 1 4 ) dτ B(t, x) • ∇ u ⊥ G(t, x, u) dx du = 0 , (4.12)
because the integrals over [0, 1] all vanish. Now, we integrate every remarks we just did. We also notice that a decomposition of the type (1.22) cannot be unique because of choices concerning the place where oscillations are put: in the superscript ǫ or in variable t ǫ . This will also prescribe what we set now: we look for

G ǫ 1 = G ǫ 1 (t, x, u) ∈ L ∞ 0, T ; L 2 (R 6 ) , k ǫ = k ǫ (t, τ, x, v) ∈ L ∞ 0, T ; L ∞ #1 (R + ; L 2 (R 6 )) , (4.13)
such that, for any τ ,

f ǫ (t, x, v) = G t, x, ⊔ ⊔ ⊔(τ, v) + ǫ G ǫ 1 t, x, ⊔ ⊔ ⊔(τ, v) + ǫ l(t, τ, x, v) + ǫ ∂k ǫ ∂τ (t, τ, x, v) + (v × M) • ∇ v k ǫ (t, τ, x, v) . (4.14)
By property (4.8) we know that G ǫ 1 and k ǫ exist and are unique. From now on, we shall denote

(G • ⊔ ⊔ ⊔)(t, τ, x, v) = G t, x, ⊔ ⊔ ⊔(τ, v) , (4.15)
leading, for instance, to the writing of

F = G • ⊔ ⊔ ⊔.
With this notation, we can claim that

(G • ⊔ ⊔ ⊔) + ǫ (G ǫ 1 • ⊔ ⊔ ⊔) ∈ Ker ∂ ∂τ + (v × M) • ∇ v , ǫ l + ǫ ∂k ǫ ∂τ + (v × M) • ∇ v k ǫ ∈ Im ∂ ∂τ + (v × M) • ∇ v , (4.16)
and, as a consequence of (4.8), decomposition (4.14) exists and is unique since G and l have already been identified as the unique couple satisfying (2.10) and (4.4).

In the previous section, when building the usual Macro-Micro decomposition, we projected the equation satisfied by f ǫ onto the kernel and the range of the involved operator. Here, this cannot be done. Indeed, here, the involved operator is ∂ ∂τ + (v × M) • ∇ v and the function which has to be used in equation (1.15) in order to extract some informations on G ǫ 1 and k ǫ is

f ǫ (t, x, v) = (G • ⊔ ⊔ ⊔)(t, t ǫ , x, v) + ǫ (G ǫ 1 • ⊔ ⊔ ⊔)(t, t ǫ , x, v) + ǫ l(t, t ǫ , x, v) + ǫ ∂k ǫ ∂τ (t, t ǫ , x, v) + (v × M) • ∇ v k ǫ (t, t ǫ , x, v) , (4.17)
which corresponds to (4.14) with τ = t ǫ .

The pathway towards Two-Scale Macro and Micro equations

In this section we summarize the way we follow to obtain the equations for G ǫ 1 and k ǫ . They make up the Two-Scale Macro and Micro equations. Once τ was replaced by t ǫ in (4.14), we face with functional spaces where operator (4.7) makes no sense. In order to bypass this difficulty, we use a weak formulation of (1.15) with oscillating test functions

(ψ) ǫ = (ψ) ǫ (t, x, v) = ψ(t, t ǫ , x, v), (4.18)
where ψ = ψ(t, τ, x, v) is regular, 1-periodic in τ and with compact support on [0,

T ) × R 3 × R 3 for any fixed τ ∈ [0, 1]. Writing (F ǫ 1 ) ǫ = (F ǫ 1 ) ǫ (t, x, v) = F ǫ 1 (t, t ǫ , x, v) = (G ǫ 1 • ⊔ ⊔ ⊔)(t, t ǫ , x, v) , (4.19)
and using the same convention for other functions, the weak formulation with oscillating test functions reads

T 0 R 6 (F ) ǫ + ǫ (F ǫ 1 ) ǫ + ǫ (l) ǫ + ǫ(h ǫ ) ǫ ∂ψ ∂t ǫ + 1 ǫ ∂ψ ∂τ ǫ + v • (∇ x ψ) ǫ + E + v × (B + M ǫ ) • (∇ v ψ) ǫ dx dv dt = - R 6 f 0 (x, v) ψ(0, 0, x, v) dx dv , (4.20)
where F is linked with G by (2.8), F ǫ 1 with G ǫ 1 by (4.19), l is defined in (4.4) and h ǫ is linked with k ǫ by

h ǫ = ∂k ǫ ∂τ + (v × M) • ∇ v k ǫ . (4.21)
The way to deduce the equation for G ǫ 1 and for k ǫ consists in using, in a first computation, test functions

ψ ∈ Ker ∂ ∂τ + (v × M) • ∇ v , i.e. of the form ψ = (γ • ⊔ ⊔ ⊔) , (4.22)
for regular functions γ = γ(t, x, u) compactly supported in [0, T ) × R 3 × R 3 . Then, in a second computation, test functions belonging to Im

∂ ∂τ + (v × M) • ∇ v , or writing ψ = ∂κ ∂τ + (v × M) • ∇ v κ , (4.23)
for some regular functions κ = κ(t, τ, x, v) with compact support in [0, T ) × R 3 × R 3 for every τ ∈ [0, 1], will be chosen.

The Two-Scale Macro equation

We begin by going further in the writing of weak formulation (4.20) by using the link between G and l.

We recall this weak formula:

T 0 R 6 (F ) ǫ ∂(ψ) ǫ ∂t + v • ∇ x (ψ) ǫ + E + v × (B + M ǫ ) • ∇ v (ψ) ǫ dx dv dt + ǫ T 0 R 6 (F ǫ 1 ) ǫ ( ∂ψ ∂t ) ǫ + 1 ǫ ( ∂ψ ∂τ ) ǫ + v • (∇ x ψ) ǫ + E + v × (B + M ǫ ) • (∇ v ψ) ǫ dx dv dt + ǫ T 0 R 6 (l) ǫ ∂(ψ) ǫ ∂t + (v × M ǫ ) • ∇ v (ψ) ǫ dx dv dt + ǫ T 0 R 6 (l) ǫ v • (∇ x ψ) ǫ + (E + v × B) • (∇ v ψ) ǫ dx dv dt + ǫ T 0 R 6 (h ǫ ) ǫ ( ∂ψ ∂t ) ǫ + 1 ǫ ( ∂ψ ∂τ ) ǫ + v • (∇ x ψ) ǫ + E + v × (B + M ǫ ) • (∇ v ψ) ǫ dx dv dt = - R 6 f 0 (x, v) ψ(0, 0, x, v) dx dv .
(5.1)

Performing an integration by parts in the first term of (5.1) gives

T 0 R 6 (F ) ǫ ∂(ψ) ǫ ∂t + v • ∇ x (ψ) ǫ + E + v × (B + M ǫ ) • ∇ v (ψ) ǫ dx dv dt = - T 0 R 6 ( ∂F ∂t ) ǫ + 1 ǫ ( ∂F ∂τ ) ǫ + v • (∇ x F ) ǫ + E + v × (B + M ǫ ) • (∇ v F ) ǫ (ψ) ǫ dx dv dt - R 6
F (0, 0, x, v) ψ(0, 0, x, v) dx dv

= - T 0 R 6 ( ∂F ∂t ) ǫ + v • (∇ x F ) ǫ + (E + v × B) • (∇ v F ) ǫ (ψ) ǫ dx dv dt - R 6 f 0 (x, v) ψ(0, 0, x, v) dx dv , (5.2) since F ∈ Ker( ∂ ∂τ + (v × M) • ∇ v ).
We invoke the link between F and G, and we consider the change of variables

v → u = ⊔ ⊔ ⊔( t ǫ , v) which is equivalent to v = ⊔ ⊔ ⊔(- t ǫ , u) = ⊔ ⊔ ⊔ -1 ( t ǫ , u). (5.3)
We also use the following notation

(ψ • ⊔ ⊔ ⊔ -1 )(t, τ, x, u) = ψ t, τ, x, ⊔ ⊔ ⊔ -1 (τ, u) = ψ t, τ, x, ⊔ ⊔ ⊔(-τ, u) . (5.4)
Since a direct computation yields

(∇ v F ) ǫ = ∇ v (F ) ǫ = ∇ v (G • ⊔ ⊔ ⊔) ǫ = r(- t ǫ ) (∇ u G) • ⊔ ⊔ ⊔ ǫ , (5.5) equation (5.2) reads T 0 R 6 (F ) ǫ ∂(ψ) ǫ ∂t + v • ∇ x (ψ) ǫ + E + v × (B + M ǫ ) • ∇ v (ψ) ǫ dx dv dt = - T 0 R 6 ∂G ∂t + r(- t ǫ ) u • ∇ x G + r( t ǫ ) E + r(- t ǫ ) u × B • ∇ u G (ψ • ⊔ ⊔ ⊔ -1 ) ǫ dx du dt - R 6 f 0 (x, v) ψ(0, 0, x, v) dx dv = - T 0 R 6 ∂G ∂t + r(- t ǫ ) u • ∇ x G + r( t ǫ ) E + u × r( t ǫ ) B • ∇ u G (ψ • ⊔ ⊔ ⊔ -1 ) ǫ dx du dt - R 6 f 0 (x, v) ψ(0, 0, x, v) dx dv .
(5.6)

