Amirat Abdelaziz
email: abdelaziz.amirat@univ-annaba.dz

Martin Patrick
email: patrick.martin@ensam.eu

Chaoui Kamel
email: chaoui_k@univ-annaba.dz

Chatti Sami
email: sami.chatti@iul.tu-dortmund.de

MasTech : a flexible modular master programme in technology developped whithin a Tempus Project

Abstract-In today's competitive industry and in view of recent economic turbulences new frontiers of challenges emerge that require new educational paradigms accompanied by new tools and methodologies applicable to all aspects of engineering areas including the functional and organizational aspects. In accordance with the objectives stipulated by the Council of European Union work programme on the future of education and training, a Tempus project (2010-2013) has been mounted to develop a novel model for modular programmes to be used in education of technology specialities at master level. The model is implemented in manufacturing technology and management area and has general applicability for technology education in several fields. The main feature of this project consists in flexibility, adaptability, dynamic interactivity while consolidating theoretical and practical skills. MasTech is the name of a flexible modular master two-year programme in technology being developed according to the Bologna process that is to be adapted to the particular conditions of the universities in Algeria, Morocco and Tunisia. Three European Universities (Sweden, Germany, France) are involved in the project. This paper introduces MasTech and describes the different steps that have been followed to develop the master programme taking into account both academic and industrial needs and priorities. Results are expressed in terms of a professional master programme that has been submitted for accreditation.

Keywords-Tempus, Master, Manufacturing, Flexible Modular

I. INTRODUCTION

The Trans-European Mobility scheme for university (Tempus), is European Union's programme which supports the modernization of higher education in Partner Countries of Eastern Europe, Central Asia, the Western Balkans and the Mediterranean region, mainly through university cooperation projects [1]. Tempus was first adopted in 1990 following the fall of the Berlin Wall in order to respond to the modernization of higher education in Central and Eastern European countries. Then every 5 years, it was extended to allow more and more countries to be included in the programme.

The aim of this paper is to present MasTech, flexible modular master programme in technology supported by Tempus IV (2007)(2008)(2009)(2010)(2011)(2012)(2013). It describes the different work packages that have been proposed to lead to the development of the master programme allowing the creation of a Professional Master in Technology of production Engineering.

II. MASTECH PROJECT PRESENTATION

The flexible modular master programme in technology baptized MasTech is Tempus programme for three years starting from 15 October 2010 to 14 October 2013 [2]. It is in the form of Joint Projects with Curricular Reform -English under the code of 511277-TEMPUS-1-2010-1-DE-TEMPUS-JPCR. The domination is defined as follows in Table 1.

The applicant of MasTech project is the Technische Universität Dortmund, Germany which plays a great role as a coordinator of 10 partners, 3 universities from Europ and 6 universities from 3 Maghreb countries. The 10th partner comes from industry, Table 2.

III. MasTech PROJECT OBJECTIVES

According to the directives of the Tempus programmes, MasTech project aims for 7 main objectives:

-Developing a novel modular curriculum for a two-year Master programme in Technology reforming the higher education and in the same time adapting to the particular conditions of each partner country (PC).

-Encouraging the mobility of teachers as well as students between universities in the partner countries -Providing the EU state of the art education in the technological fields that results in more flexibility in learning and practical qualification.

-Enhancing transparency and comparability of the PC educational systems and facilitating the recognition of studies abroad.

-Restructuring the education system in technology to enhance the competitiveness of PC universities within their own countries and in the global market place.

-Focusing the education in technological fields specific for partner countries industries, thus making easier the access to the labour market.

-Fortifying the university-enterprises relationship IV. MasTech WORK PACKAGES 8 work packages have been determined to develop MasTech project. Each work package has been proposed with its deliverables [START_REF]eForm_TUD_tempus_MasTech_2010_en Tempus IV-3rd call for proposals Joint projects[END_REF] that have been programmed within a predetermined time, Table 3. Each work package is put under a leading partner to coordinate the development of its content by all the partners.