Using now equation (2.10) satisfied by G, we finally obtain

T 0 R 6 (F ) ǫ ∂(ψ) ǫ ∂t + v • ∇ x (ψ) ǫ + E + v × (B + M ǫ ) • ∇ v (ψ) ǫ dx dv dt = - T 0 R 6 r(- t ǫ ) u • ∇ x ⊥ G + r( t ǫ ) E ⊥ + u × r( t ǫ ) B ⊥ • ∇ u ⊥ G (ψ • ⊔ ⊔ ⊔ -1 ) ǫ dx du dt - R 6 f 0 (x, v) ψ(0, 0, x, v) dx dv .
(5.7)

Notice that the last term containing f 0 will vanish with the right hand side member of (5.1). Now, we study the third term of (5.1). For this purpose, we first notice that

∂(l • ⊔ ⊔ ⊔ -1 ) ∂τ = ( ∂l ∂τ ) • ⊔ ⊔ ⊔ -1 + (r(-τ ) u) × M • (∇ v l) • ⊔ ⊔ ⊔ -1 = -r(-τ ) u • ∇ x ⊥ G -r(τ ) E + u × r(τ ) B • ∇ u ⊥ G , (5.8)
hence, integrating by parts the third term of (5.1) and making the change of variables defined by (5.3), we get

ǫ T 0 R 6 (l) ǫ ∂(ψ) ǫ ∂t + (v × M ǫ ) • ∇ v (ψ) ǫ dx dv dt = - T 0 R 6 ǫ ( ∂l ∂t ) ǫ + ( ∂l ∂τ ) ǫ + (v × M) • (∇ v l) ǫ (ψ) ǫ dx dv dt - R 6 (l) ǫ (0, x, v) (ψ) ǫ (0, x, v) dx dv , = - T 0 R 6 ǫ ∂(l • ⊔ ⊔ ⊔ -1 ) ∂t ǫ + ∂(l • ⊔ ⊔ ⊔ -1 ) ∂τ ǫ (ψ • ⊔ ⊔ ⊔ -1 ) ǫ dx du dt = - T 0 R 6 ǫ ∂(l • ⊔ ⊔ ⊔ -1 ) ∂t ǫ (ψ • ⊔ ⊔ ⊔ -1 ) ǫ dx du dt + T 0 R 6 r(- t ǫ ) u • ∇ x ⊥ G + r( t ǫ ) E + u × r( t ǫ ) B • ∇ u ⊥ G (ψ • ⊔ ⊔ ⊔ -1 ) ǫ dx du dt , (5.9)
since l(0, 0, x, v) = 0 (see (4.4)). The last term of (5.9) and the first term in the right hand side of (5.7) will cancel each other.

Hence (5.1) has to be replaced by

- T 0 R 6 ǫ ∂(l • ⊔ ⊔ ⊔ -1 ) ∂t ǫ (ψ • ⊔ ⊔ ⊔ -1 ) ǫ dx du dt + ǫ T 0 R 6 (F ǫ 1 ) ǫ ( ∂ψ ∂t ) ǫ + 1 ǫ ( ∂ψ ∂τ ) ǫ + v • (∇ x ψ) ǫ + E + v × (B + M ǫ ) • (∇ v ψ) ǫ dx dv dt + ǫ T 0 R 6 (l) ǫ v • (∇ x ψ) ǫ + (E + v × B) • (∇ v ψ) ǫ dx dv dt + ǫ T 0 R 6 (h ǫ ) ǫ ( ∂ψ ∂t ) ǫ + 1 ǫ ( ∂ψ ∂τ ) ǫ + v • (∇ x ψ) ǫ + E + v × (B + M ǫ ) • (∇ v ψ) ǫ dx dv dt = 0 .
(5.10)

Now, choosing ψ as in (4.22) yields

ǫ T 0 R 6 (G ǫ 1 • ⊔ ⊔ ⊔) ǫ ( ∂γ ∂t ) • ⊔ ⊔ ⊔ ǫ + v • (∇ x γ) • ⊔ ⊔ ⊔ ǫ + (E + v × B) • ∇ v (γ • ⊔ ⊔ ⊔) ǫ dx dv dt - T 0 R 6 ǫ ∂(l • ⊔ ⊔ ⊔ -1 ) ∂t ǫ γ dx du dt + T 0 R 6 ǫ (h ǫ ) ǫ ∂(γ • ⊔ ⊔ ⊔) ∂t ǫ dx dv dt + T 0 R 6 ǫ (l) ǫ + ǫ (h ǫ ) ǫ v • ∇ x (γ • ⊔ ⊔ ⊔) ǫ + (E + v × B) • ∇ v (γ • ⊔ ⊔ ⊔)
ǫ dx dv dt = 0 .

(5.11)

Making in the first, third and fourth terms the change of variables v → ⊔ ⊔ ⊔( t ǫ , v), replacing h ǫ by its expression in terms of k ǫ and dividing (5.11) by ǫ finally gives

T 0 R 6 G ǫ 1 ∂γ ∂t + r(- t ǫ ) u • ∇ x γ + r( t ǫ ) E + u × r( t ǫ ) B • ∇ u γ dx du dt - T 0 R 6 ∂(l • ⊔ ⊔ ⊔ -1 )
∂t ǫ γ dx du dt

+ T 0 R 6 ( ∂k ǫ ∂τ ) • ⊔ ⊔ ⊔ -1 ǫ + r(- t ǫ ) M • (∇ v k ǫ ) • ⊔ ⊔ ⊔ -1 ǫ ∂γ ∂t dx du dt + T 0 R 6 (l • ⊔ ⊔ ⊔ -1 ) ǫ + ( ∂k ǫ ∂τ ) • ⊔ ⊔ ⊔ -1 ǫ + r(- t ǫ ) M • (∇ v k ǫ ) • ⊔ ⊔ ⊔ -1 ǫ × r(- t ǫ ) u • ∇ x γ + r( t ǫ ) E + u × r( t ǫ ) B • ∇ u γ dx du dt = 0 , (5.12)
which is the Two-Scale Macro equation of the model.

The Two-Scale Micro equation

In this section, we use in the weak formulation (4.20) oscillating test functions defined by (4.23). The computation leading to formula (5.10) is valid for any oscillating function ψ so we use it as a starting point for finding the Micro equation.

Recalling that F ǫ 1 ∈ Ker(

∂ ∂τ + (v × M) • ∇ v )
, the second term in (5.10) yields

ǫ T 0 R 6 (F ǫ 1 ) ǫ ( ∂ψ ∂t ) ǫ + 1 ǫ ( ∂ψ ∂τ ) ǫ + v • (∇ x ψ) ǫ + E + v × (B + M ǫ ) • (∇ v ψ) ǫ dx dv dt = ǫ T 0 R 6 (F ǫ 1 ) ǫ ∂(ψ) ǫ ∂t + (v × M ǫ ) • ∇ v (ψ) ǫ dx dv dt + ǫ T 0 R 6 (F ǫ 1 ) ǫ v • (∇ x ψ) ǫ + (E + v × B) • (∇ v ψ) ǫ dx dv dt = -ǫ T 0 R 6 ( ∂F ǫ 1 ∂t ) ǫ (ψ) ǫ dx dv dt + ǫ R 6 F ǫ 1 (0, 0, x, v) ψ(0, 0, x, v) dx dv + ǫ T 0 R 6 (F ǫ 1 ) ǫ v • (∇ x ψ) ǫ + (E + v × B) • (∇ v ψ) ǫ dx dv dt = -ǫ T 0 R 6 ( ∂F ǫ 1 ∂t ) ǫ ( ∂κ ∂τ ) ǫ + (v × M) • (∇ v κ) ǫ dx dv dt + ǫ R 6 F ǫ 1 (0, 0, x, v) ∂κ ∂τ (0, 0, x, v) + (v × M) • ∇ v κ(0, 0, x, v) dx dv + ǫ T 0 R 6 (F ǫ 1 ) ǫ v • ( ∂∇ x κ ∂τ ) ǫ + (v × M) • (∇ x ∇ v κ) ǫ v + (E + v × B) • ( ∂∇ v κ ∂τ ) ǫ + (E × M + v × B × M) • (∇ v κ) ǫ + (E + v × M) • (∇ 2 v κ) ǫ (v × M) dx dv dt . (6.1)
The first term of (5.10) reads

- T 0 R 6 ǫ ∂(l • ⊔ ⊔ ⊔ -1 ) ∂t ǫ (ψ • ⊔ ⊔ ⊔ -1 ) ǫ dx dv dt = -ǫ T 0 R 6 ∂(l • ⊔ ⊔ ⊔ -1 ) ∂t ǫ ( ∂κ ∂τ • ⊔ ⊔ ⊔ -1 ) ǫ + r(- t ǫ ) v × M • (∇ v κ) • ⊔ ⊔ ⊔ -1 ǫ dx dv dt . (6.2)
The third term gives

ǫ T 0 R 6 (l) ǫ v • (∇ x ψ) ǫ + (E + v × B) • (∇ v ψ) ǫ dx dv dt = ǫ T 0 R 6 (l) ǫ v • ( ∂∇ x κ ∂τ ) ǫ + (v × M) • (∇ x ∇ v κ) ǫ v + (E + v × B) • ( ∂∇ v κ ∂τ ) ǫ + (E × M + v × B × M) • (∇ v κ) ǫ + (E + v × B) • (∇ 2 v κ) ǫ (v × M) dx dv dt . (6.3)
Concerning the last term of (5.10), we have

T 0 R 6 (h ǫ ) ǫ ( ∂ψ ∂t ) ǫ + 1 ǫ ( ∂ψ ∂τ ) ǫ dx dv dt = T 0 R 6 (h ǫ ) ǫ ∂(ψ) ǫ ∂t dx dv dt = - T 0 R 6 ∂(h ǫ ) ǫ ∂t (ψ) ǫ dx dv dt + R 6 h ǫ (0, 0, x, v) ψ(0, 0, x, v) dx dv = - T 0 R 6 ( ∂h ǫ ∂t ) ǫ + 1 ǫ ( ∂h ǫ ∂τ ) ǫ ( ∂κ ∂τ ) ǫ + (v × M) • (∇ v κ) ǫ dx dv dt + R 6 h ǫ (0, 0, x, v) ∂κ ∂τ (0, 0, x, v) + (v × M) • ∇ v κ(0, 0, x, v) dx dv = - T 0 R 6 ( ∂ 2 k ǫ ∂t ∂τ ) ǫ + (v × M) • ( ∂∇ v κ ∂t ) ǫ + 1 ǫ ( ∂ 2 k ǫ ∂τ 2 ) ǫ + 1 ǫ (v × M) • ( ∂∇ v κ ∂τ ) ǫ × ( ∂κ ∂τ ) ǫ + (v × M) • (∇ v κ) ǫ dx dv dt + R 6 ∂k ǫ ∂τ (0, 0, x, v) + (v × M) • ∇ v k ǫ (0, 0, x, v) × ∂κ ∂τ (0, 0, x, v) + (v × M) • ∇ v κ(0, 0, x, v) dx dv , (6.4)
on the one hand, and

T 0 R 6 (h ǫ ) ǫ v • (∇ x ψ) ǫ + E + v × (B + M ǫ ) • (∇ v ψ) ǫ dx dv dt = T 0 R 6 ( ∂k ǫ ∂τ ) ǫ + (v × M) • (∇ v k ǫ ) ǫ v • ( ∂∇ x κ ∂τ ) ǫ + (v × M) • (∇ x ∇ v κ) ǫ v + (E + v × B) • ( ∂∇ v κ ∂τ ) ǫ + (E × M + (v × B) × M) • (∇ v κ) ǫ + (E + v × B) • (∇ 2 v κ) ǫ (v × M) + 1 ǫ (v × M) • ( ∂∇ x κ ∂τ ) ǫ + (v × M) × M • (∇ v κ) ǫ + (v × M) • (∇ 2 v κ) ǫ (v × M) dx dv dt , (6.5)
on the other hand. Remarking that

(v × M) × M • (∇ v κ) ǫ = -(v × M) • (∇ v κ) ǫ × M = -v • (∇ v κ) ǫ , (6.6)
and dividing by ǫ, we finally get the weak formulation of the Two-Scale Micro equation:

- T 0 R 6 ( ∂ 2 k ǫ ∂t ∂τ ) ǫ + (v × M) • ( ∂∇ v k ǫ ∂t ) ǫ + 1 ǫ ( ∂ 2 k ǫ ∂τ 2 ) ǫ + 1 ǫ (v × M) • ( ∂∇ v k ǫ ∂τ ) ǫ ( ∂κ ∂τ ) ǫ + (v × M) • (∇ v κ) ǫ dx dv dt + R 6 ∂k ǫ ∂τ (0, 0, x, v) + (v × M) • ∇ v k ǫ (0, 0, x, v) ∂κ ∂τ (0, 0, x, v) + (v × M) • ∇ v κ(0, 0, x, v) dx dv + T 0 R 6 ( ∂k ǫ ∂τ ) ǫ + (v × M) • (∇ v k ǫ ) ǫ v • ( ∂∇ x κ ∂τ ) ǫ + (v × M) • (∇ x ∇ v κ) ǫ v + (E + v × B) • ( ∂∇ v κ ∂τ ) ǫ + (E × M + (v × B) × M) • (∇ v κ) ǫ + (E + v × B) • (∇ 2 v κ) ǫ (v × M) + 1 ǫ (v × M) • ( ∂∇ x κ ∂τ ) ǫ -v • (∇ v κ) ǫ + (v × M) • (∇ 2 v κ) ǫ (v × M) dx dv dt - T 0 R 6 ∂(l • ⊔ ⊔ ⊔ -1 ) ∂t ǫ ( ∂κ ∂τ • ⊔ ⊔ ⊔ -1 ) ǫ + r(- t ǫ ) v × M • (∇ v κ) • ⊔ ⊔ ⊔ -1 ǫ dx dv dt + T 0 R 6 (l) ǫ v • ( ∂∇ x κ ∂τ ) ǫ + (v × M) • (∇ x ∇ v κ) ǫ v + (E + v × B) • ( ∂∇ v κ ∂τ ) ǫ + (E × M + (v × B) × M) • (∇ v κ) ǫ + (E + v × B) • (∇ 2 v κ) ǫ (v × M) dx dv dt - T 0 R 6 ( ∂F ǫ 1 ∂t ) ǫ ( ∂κ ∂τ ) ǫ + (v × M) • (∇ v κ) ǫ dx dv dt + R 6 F ǫ 1 (0, 0, x, v) ∂κ ∂τ (0, 0, x, v) + (v × M) • ∇ v κ(0, 0, x, v) dx dv + T 0 R 6 (F ǫ 1 ) ǫ v • ( ∂∇ x κ ∂τ ) ǫ + (v × M) • (∇ x ∇ v κ) ǫ v + (E + v × B) ( ∂∇ v κ ∂τ ) ǫ + (E × M + (v × B) × M) • (∇ v κ) ǫ + (E + v × M) • (∇ 2 v κ) ǫ (v × M) dx dv dt = 0 . (6.7)
Remark 6.1. First, system (5.12)-(6.7) clearly has solutions, since functions G ǫ 1 and k ǫ as introduced in (4.14) by using (4.8) were shown to satisfy (5.12)-(6.7). Then, we are aware that the Two-Scale Macro-Micro system, (5.12)-(6.7), has not necessarily a unique solution. The reason for the uniqueness lack is the replacement of τ by t ǫ . In future work we will look for an additional condition for G ǫ 1 and k ǫ leading to the uniqueness of the whole solution (G ǫ 1 , k ǫ ). Such a condition will be useful for the conception of the numerical scheme for the Two-Scale Macro-Micro model in order to be sure that when ǫ ∼ 1 we will approximate the solution of the starting Vlasov equation (1.15). 7 Study of the asymptotic behaviour

The Two-Scale Macro-Micro problem

The Two-Scale Macro-Micro problem, i.e. the system of equations (2.10)-(4.4)-(5.12)-(6.7), is more complicated to solve than the original problem (1.15), but has two advantages over the latter. First, the Two-Scale Macro-Micro model trivially reduces to the limit model (2.10) by taking ǫ → 0 in (2.10)-(4.4)-(5.12)-(6.7). Second, the Two-Scale Macro-Micro model decomposes the original solution

f ǫ into a Macro part, G • ⊔ ⊔ ⊔ + ǫ G ǫ 1 • ⊔ ⊔ ⊔,
and a Micro part, ǫ l + ǫ h ǫ . This question is relevant when doing the transition from the very small ǫ regime to the one of ǫ ∼ 1, since our model describes separately the evolution at the macroscopic time scale (of

G • ⊔ ⊔ ⊔ and G ǫ 1 • ⊔ ⊔ ⊔)
which contains essential oscillation through ⊔ ⊔ ⊔, and the evolution of oscillation corrections (l and h ǫ ) at the microscopic time scale. Now it is easy to see that, under the hypothesis of uniqueness of the solution of equations (5.12) and (6.7) (see Remark 6.1), the Two-Scale Macro-Micro problem (2.10)-(4.4)-(5.12)-(6.7) is equivalent to the original problem (1.15). Indeed, let f ǫ be the solution of (1.15). We have seen, by using Lemma B.1, that the decomposition (4.14) exists and is unique. Then, by the calculations we did in the previous sections, we obtain that G is solution to (2.10), l is given by the formula (4.4), G ǫ 1 is solution to (5.12) and k ǫ to (6.7). In particular, we have proved that the equation system (5.12)-(6.7) has solution.