V. MasTech PROGRAMME STRUCTURE

The MasTech structure proposes 4 mains parts, the basics modulus (BM), the specialization modulus (SM), the project and laboratory works (PLW) and a Master thesis (MT). The basics modulus are compulsory and taught in one semester whereas the specialization modulus are divided into 2 parts. Part I is given in the second semester and is compulsory. Part II concerns only the specialization that characterizes every partner in the projects. Table 4 shows the MasTech Structure that indicates the repartition of basic modulus over the 3 teaching semester and the specialization modulus. In the third semester in addition to some basics modulus each partner offers its specialization modulus. It is worth noting that 5 modulus are offered in each semester. The 4th semester is dedicated to the preparation of the thesis. The repartition of the credits allows the mobility of the student easily in any semester. There are 6 specialization modulus offered by each partner as shown in Figure 2. The modulus can be composed of two or more than two subjects closely related one to each other. For each subject, course content has been developed with its syllabus. This is an important step in developing the Master programme in Technology for each university partner according to the specification of accreditation of each country. Then, for each syllabus, course content, work laboratory and project are developed.

FIGURE III. SCHEMATIC STEPS OF THE DEVELOPPEMENT OF MASTECH

The courses are organized into slides in English and checked in order to set then within the website platform [4] that has been developed for this purpose. It is worth noting that there are 36 subjects being developed to be introduced into the web site platform courses. Figure 3 illustrates the different steps for the development of a MasTech Master Programme in Technology.

Figure 4 shows a sampling overview of the project content.

VI. MasTech MAIN RESULTS During the last three years an intensive investigation has been made to reach the main objectives of this project. From the beginning a kick off meeting was the real starter meeting that allowed to sort out the strategy to adopt in conducting the project. Each country partner was represented by at least two persons. Three main sections have been programmed. The first section consisted in introducing the partners and presenting the project by giving a short background on how the project has been mounted, its objectives, the content expressed by the deliverables and the time table over the 3 years. The second section has been accorded to the financial and administrative matters in which the budget, the grant distribution on work packages, the budget categories, the cost reporting the rates and all financial points have been exposed and clarified. Sincere and trustful discussions longed very late to make every thing clear and throw away any misunderstood idea. The third section was devoted to WP1.1 and WP1.2 where each partner has presented the part of work on which he is responsible. It is interesting to remind that WP1.1 consisted in Academic transfer and comparison of PC/EU education" . The next step was therefore to develop the syllabus for each course that have validated after a large discussion to make them coherent, pertinent and reliable as well to avoid any repeated chapters from one course to another. The validation of the syllabus allowed in one hand to present a proposal of the Master programme according to a template for each partner country and in the other hand to develop the courses. As far as the UBMA is concerned a professional Master programme in technology of production engineering (MsTPM) is submitted in February 2013 and accepted for accreditation by the Algerian university national commission end of Jully 2013. The courses are being developed in the form of slides to be set within the MasTech Platform [4]. This Master is supported by all MasTech Tempus partners and 3 main companies in Annaba region. Table 5 shows the 3 organized structure of the Master programme in Technology of production engineering [START_REF] Chaoui | Technologies de Production Mécanique :Un premier master professionnalisant en mécanique à l'Université d'Annaba[END_REF].

At the end of the project, it is important to note that 3 summer schools have been organized in the last semester in the three Maghreb countries on 18 selected students (3 per each university) in order to appreciate and valid the courses.

It is also worth noting that there have been many difficulties encountered during the development of the project. The respect of the time table is very dependant on the availability of the members and the syllabus and courses developers. The procedure of payment of travel and staff cost together student cost was the main problem to solve from the beginning. The contribution of developers in setting the courses within the Mastech platform is fairly sensitive to the their comprehension and some preparation. All MasTech members are very worried about the acquisition of equipment. The procedure is very difficult as it depends on each government administration.

CONCLUSION

MasTech or the flexible modular master in technology allowed developing and implementing a master programme encouraging the mobility of teachers as well as students between Maghreb universities. It is a new modular curriculum for an innovative two-year master programme of excellence in technology reforming the higher education. The Master programme consists in basic and specialty modules. The success of this Master is due to a strong consortium with partners having long experience in TEMPUS cooperation.

Figure 1

 1 Figure1shows the repartition of required staff days for the fulfillment of the project. 6 ordinary meetings have been programmed along the three years to assess the progression of the work packages where each leading partner is due to present the state of development of the corresponding work package.