Conversely, assume that the solution of the system (5.12)-(6.7) is unique. Let (G, l, G ǫ 1 , k ǫ ) be the solution of (2.10)-(4.4)-(5.12)-(6.7). Then constructing f ǫ by (4.17), we will obtain the solution of (1.15), since this problem has unique solution.

Macro-Micro decomposition vs Two-Scale Macro-Micro decomposition

As it has been mentioned in the previous sections, the Two Indeed, let us recall that the unique decomposition made in (4.14) is of the kind

f ǫ (t, x, v) = F (t, τ, x, v) + F ǫ 1 (t, τ, x, v) + H ǫ 1 (t, τ, x, v), (7.1)
for some

F ǫ 1 ∈ Ker ∂ ∂τ + (v × M) • ∇ v and H ǫ 1 ∈ Im ∂ ∂τ + (v × M) • ∇ v .
Then, integrating (7.1) in τ , we obtain using (2.12)

f ǫ (t, x, v) = f (t, x, v) + 1 0 F ǫ 1 (t, τ, x, v) dτ + 1 0 H ǫ 1 (t, τ, x, v) dτ. (7.2)
Now, changing the variable v into (v || , v ⊥ , α) defined in (3.1), and using Lemma B.2, we obtain that the

function (v ⊥ , α) → 1 0 F ǫ 1 t, τ, x, v || e 1 + v ⊥ (cos α e 2 + sin α e 3 ) dτ is in Ker ∂ ∂α . Similarly, (v ⊥ , α) → 1 0 H ǫ 1 t, τ, x, v || e 1 + v ⊥ (cos α e 2 + sin α e 3 ) dτ is in Im ∂ ∂α . Next, writing (7.
2) in the new variables and recalling the decomposition of f ǫ (t, x, v) in (3.12), we deduce from (3.5) and the uniqueness of such a decomposition (stated in Lemma A.1(iii)) that

m ǫ 1 (t, x, v || , v ⊥ ) = 1 0 F ǫ 1 t, τ, x, v || e 1 + v ⊥ (cos α e 2 + sin α e 3 ) dτ, n ǫ (t, x, v || , v ⊥ , α) = 1 0 H ǫ 1 t, τ, x, v || e 1 + v ⊥ (cos α e 2 +
sin α e 3 ) dτ.

(7.3)

Convergence of the Two-Scale Macro-Micro problem

Let us recall that our Two-Scale Macro-Micro decomposition (4.14) is based on the convergence result in (4.2). More precisely, keeping in mind that G + ǫ G + ǫ l is the first order approximation of f ǫ , (4.14) can be seen as a Macro-Micro decomposition of f ǫ at the first order level of approximation. Next, we justify this approximation using the Two-Scale convergence.

Theorem 7.2. We assume that

f 0 ∈ L 2 (R 6 ), E ∈ W 1,∞ (R 3 ), B ∈ W 1,∞ (R 3 ), and ∂l/∂t, ∇ x,v l ∈ L ∞ 0, T ; L ∞ (0, 1; L 2 (R 6
)) . Then, when ǫ → 0, the solutions G ǫ 1 • ⊔ ⊔ ⊔ of (5.12) Two-Scale converge to G ∈ L ∞ 0, T ; L 2 (R 6 ) , the solution of (4.5). When ǫ → 0, the solutions k ǫ of (6.7) Two-Scale converge to 0.

Proof. Recall from Theorem 1.5 of [START_REF] Frénod | Two-scale expansion of a singularly perturbed convection equation[END_REF] that when ǫ → 0,

1 ǫ f ǫ -G • ⊔ ⊔ ⊔ -l ǫ>0 Two-Scale converges to G • ⊔ ⊔ ⊔ ∈ Ker ∂ ∂τ + (v × M) • ∇ v . Then, since (4.17) implies that 1 ǫ f ǫ -G • ⊔ ⊔ ⊔ -l = G ǫ 1 • ⊔ ⊔ ⊔ + h ǫ , and since G ǫ 1 • ⊔ ⊔ ⊔ ∈ Ker ∂ ∂τ + (v × M) • ∇ v and h ǫ ∈ Im ∂ ∂τ + (v × M) • ∇ v , the theorem's conclusion is clearly true.
Remark 7.3. Thanks to the previous results, the Two-Scale Macro-Micro model we have built satisfies the expected asymptotic behaviour symbolized by the arrows in the top layer of diagram (1.14). Concerning the time discretization of (2.10)-(4.4)-(5.12)-(6.7), we intend to proceed as discussed in Remark 3.2 for the classical Macro-Micro formulation (see [START_REF] Bennoune | Uniformly stable numerical schemes for the Boltzmann equation preserving the compressible Navier-Stokes asymptotics[END_REF][START_REF] Crouseilles | An asymptotic preserving scheme based on a micro-macro decomposition for Collisional Vlasov equations: diffusion and high-field scaling limits[END_REF][START_REF] Lemou | A New Asymptotic Preserving Scheme Based on Micro-Macro Formulation for Linear Kinetic Equations in the Diffusion Limit[END_REF]), in order to obtain Asymptotic-Preserving schemes, symbolized by the arrows in the bottom layer of diagram (1.14).

Appendices

In Appendix A, we characterize the projection onto Ker( ∂ ∂α ). We establish that it consists in computing its average in α and projecting it onto Im( ∂ ∂α ) consists in substracting from it its average value. Then, in the second Appendix, we do the same with operator ∂ ∂τ + (v × M) • ∇ v . We also establish the link that exists between these two operators.

A About the operator ∂ ∂α

Lemma A.1. The unbounded operator

∂ ∂α : L 2 #2π (R; L 2 (R + ; v ⊥ dv ⊥ )) → L 2 #2π (R; L 2 (R + ; v ⊥ dv ⊥ )) has a closed range in L 2 #2π (R; L 2 (R + ; v ⊥ dv ⊥ )) and (i) Ker ∂ ∂α ⊥ = Im ∂ ∂α , (ii) For any function f ∈ L 2 #2π (R; L 2 (R + ; v ⊥ dv ⊥ )), the projection of f on Ker ∂ ∂α is the function P f : (v ⊥ , α) -→ 1 2π 2π 0 f (v ⊥ , θ) dθ , (A.1) (iii) L 2 #2π (R; L 2 (R + ; v ⊥ dv ⊥ )) = Ker ∂ ∂α ⊕ Im ∂ ∂α . Proof. (i) Let q ∈ Ker ∂ ∂α ⊥ .
Then, for any g ∈ Ker ∂ ∂α , i.e. g = g(v ⊥ ), we have

2π 0 +∞ 0 q(v ⊥ , α) g(v ⊥ ) v ⊥ dv ⊥ dα = 0 . (A.2)
Remarking that q can be written as

q(v ⊥ , α) = ∂ ∂α α 0 q(v ⊥ , θ) dθ , (A.3)
we only have to prove that the function K defined by

K(v ⊥ , α) = α 0 q(v ⊥ , θ) dθ , (A.4) is in L 2 #2π (R; L 2 (R + ; v ⊥ dv ⊥ )), i.e. K is 2π-periodic in α and 2π 0 +∞ 0 K(v ⊥ , α) 2 v ⊥ dv ⊥ dα < +∞.
First, we have

K(v ⊥ , α + 2π) -K(v ⊥ , α) = 2π 0 q(v ⊥ , θ) dθ = 0 , (A.5)
by (A.2). Second, using Jensen's inequality, we remark that This can be done using a similar way as in the proof of item (i). This concludes the proof of the Lemma.

K(v ⊥ , α) 2 ≤ |α| α 0 q(v ⊥ , θ) 2 dθ ≤ 2π 2π 0 q(v ⊥ , θ
B The operator

∂ ∂τ + (v × M) • ∇ v Lemma B.1. Let the unbounded operator ∂ ∂τ + (v × M) • ∇ v : L ∞ 0, T ; L ∞ #1 (R + ; L 2 (R 6 )) → L ∞ 0, T ; L ∞ #1 (R + ; L 2 (R 6
)) . Then we have

L ∞ 0, T ; L ∞ #1 (R + ; L 2 (R 6 )) = Ker ∂ ∂τ + (v × M) • ∇ v ⊕ Im ∂ ∂τ + (v × M) • ∇ v . (B.1) Proof. Let f ∈ L ∞ 0, T ; L ∞ #1 (R + ; L 2 (R 6 )) . Through the change of variables    v || ∈ R such that v || = v • e 1 , v ⊥ ∈ R + such that v ⊥ = v
∂ ∂β : L ∞ #1 (R + ; L 2 #2π (R; L 2 (R × R + ; v ⊥ dv || dv ⊥ ))) → L ∞ #1 (R + ; L 2 #2π (R; L 2 (R × R + ; v ⊥ dv || dv ⊥ ))), (B.4)
we know, by Lemma A.1, that L 2 #2π (R;

L 2 (R + ; v ⊥ dv ⊥ )) = Ker ∂ ∂β ⊕Im ∂ ∂β . Hence, because L ∞ #1 (R + ; L 2 #2π (R; L 2 (R × R + ; v ⊥ dv || dv ⊥ ))) ⊂ L 2 #1 (R + ; L 2 #2π (R; L 2 (R × R + ; v ⊥ dv || dv ⊥ )
)), we can write f = P f + ( f -P f ), where P f is defined as in (A.1). Since f was chosen in L ∞ #1 (R + ; L 2 #2π (R; L 2 (R × R + ; v ⊥ dv || dv ⊥ ))), it is easy to see that P f belongs to this same space. Obviously f -P f ∈ L ∞ #1 (R + ; L 2 #2π (R; L 2 (R× R + ; v ⊥ dv || dv ⊥ ))) and thus (B.1) is proved.

  ),(3.18),(3.21),(3.22) is unique. On the other hand, it can be easily shown that the classical Macro-Micro model (3.10), (3.17), (3.18), (3.21), (3.22) is equivalent to the original problem (1.15). Remark 3.2 (Conceiving Asymptotic-Preserving schemes). In the light of Remark 3.1, the reformulated problem (3.10), (3.17), (3.18), (3.21), (3.22) is Asymptotic-Preserving since formally, we obtain from (3.12) and (3.15) that m ǫ 1 → 0 and n ǫ → 0 when ǫ → 0 and thus, the classical Macro-Micro model reduces to (3.10).

Remark 6 . 2 .

 62 Problem (2.10)-(4.4)-(5.12)-(6.7) plays the role of problem E ǫ U ǫ = 0 in diagram (1.14).

Property 7 . 1 .

 71 -Scale Macro-Micro decomposition model (2.10)-(4.4)-(5.12)-(6.7) is linked to the usual Macro-Micro model (3.10), (3.17), (3.18), (3.21), (3.22). Integrating in τ the Two-Scale Macro-Micro decomposition leads to the classical Macro-Micro model developed in Section 3.2.

fff

  ⊥ , θ) 2 v ⊥ dv ⊥ dθ < +∞ , (A.7) since q ∈ L 2 #2π (R; L 2 (R + ; v ⊥ dv ⊥ )). Thus q ∈ Im inclusion is obvious, (i) is proved and the range of the operator is indeed closed.(ii) It is clear that f ∈ L 2 #2π (R; L 2 (R + ; v ⊥ dv ⊥ )) implies that P f ∈ L 2 #2π (R; L 2 (R + ; v ⊥ dv ⊥ )).Then, using the projection characterization, we have for any functionψ ∈ L 2 #2π (R; L 2 (R + ; v ⊥ dv ⊥ )) such that ∂ψ ∂α (v ⊥ , θ) dθ ψ(v ⊥ ) v ⊥ dv ⊥ dα = (v ⊥ , α) dα ψ(v ⊥ ) v ⊥ dv ⊥ -(v ⊥ , θ) dθ ψ(v ⊥ ) v ⊥ dv ⊥ = 0 .Hence P f defined in (A.1) is the projection of f on Ker ∂ ∂α .(iii) Using item (i) we have only to prove that anyf ∈ L 2 #2π (R; L 2 (R + ; v ⊥ dv ⊥ )) writes as a sum f = g + h with g ∈ Ker ∂ ∂α and h ∈ Im ∂ ∂α . Given f ∈ L 2 #2π (R; L 2 (R + ; v ⊥ dv ⊥ )), let g be the projection of f on Ker ∂ ∂α and let h = f -g. Then, by item (ii) we only need to show that h ∈ Im ∂ ∂α .

2 2 + v 2 3 ,

 3 α ∈ [0, 2π] such that v 2 = v ⊥ cos α and v 3 = v ⊥ sin α, (B.2) f where f (t, τ, x, v || , v ⊥ , α) = f (t, τ, x, v || e 1 + v ⊥ (cos α e 2 + sin α e 3 )) is in L ∞ 0, T ; L ∞ #1 (R + ; L 2 #2π (R; L 2 (R 3 × R × R + ; v ⊥ dx dv || dv ⊥ ))) while the operator ∂ ∂τ + (v × M) • ∇ v writes ∂ (t, σ, x, v || , v ⊥ , β) = f (t, (σ + β)/2, x, v || , v ⊥ , π(σ -β)) stays in L ∞ 0, T ; L ∞ #1 (R + ; L 2 #2π (R; L 2 (R 3 × R × R + ; v ⊥ dx dv || dv ⊥ ))) and the operator ∂ on, let us fix (t, x) ∈ [0, T ] × R 3 . Considering

Lemma B. 2 . 1 0f 1 0f 1 0

 2111 Let f ∈ L ∞ 0, T ; L ∞ #1 (R + ; L 2 (R 6)) and let f be defined as in the proof of Lemma B.1. Then(i) if f ∈ Ker ∂ ∂τ + (v × M) • ∇ v then for any fixed (t, x, v || ) ∈ [0, T ] × R 3 × R we have F : (v ⊥ , α) → (t, τ, x, v || , v ⊥ , α) dτ ∈ Ker ∂ ∂α , (ii) if f ∈ Im ∂ ∂τ + (v × M) • ∇ v then for any fixed (t, x, v || ) ∈ [0, T ] × R 3 × R we have F : (v ⊥ , α) → (t, τ, x, v || , v ⊥ , α) dτ ∈ Im ∂ ∂α . Proof. (i) Let f ∈ Ker ∂ ∂τ + (v × M) • ∇ v and fix (t, x, v || ) ∈ [0, T ] × R 3 × R. Thus, by the proof of Lemma B.1, f ∈ Ker ∂ τ, x, v || , v ⊥ , α) dτ τ, x, v || , v ⊥ , α) dτ = 0 , (B.5)by the 1-periodicity in τ of f . Thus, F ∈ Ker ∂ ∂α .(ii) Now let f ∈ Im ∂ ∂τ + (v × M) • ∇ v . Then there exists a function h ∈ L ∞ 0, T ; L ∞ #1 (R + ; L 2 (R 6 )) such that f = ∂ h ∂τ -2π ∂ h ∂α . Fixing (t, x, v || ) ∈ [0, T ] × R 3 × R and integrating in τ from 0 to 1, we obtain, using that h is 1-periodic in τ , F (v ⊥ , v || ) = -2π∂ ∂α h(t, τ, x, v || , v ⊥ , α) dτ, leading to F ∈ Im ∂ ∂α and thus concluding the Lemma.


Antoine Mandel

Herbert Gintis

Stochastic stability in the Scarf economy

Keywords: General Equilibrium, Exchange economies, Bargaining Games, Stochastic Stability. JEL Codes: D51, C62, C63, C78

]. We then characterize, in the Scarf economy, a class of dynamics for which the Walrasian equilibrium is the only stochastically stable state. Hence, we provide dynamic foundations for general equilibrium for one of the best-known example of instability of the tâtonement process.

Introduction

The Scarf economy [START_REF] Scarf | Some examples of global instability of competitive equilibrium[END_REF] is the paradigmatic example of the failure of the tâtonement process to provide a generically valid process for the convergence of an economy to its general equilibrium. In a series of recent contribution (see [START_REF] Gintis | The Dynamics of General Equilibrium[END_REF]], [START_REF] Gintis | The Dynamics of Pure Market Exchange[END_REF]), Gintis has revisited this issue of "walrasian dynamics" in the Scarf economy using computer simulations where agents repeatedly perform the following sequence of operations: they receive their initial endowment, engage in bilateral trades on the basis of private prices, consume, and update their private prices on the basis of the utility these prices yielded during the period. In other words, [START_REF] Gintis | The Dynamics of Pure Market Exchange[END_REF]] investigates evolutionary dynamics in bargaining games played in the Scarf economy by agents who use private prices as strategies. He reports surprising results of convergence to equilibrium. The aim of this note is to provide an analytical counterpart to these results.

Namely, we use the notion of stochastic stability (see [START_REF] Peyton-Young | The Evolution of Conventions[END_REF]) to characterize the asymptotic properties of a stylized version of Ginitis' model. We show that the general equilibrium is the only stochastically stable state of this model, what implies that, independently of the initial conditions, all the agents should eventually adopt the equilibrium price and obtain equilibrium allocation.

Our results mainly builds on the assumption that out-of-equilibrium trading is efficient in the same sense as in the Hahn process (see [START_REF] Hahn | A theorem on non-tatonnement stability[END_REF]): after trade there are not both unsatisfied suppliers and unsatisfied demanders for any given good. We also assume that agents strategically restrict their outof-equilibrium trade whenever it is profitable to do so. In this setting, it turns out that price movement towards equilibrium are always favorable to a majority of agents. As "price-setting power" is uniformly distributed, given that each agent has its own private price to update, this progressively leads to the general adoption of the equilibrium price.