Flexible flexibility in conduction

  FIGURE I. REPARTITION OF MASTECH WORK PACKAGESONTO PARTNERS

  FIGURE IV. SAMPLING OVERVIEW OF MASTECH CONTENT

TABLE II .

 II LIST OF MASTECH INSTITUTION PARTNERS

TABLE III .

 III

LIST OF MASTECH WORK PACKAGES

TABLE IV

 IV

	FIGURE II. SPECIALIZATION MODULUS WITHIN MASTECH
	AND THE CORRESPONDING PARTNER
				Number of Staff Days	Partners Involved
	Work Packages	WP.1 WP.2 WP.3 WP.4 WP.5 WP.6 WP.7 WP.8 Work Package Number	229 891 428 351 253 128 130 406 Staff Days per Package	26 37 102 18 68 16 78 26 74 28 117 58 42 60 34 24 112 54 18 106 46 24 106 50 12 110 36 18 36 12 24 11 40 52 32 22 10 10 18 10 10 22	16 78 76 71 18 24 22 18 18 10 46 32 22 26 44 42	4 1 2 3 5 6 7 8 9 10 1 2 3 4 5 6	2 1 3 4 5 6 7 8 9 10 9 1 2 3 4	8 1 2 3 4 5 6 7 9 10 1 2 3 6	7 2 3 4 5 6 8 5 1 2 10 9 1	Modular Master Programme in Technologie
				10 8 8 10	12 11 11 12	22 25 27 28	52 60 62 20	7 8 9 10	5 6 7 8	4 5 7 8	3 4 6 7	Flexible
				12	13	25			10	9	8
				10	6					10	9
				0							10
	.	MASTECH PROGRAMME STRUCTURE

TABLE V

 V

	. FOLIO OF THE ORGANIZED PROGRAMME
	OF MsTPM (3 SEMESTERS, 4TH SEMESTER
	FOR MASTER THESIS [5]

ACKNOWLEDGMENT

The Authors would like first to thank the European Union for the financial support, all the partners of the MasTech project without whom this project could not be realized.

The project of the professional Matser programme in Technology of Production Engineering submitted by the University of Badji Mokhtar Annaba for accreditation in Algeria is supported by 3 Algerian companies: E.P.E Ferrovial Annaba, ALEMO Constantine and SOMEMI Jijel. The UBMA Team is very thankful to them.


A Gloria
email: agloria@ulb.ac.be

F Otto

Quantitative estimates on the periodic approximation of the corrector in stochastic homogenization

1 Introduction, setting, and main results

Introduction

This article corresponds to a course given by F. Otto at the summer school CEMRACS 2013 in Luminy, France.

The qualitative theory of stochastic homogenization dates back to the seminal contributions of Papanicolaou and Varadhan [START_REF] Papanicolaou | Boundary value problems with rapidly oscillating random coefficients[END_REF], and Kozlov [START_REF] Kozlov | The averaging of random operators[END_REF]. Let D a bounded domain of R d , f ∈ H -1 (D), and A be a stationary ergodic random field. Then the weak solution u ε ∈ H 1 0 (D) of

-∇ • a( • ε )∇u ε = f
almost surely weakly converges in H 1 (D) to the unique weak solution

u 0 ∈ H 1 0 (D) of -∇ • a hom ∇u 0 = f, (1)
where a hom is a deterministic matrix (independent of f ) called the homogenized matrix. As far as quantitative estimates are concerned there are only few contributions in the literature. A first general comment is that ergodicity alone is not enough to obtain convergence rates, so that mixing properties have to be assumed on the random coefficients a. Besides the optimal estimates in the one-dimensional case by Bourgeat and Piatnitskii [START_REF] Bourgeat | Estimates in probability of the residual between the random and the homogenized solutions of one-dimensional second-order operator[END_REF], the first and still unsurpassed contribution in the linear case is due to Yurinskiȋ who proved in [16, (0.10)] that for d > 2 and for mixing coefficients with an algebraic decay (not necessarily integrable), there exists γ > 0 such that