Hence, the main contribution of the paper is to explain the behavior observed in Gintis' simulations. The formalism we develop might also pave the way for the proof of more general results of convergence to general equilibrium in evolutionary models, for which related contributions (see [START_REF] Serrano | Mistakes in Cooperation: the Stochastic Stability of Edgeworth's Recontracting[END_REF], [START_REF] Vega-Redondo | The Evolution of Walrasian Behavior[END_REF]) provide evidence.

The paper is organized as follows: in section 2, we explicit the markov chain structure of Gintis evolutionary bargaining models and show they are models of evolution with noise in the sense of [START_REF] Ellison | Basins of Attraction, Long Run Stochastic Stability and the Speed of Step-by-step Evolution[END_REF]]. In section 3, we characterize out-of-equilibrium trading in the Scarf economy and give sufficient conditions on the price updating mechanism to ensure the stochastic stability of equilibrium. Section 4 offers our conclusion.

Evolutionary dynamics in exchange economies

We aim at investigating evolutionary dynamics in exchange economies where each agent carries a private vector of prices (i.e has a private valuation of goods), uses these private prices in order to determine acceptable trades, update them by imitating those of peers who were more successful in the trading process, and randomly mutate them in some instances.

More precisely, let us consider an exchange economy with L goods1 , N types of agents2 and M agents of each type3 . All the agents have Q := R L + as consumption set. Agents of type i are characterized by an utility function u i : Q → R and a vector of initial endowment ω i ∈ Q. Moreover, agent (i, j) (the jth agent of type i) is endowed with a normalized vector p i,j of private prices chosen in a finite subset P of the unit simplex of

R L + , S := {p ∈ R L + | L =1 p = 1}.
The population of agents is then characterized by a vector π ∈ Π = P M ×N .

Repeated bilateral trades between agents define a trading process, which allocates as a function of agents private prices the total resources of the economy. This process might involve some randomness in order to cope with rationing in out of equilibrium situations. In all generality, we can represent the trading process by a transition measure T from Π to Ξ which associates to a population of prices π ∈ Π, a probability distribution T π on the set of allocations4 Ξ defined as:

Ξ = {ξ ∈ Q N ×M | N i=1 M j=1 ξ i,j = M N i=1 ω i }, (1)
where ξ i,j represents the allocation to the jth agent of type i.

Private prices are then updated through an imitation process: agents imitate peers of the same type taking into consideration the utility gained through trading. In all generality, we can represent this imitation process as associating to a population of prices π ∈ Π and to an allocation ξ ∈ Ξ, a probability distribution I (π,ξ) on Π (which gives the distribution of prices after updating).

We are then concerned with the dynamics of private prices generated by the sequential iteration of trading and imitation processes. That is the process in which initial endowments are reinitialized at the beginning of each step, agents trade according to their private prices and update these as a function of the utility gained. This corresponds to the Markovian dynamics on Π defined by the transition matrix F such that:

F π,π = ξ ∈Ξ I (π,ξ) (π ) dT π (ξ) (2)
If agents then randomly and independently mutate (i.e randomly choose a new price in P ) with probability > 0, the dynamics are modified according to:

F π,π = ρ ∈Π R ρ,π dF π,ρ = ρ∈Π R ρ,π F π,ρ (3)
where ,π ) and δ(ρ, π ) denotes the number of mutations, that is the cardinal of the set {(i, j) | ρ i,j = π i,j }.

R (ρ, π ) = (1 -) M N -δ(ρ,π ) × ( |P | -1 ) δ(ρ
The family (F ) ≥0 then is a model of evolution in the sense of [START_REF] Ellison | Basins of Attraction, Long Run Stochastic Stability and the Speed of Step-by-step Evolution[END_REF]]), that is satisfies the following conditions:

1. F is ergodic for each > 0, 2. F is continuous in and F 0 = F, 3. there exists5 a function c : P n×m × P n×m → N such that for all π, π ∈

P n×m , lim →0 F (π,π )
c(π,π ) exists and is strictly positive.

Condition (1) implies in particular that for each > 0, F has a unique invariant distribution ψ . A population π ∈ P n×m is then called stochastically stable if lim →0 ψ (π) > 0.

This notion of stochastic stability can be used for the analysis of the stability of the equilibria of the underlying exchange economy thanks to the identification of an equilibrium price p with the population π such that every agent uses price p (that is such that for all (i, j), one has π i,j = p). The equilibrium associated with the price p can then be called stochastically stable if π is. The interesting case is this where π is the only stochastically stable population which implies that lim →0 ψ (π) = 1 and that for vanishingly small perturbations the process eventually settles in π independently of the initial conditions, in other words converges to equilibrium.

3 Stochastic stability in the Scarf Economy

The Economy

We shall investigate the stochastic stability of equilibrium in the [START_REF] Scarf | Some examples of global instability of competitive equilibrium[END_REF]] economy, which is probably the best known example of non-stability of the tâtonnement process. In this economy, there are three goods and three type of agents whose respective utility, endowment and demand are:

u 1 (x 1 , x 2 , x 3 ) = min(x 1 , x 2 ), ω 1 = (1, 0, 0), d 1 (p 1 , p 2 , p 3 ) = p 1 p 1 + p 2 (1, 1, 0) u 2 (x 1 , x 2 , x 3 ) = min(x 2 , x 3 ), ω 2 = (0, 1, 0), d 2 (p 1 , p 2 , p 3 ) = p 2 p 2 + p 3 (0, 1, 1) u 3 (x 1 , x 2 , x 3 ) = min(x 1 , x 3 ), ω 3 = (0, 0, 1), d 3 (p 1 , p 2 , p 3 ) = p 3 p 1 + p 3 (1, 0, 1)
This economy has an unique equilibrium for the price p

= ( 1 3 , 1 3 , 1 3
). Whereas, it is well-known that the law of demand does not hold and that the tâtonnement process does not converge but follows periodic orbits (see [START_REF] Negishi | The Stability of a Competitive Economy: A Survey Article[END_REF]), simulations of evolutionary dynamics in the Scarf economy by [START_REF] Gintis | The Dynamics of General Equilibrium[END_REF]] suggest convergence to equilibrium might hold for dynamics of the kind introduced in the preceding section. Accordingly, we provide below a proof of the stochastic stability of equilibrium in a stylized version of [START_REF] Gintis | The Dynamics of General Equilibrium[END_REF]] model.

Out-of-equilibrium trading

We first characterize the outcome of out-of-equilibrium trading in an economy with a single agent of each type and a public price. This characterization builds on two premises. First, we shall assume that trading is efficient in the sense put forward by the Hahn process (see [START_REF] Hahn | A theorem on non-tatonnement stability[END_REF]] and [START_REF] Fisher | Disequilibrium Foundations of Equilibrium Economics[END_REF]] for an in-depth discussion): after trade there are not both unsatisfied suppliers and unsatisfied demanders for any given good. Second, as in [START_REF] Gintis | The Dynamics of General Equilibrium[END_REF]], we consider that an agent allocates its initial endowment in priority to the fulfillment of its own demand at the expense of unfulfilled 'outside" demand. Hence agents strategically restrict their out-of-equilibrium trade in order to maximize their own utility. In the Scarf economy, the aggregate excess demand at a price (p 1 , p 2 , p 3 ) is given by

Z(p 1 , p 2 , p 3 ) = ( p 1 p 1 + p 2 + p 3 p 1 + p 3 -1, p 1 p 1 + p 2 + p 2 p 2 + p 3 -1, p 2 p 2 + p 3 + p 3 p 1 + p 3 -1)
and one has:

• Excess demand for good 1 if p 1 p 1 + p 2 + p 3 p 1 + p 3 > 1 or equivalently p 3 > p 2 .
• Excess demand for good 2 if

p 1 p 1 + p 2 + p 2 p 2 + p 3 > 1 or equivalently p 1 > p 3 .
• Excess demand for good 3 if

p 2 p 2 + p 3 + p 3 p 1 + p 3 > 1 or equivalently p 2 > p 1 .
The outcome of trading at price (p 1 , p 2 , p 3 ), is then completely determined by the assumptions that trading is efficient and that agents strategically restrict their out-of-equilibrium trade. There are essentially three types of situation:

1. The price is such that there is rationing on two markets: for example, p satisfies p 2 > p 1 > p 3 , so that there is rationing on good 2 and 3 markets.

In this setting, agent 3 won't be rationed because he can strategically restrict its sales in order to fulfill its own demand of good 3 and because good 1 is not rationed at all. Hence, the trading process shall yield agent 3 the allocation p 3 p 1 + p 3

(1, 0, 1). Agent 2 will then be rationed in good 3, as he might only be allocated the remaining p 1 p 1 + p 3 units of good 3, rather than the p 2 p 2 + p 3 units he demands. Consequently, agent 2 has no interest in retaining more than p 1 p 1 + p 3 units of good 2 and can supply p 3 p 1 + p 3 units of that good to agent 1. There are then two cases:

(a) If p 1 p 1 + p 2 ≤ p 3 p 1 + p 3
, that is p 2 1 ≤ p 3 p 2 , agent 1 is rationed and the agents' allocations and utilities are given by: agent allocation utility

1 p 1 p 1 + p 2 (1, 1, 0) p 1 p 1 + p 2 2 ( p 2 p 1 + p 2 - p 3 p 1 + p 3 , p 2 p 1 + p 2 , p 1 p 1 + p 3 ) p 1 p 1 + p 3 3 p 3 p 1 + p 3 (1, 0, 1) p 3 p 1 + p 3 (b) If p 1 p 1 + p 2 > p 3 p 1 + p 3
, that is p 2 1 > p 3 p 2 , both agents 1 and 2 are rationed and the agents' allocations and utilities are given by: 5 Documents de Travail du Centre d'Economie de la Sorbonne -2012.66 agent allocation utility

1 (1 - p 2 p 1 p 3 p 1 + p 3 , p 3 p 1 + p 3 , 0) p 3 p 1 + p 3 2 (( p 2 p 1 -1) p 3 p 1 + p 3 , p 1 p 1 + p 3 , p 1 p 1 + p 3 ) p 1 p 1 + p 3 3 p 3 p 1 + p 3
(1, 0, 1).

p 3 p 1 + p 3 2.
The price is such that there is rationing on a single market, for example p satisfies p 1 > p 2 > p 3 , so that there is rationing on good 2 market only.

In this setting, agent 3 won't be rationed because there is no rationing in either of the good he demands, agent 2 won't be rationed either because there is no rationing in good 1 and he can strategically restrict its sales of good 2 in order to fulfill its own demand. The agents' allocations and utilities are given by: agent allocation utility

1 ( p 1 p 1 + p 3 , p 3 p 2 + p 3 , p p 1 + p 3 - p 2 p 2 + p 3 ) p 3 p 2 + p 3 2 ( p 2 p 2 + p 3 (0, 1, 1) p 2 p 2 + p 3 3 p 3 p 1 + p 3 (1, 0, 1) p 3 p 1 + p 3
Up to a permutation of indices, the other cases are similar. There is either rationing on two markets (if p 3 > p 2 > p 1 or p 1 > p 3 > p 2 ,) or rationing on a single market (if p 3 > p 1 > p 2 or p 2 > p 3 > p 1 ). One can then define the outof-equilibrium allocation rule in the Scarf economy as the mapping x : P → Q 3 such that: 6 Documents de Travail du Centre d'Economie de la Sorbonne -2012.66

x 1 (p 1 , p 2 , p 3 ) =                                        (1 - p 2 p 1 p 3 p 1 + p 3 , p 3 p 1 + p 3 , 0) if p 2 > p 1 > p 3 and p 2 1 > p 3 p 2 ( p 1 p 1 + p 3 , p 3 p 2 + p 3 , p 1 p 1 + p 3 - p 2 p 2 + p 3 ) if p 1 > p 2 > p 3 ( p 1 p 3 + p 1 , p 2 p 2 + p 1 , p 1 p 3 + p 1 - p 2 p 3 + p 2 ) if p 1 > p 3 > p 2 and p 2 3 > p 2 p 1 ( p 3 p 3 + p 2 , p 3 p 3 + p 2 , ( p 1 p 3 -1) p 2 p 3 + p 2 ) if p 1 > p 3 > p 2 and p 2 3 ≤ p 2 p 1 p 1 p 1 + p 2 (1, 1, 0) otherwise. x 2,σ(i) (p 1 , p 2 , p 3 ) = x 1,i (p σ(1) , p σ(2) , p σ(3) ) where σ(1) = 2, σ(2) = 3, σ(3) = 1, x 3,τ (i) (p 1 , p 2 , p 3 ) = x 1,i (p τ (1) , p τ (2) , p τ (3) ) where τ (1) = 3, τ (2) = 1, τ (3) = 2, (4)

Trading process for a population of agents

We now want to define the outcome of out-of-equilibrium trading for a population of agents using private prices. In Gintis' model, agents assess the value of trades using their private prices and perform those which have a non-negative value. However, tracking all such possible trades lead to excessive combinatorial complexity. In order to provide a parsimonious analysis, we rather consider the benchmark situation where agents only trade with peers using the same price. This restriction in fact discards "lucky" trades which would increase the value of one's stock. It is also standard in the non-tâtonnement literature (see e.g [START_REF] Fisher | Disequilibrium Foundations of Equilibrium Economics[END_REF]). An alternative interpretation of this restriction of trading to peers using the same price is to consider that different market places or trading posts coexist, in each of which exchanges are performed according to one price, and that the private price of an agent is a marker of the market he is "affiliated" to. The choice of a private price by an agent can then be seen as a form of voting with one's feet for a set of exchange ratios.

Formally, we place ourselves in the framework of section 2, with N = 3, utilities and endowments being those of the Scarf economy and we assume P contains the equilibrium price p

= ( 1 3 , 1 3 , 1 3
). We also let µ i (π, p) denote the number of agents of type i using price p in the population π, µ(π, p) = N i=1 µ i (π, p) denotes the total number of agents using price p in the population π, and µ(π, p) := min(µ 1 (π, p), µ 2 (π, p), µ 3 (π, p)) denotes the minimal number of agents using price p among the different types in the population π.

In this setting the assumptions that agents only trade with peers using the same price and that they strategically restrict their trades amount to consider that agents of type i using price p collectively offer µ i (π, p)(ω i -x i (p)) units of good i to their peers of other types using price p. To reduce the combinatorial complexity of the analysis, we shall restrict attention to the simplest allocations consistent with these constraints, namely those where µ(π, p) among the agents using price p receive the allocation x(p), while the remaining retain their initial allocation and do not trade. The random component of the trading process is hence restricted to the designation of trading and non-trading agents. This amounts to assume that for all population π ∈ Π and type of agent i6 , one has:

T π    ξ ∈ Ξ | card{j | π i,j = p ∧ ξ i,j = x i (p)} = µ(π, p) card{j | π i,j = p ∧ x i,j = x i (p)} + card{j | π i,j = p ∧ ξ i,j = ω i } = µ i (π, p)    = 1 (I)
We hence obtain a simplified representation of the trading process of [START_REF] Gintis | The Dynamics of General Equilibrium[END_REF]], by assuming that trading is efficient and takes place only among subgroups consisting of a similar number of agents of each type.

Imitation process

In order to characterize the imitation process, we have to specify the probability of the event that the jth agent of type i adopt the price of the j th agent of type i . We denote this event by {(i, j) → (i , j )} and assume that for all pair of agents (i, j) and (i , j ), one has:

I (π,ξ) {(i, j) → (i , j )} > 0 ⇔            (i) i = i and ξ i,j = ω i and u i (ξ i,j ) ≤ u i (ξ i,j ) or (ii) µ(π, π i ,j ) = max p∈P µ(π, p) or (iii) (i, j) = (i , j ) (II)
Interpreted in conjunction with assumption (I), condition (i) states that agents which have traded have a positive probability to copy the prices of a peer who has been more successful in trading7 .

Condition (ii) states that agents always have a positive probability to copy the most widely used price in the population. This prevents in particular that the system gets stuck in the extremely inefficient situation where each type of agent uses a different price.

Condition (iii) states that an agent can conserve his price with some probability.