|u ε -u hom | 2 ε γ . (2)
In the case of discrete elliptic equations, the quantitative theory is much more developed and started with the inspiring unpublished work by Naddaf and Spencer [START_REF] Naddaf | Estimates on the variance of some homogenization problems[END_REF], who understood the right notion of mixing condition for stochastic homogenization in the form of a spectral gap estimate for the Glauber dynamics. In [START_REF] Gloria | An optimal variance estimate in stochastic homogenization of discrete elliptic equations[END_REF][START_REF] Gloria | An optimal error estimate in stochastic homogenization of discrete elliptic equations[END_REF] we proved the first optimal quantitative estimates on the corrector, on the variance of the spatial averages of the energy density of the corrector, and estimates on the difference between the homogenized coefficients and their approximations using a massive term in the corrector equation. In collaboration with Neukamm in [START_REF] Gloria | Quantification of ergodicity in stochastic homogenization: optimal bounds via spectral gap on Glauber dynamics[END_REF] we then obtained optimal estimates in any dimension on the corrector, on its periodic and massive approximations, and on the approximations of the homogenized coefficients by periodization or using a massive term. Marahrens and the second author proved in [START_REF] Marahrens | Annealed estimates on the Green's function[END_REF] optimal annealed estimates on the Green function, which in turn allowed us and Neukamm to prove an optimal quantitative two-scale expansion of u ε for all d ≥ 2, thus improving considerably [START_REF] Bourgeat | Estimates in probability of the residual between the random and the homogenized solutions of one-dimensional second-order operator[END_REF] in the discrete setting. More recent results go beyond variance estimates and address statistical properties of solutions such as central limit theorems (CLT). In particular, for the approximation of homogenized coefficients, a CLT was recently obtained by Biskup, Salvi and Wolff [START_REF] Biskup | A central limit theorem for the effective conductance: I. Linear boundary data and small ellipticity constrasts[END_REF] in the case of small ellipticity ratio (that is, for a coefficient field a close to identity) for independent and identically distributed coefficients, see also Rossignol [START_REF] Rossignol | Noise-stability and central limit theorems for effective resistance of random electric networks[END_REF]. An optimal quantitative version of this CLT (which estimates in particular the Wasserstein distance to normality) is proved by Nolen and the first author in [START_REF] Gloria | Quantitative central limit theorem for the effective conductance[END_REF] (without the smallness assumption on the ellipticity ratio). The route towards a CLT for the solution u ε itself started for d > 2 with the contribution by Mourrat and the second author on the structure of the asymptotic covariance of the corrector, see [START_REF] Mourrat | Correlation structure of the corrector in stochastic homogenization[END_REF].

The present contribution concerns the case of continuum linear elliptic equa-tions. It is based on [START_REF] Gloria | Quantitative results on the corrector equation in stochastic homogenization of linear elliptic equations[END_REF], which is a complete continuum version of [START_REF] Gloria | An optimal variance estimate in stochastic homogenization of discrete elliptic equations[END_REF][START_REF] Gloria | An optimal error estimate in stochastic homogenization of discrete elliptic equations[END_REF] (with in addition optimal results for d = 2). In the present contribution we establish quantitative results on the periodic approximation of the corrector equation for the stochastic homogenization of linear elliptic equations in divergence form, when the diffusion coefficients satisfy a spectral gap estimate in probability, and for d > 2. The main difference with respect to the first part of [START_REF] Gloria | Quantitative results on the corrector equation in stochastic homogenization of linear elliptic equations[END_REF] is that we avoid here the use of Green's functions and more directly rely on the De Giorgi-Nash-Moser theory. Let us also mention the work [START_REF] Nolen | Normal approximation for a random elliptic equation[END_REF] by Nolen, which follows [START_REF] Gloria | An optimal variance estimate in stochastic homogenization of discrete elliptic equations[END_REF] to establish estimates on the corrector, and proves a normal approximation result for a class of random coefficients in this continuum setting.

Setting

We start by introducing the relevant deterministic notions: The corrector φ(a; •) and the homogenized coefficient a L hom (a) for an arbitrary coefficient field a on the torus of side length L, which we sometimes denote as [-L 2 , L 2 ) d to single out the point 0.

Definition 1.

Space of coefficient fields. For a given side-length L let Ω be the space of all [-L 2 , L 2 ) d -periodic fields of d × d matrices a that are uniformly elliptic in the sense

∀ x ∈ - L 2 , L 2 d , ξ ∈ R d λ|ξ| 2 ≤ ξ • a(x)ξ, |a(x)ξ| ≤ |ξ|,
where λ > 0 is a number fixed throughout the article.