As far as the relationships between agents' imitation behavior is concerned, we shall assume there is independent inertia, that is the imitation probabilities are independent. Namely, for all agents (i, j), (i , j ), (i , j ), (i , j ) with (i, j) = (i , j ), one has:

I (π,ξ) {(i, j) → (i , j ) ∧ (i , j ) → (i , j )} = I (π,ξ) {(i, j) → (i , j )} × I (π,ξ) {(i , j ) → (i , j )} (III)

Stochastic stability of equilibrium in the Scarf economy

Note that the population π where each agent use the equilibrium price p (i.e for all (i, j), π i,j = p) is an invariant distribution of the process F under assumption (II) and that under assumption (I) it can naturally be identified with the equilibrium of the Scarf economy as the trading process T π then allocates to each agent the corresponding equilibrium allocation with probability one. Conditions (I), (II) and (III) are in fact sufficient to prove our main result:

Theorem 1 Under conditions (I), (II), and (III), the population π is the only stochastically stable state of the dynamics (F ) ≥0 in the Scarf economy

Proof: The proof mainly builds on [START_REF] Ellison | Basins of Attraction, Long Run Stochastic Stability and the Speed of Step-by-step Evolution[END_REF]] radius-coradius theorem, which makes use of the following notions:

• A path from π ∈ P n×m to π ∈ P n×m is a finite sequence of states, π 1 , • • • , π K ∈ P n×m , such that π 1 = π and π K = π . The set of paths from π to π is denoted by S(π, π ). The cost of a path (π 1 , • • • , π K ) is defined as:

c(π 1 , • • • , π K ) = K-1 k=1 c(π k , π k+1 ) (5)
One can then remark that equation (3) implies that c(π, π ) = 0 whenever F π,π > 0, that is whenever there is a positive probability to reach π from π via the unperturbed process, and that c(π, π ) is bounded above by the number of distinct prices between π and π , that is c(π, π

) ≤ card {(i, j) | π i,j = π i,j }.
• The basin of attraction of the population π is the set of initial states from which the unperturbed Markov process (with transition probability F) converges to π with probability one, that is:

D(π) = {π ∈ P n×m | lim T →+∞ F T π,π = 1} (6)
• The radius r(π) of the population π is then defined as the minimal cost of a path leaving D(π). That is letting S(π, D(π) c ) := ∪ π∈D(π) c S(π, π) denote the set of paths out of D(π), one has:

r(π) = min s∈S(π,D(π) c ) c(s) (7)
• Finally, the coradius of π is defined as the maximal cost of a transition to π cr(π) = max π =π min s∈S(π,π) c(s)

Following theorem 1 in [START_REF] Ellison | Basins of Attraction, Long Run Stochastic Stability and the Speed of Step-by-step Evolution[END_REF]], in order to prove that π is stochastically stable, it suffices to prove that its radius if greater than its coradius. We shall precisely prove that: r(π) > 3 and cr(π) ≤ 3 (9)

• Let us first prove that cr(π) ≤ 3. Part (ii) of assumption (II) ensures that there always exists a zero cost path from any population to a uniform one where each agent uses the same price. So, without loss of generality, we can restrict attention to populations π such that for all (i, j), π i,j = p. Moreover as in section (3.2), we can without loss of generality restrict attention to the cases where p is such that p 2 > p 1 > p 3 or p 1 > p 2 > p 3 .

In either cases, one has u 1 (x 1 (p)) < 1 2 and u 3 (x 3 (p)) < 1 2 .

Let then π be such that for all i, π i,1 = p and and for all j = 1, π i,j = π i,j = p. One clearly has c(π, π ) = 3. (10)

Moreover, according to assumption (I), any ξ such that T π (ξ) > 0 should satisfy u 1 (ξ 1,1 ) = u 1 (x 1 (π 1,1 )) = u 1 (x 1 (p)) = 1 2 and for j = 1 u 1 (ξ 1,j ) =

u 1 (x 1 (π 1,j )) = u 1 (x 1 (p)) < 1 2
, so that for all j = 1, u 1 (ξ 1,j ) < u 1 (ξ 1,1 ).

Similarly, one obtains for all j = 1, u 3 (ξ 3,j ) < u 1 (ξ 3,1 ).

Let then π be such that for all j, π 1,j = π 1,j = p, π 3,j = π 3,j = p and π 2,j = π 2,j . Using part (i) of assumption (II), one clearly has

c(π, π ) = 0 (11)
Finally, as it is clear that µ(π , p) = max p∈P µ(π, p), one has using part (ii) of assumption (II) that c(π , π) = 0. ( 12)

Equations (10), to (12) eventually yield:

cr(π) ≤ 3. ( 13
)
This ends the first part of the proof.

• Let us then prove that r(π) > 3 :

-Assumption (II) clearly implies that F π,π = 1, so that for any π = π, one has c(π, π) = 0. Hence r(π) > 0 10 Documents de Travail du Centre d'Economie de la Sorbonne -2012.66

-Moreover equation (3) implies that for any π such that c(π, π) ≤ 2, one has for any p = p, µ(π, p) = 0 as well as p =p µ(π, p) ≤ 2. Let us consider (i, j) ∈ such that π i,j = p and ξ ∈ Ξ with T π (ξ) > 0. Assumption (I) implies that ξ i,j = ω i . As moreover, one clearly hasn't µ(π, π i,j ) = max p∈P µ(π, p), assumption (II) in fact implies that for any (i , j ) = (i, j), I (π,ξ) {(i, j) → (i , j )} = 0. Hence, any π such that c(π, π ) = 0 must satisfy µ(π , p) = 0 as well as p =p µ(π , p) ≤ 2. By recursion the same can be proven for any population π such that c(s) = 0 for some s ∈ S(π, π ). Moreover, part (ii) of assumption (II) ensures that any such π satisfies F π ,π > 0. This clearly implies that any such π and in particular π belong to D(π). We have hence proven that r(π) > 2.

-Finally, if π is such that c(π, π) ≤ 3, either one has for any p = p, µ(π, p) = 0 as well as p =p µ(π, p) ≤ 3 and one can prove as in the preceding case that π ∈ D(π), or there exists p = p such that µ(π, p) = 1. Without loss of generality, we can then assume that for all i π(i, 1) = p and for all j = 1, π(i, j) = p. Also, following remark (3.2), we can restrict attention to the cases where p is such that p 2 > p 1 > p 3 or p 1 > p 2 > p 3 . In either cases, one has u 1 (x 1 (p)) < 1 2 and u 3 (x 3 (p)) < 1 2 whereas one has respectively

u 1 (x 1 (p)) = 1 2 and u 3 (x 3 (p)) = 1 2
. Assumption (II) then implies that for any ξ ∈ Ξ with T π (ξ) > 0, one has I (π,ξ) {(1, j) → (1, 1)} = 0 and I (π,ξ) {(3, j) → (3, 1)} = 0, although one might have I (π,x) {(2, j) → (2, 1)} > 08 . This implies that for any π such that c(π, π ) = 0, one has µ 1 (π , p) ≤ 1, µ 2 (π , p) ≤ M, and µ 3 (π , p) ≤ 1, as well as µ 1 (π , p) ≥ M -1, and µ 3 (π , p) ≥ M -1. An immediate recursion show the same is true for any π such that c(s) = 0 for some s ∈ S(π, π ). Part (ii) of assumption (II) ensures that any such π satisfies F π ,π > 0. This clearly implies that any such π and in particular π belong to D(π). We have hence proven that r(π) > 3.

This ends the proof.

Basic asymptotic properties of ergodic Markov chains yield a straightforward interpretation of theorem 1 : as the mutation rate tends towards zero, the frequency with which the system lies in equilibrium tends towards one.

Concluding remarks

We have shown that in a framework where trading is governed by private prices and strategic behavior, agents who update their private prices by imitation and random mutation will eventually adopt the equilibrium price (but for vanishingly small perturbations) and obtain their equilibrium allocation in the Scarf economy. Hence, we provide some analytical support to the result obtained in series of simulations performed by Gintis: [START_REF] Gintis | The Dynamics of General Equilibrium[END_REF]] and [START_REF] Gintis | The Dynamics of Pure Market Exchange[END_REF]]. One should however take note that this result has only been obtained at the expense of a considerable simplification of Gintis' model of market exchange. It is also the case that our results rely crucially on the fact that mutations are drawn from an uniform distribution and hence the remark in [START_REF] Fudenberg | Evolutionary dynamics with aggregate shocks[END_REF]] is particularly relevant in our setting: "Intuitively, the likelihood that a Wiener process will be able to "swim upstream" k meters against a deterministic flow depends both on the distance k and on the strength of the flow, while the probability that a discrete-time system jumps k or more meters "over the flow" in a single period depends on k but not on the strength of the flow. This explains the differences in the generality of the models' conclusions, and suggests that long-run behavior may depend on the precise form of the deterministic dynamics in any model with continuous sample paths." As a matter of fact, we would conjecture that our results hold for weaker sets of assumption than those put forward here but our current results suggest that the problem becomes more and more complex from the combinatorial point to view as these assumptions are relaxed. Further investigations are however necessary to determine wether the results presented here and in [START_REF] Gintis | The Dynamics of General Equilibrium[END_REF]] and [START_REF] Gintis | The Dynamics of Pure Market Exchange[END_REF]] can be turned into the conjecture that the general equilibrium of an economy always is stochastically stable for a certain class of evolutionary dynamics.

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  Documents de Travail du Centre d'Economie de la Sorbonne -2012.66

indexed by = 1 • • • L

indexed by i = 1 • • • N

indexed by j = 1 • • • M 2 Documents de Travail du Centre d'Economie de la Sorbonne -2012.66

We shall assume that Ξ is endowed with the Borel σ-algebra

This last point follows form the fact that the coefficients of F (π,π ) are polynomia in . 3 Documents de Travail du Centre d'Economie de la Sorbonne -2012.66

Documents de Travail du Centre d'Economie de la Sorbonne -2012.66

card(E) denotes the cardinal of the set E.

We could as well assume that all agents can copy a peer's price but we should then assume that they compare the utility each price yeilded to agents which have actually traded.

That is agents of type 2 might adopt price p but not agents of type 1 and 3.