Corrector. For given a ∈ Ω, the corrector φ(a;

•) is an [-L 2 , L 2 ) d -periodic function defined through the elliptic equation -∇ • a(∇φ(a; •) + ξ) = 0 and ˆ[-L 2 , L 2 ) d φ(a; •) = 0, (3)
where ξ ∈ R d with |ξ| = 1 is a direction which is fixed throughout the article.

For further reference we note that φ is "stationary" in the sense of

φ(a(• + z), x) = φ(a, x + z) (4)
for all points x ∈ R d , coefficient fields a ∈ Ω, and shift vectors z ∈ R d .

Homogenized coefficient. The homogenized coefficient in directions ξ, ξ ′ is defined via

ξ ′ • a L hom (a)ξ := L -d ˆ[-L 2 , L 2 ) d ξ ′ • a(∇φ(a; •) + ξ), (5)
where ξ ′ with |ξ ′ | = 1 is a direction which is fixed throughout the article.

We now introduce our example of an ensemble on the space of coefficient fields on the torus.

Definition 2. By the "Poisson ensemble" we understand the following probability measure on Ω: Let the configuration of points X := {X n } n=1,••• ,N on the torus be distributed according to the Poisson point process with density one. This means the following

• For any two disjoint (Lebesgue measurable) subsets D and D ′ of the torus we have that the configuration of points in D and the configuration of points in D ′ are independent. In other words, if ζ is a function of X that depends on X only through X |D and ζ ′ is a function of X that depends on X only through X |D ′ we have

ζζ ′ 0 = ζ 0 ζ ′ 0 , (6)
where • 0 denotes the expectation w. r. t. the Poisson point process.

• For any (Lebesgue measurable) subset D of the torus, the number of points in D is Poisson distributed; the expected number is given by the Lebesgue measure of D.

Note that N is random, too. With any realization X = {X n } n=1,••• ,N of the Poisson point process, we associate the coefficient field a ∈ Ω via

a(x) = λ if x ∈ N n=1 B 1 (X n ) 1 else id. ( 7
)
Here and throughout the article, balls like B 1 (X n ) refer to the distance function of the torus. This defines a probability measure on Ω by "push-forward" of • 0 . We denote the expectation w. r.

ζ(a(• + z)) = ζ . (8)
Spectral Gap Estimate. The Poisson ensemble satisfies a Spectral Gap Estimate by which we understand the following: There exists a radius R only depending on d such that for any function ζ : Ω → R, we have

(ζ -ζ ) 2 ≤ z∈Z d ∩[-L 2 , L 2 ) d (osc B R (z) ζ) 2 . (9)
Here, for a (Lebesgue measurable) subset D of the torus, the (essential) oscillation osc D ζ of ζ with respect to D is a random variable defined through

(osc D ζ)(a) = sup{ζ(ã)|ã ∈ Ω with ã = a outside D} -inf{ζ(ã)|ã ∈ Ω with ã = a outside D}. (10

Main results

The main result of the article is a Central Limit Theorem-type scaling of the variance of the homogenized coefficient in terms of the system volume L d .

Theorem 1. Suppose • is stationary and satisfies the Spectral Gap Estimate. Then we have the following estimate on the variance of the periodic approximation of the homogenized coefficient

(ξ ′ • a L hom ξ -ξ ′ • a L hom ξ ) 2 ≤ C(d, λ)L -d .
In this article, we prove Theorem 1 only for d > 2. We shall derive it from the following result of independent interest, which is only true for d > 2.

Proposition 1. Let d > 2 and suppose • is stationary and satisfies the Spectral Gap Estimate. Then all moments of the corrector are bounded independently of L, that is, for any 1 ≤ p < ∞ we have

φ 2p ≤ C(d, λ, p).
Here and in the entire text, we write φ 2p for (φ 2 ) p , so that expressions like above make sense also for a non-integer exponent p.

Auxiliary results

We need the following L p (Ω)-version of the Spectral Gap Estimate.

Lemma 2. Let • satisfy the Spectral Gap Estimate. Then it satisfies an L p (Ω)-version of a Spectral Gap Estimate in the following sense: Let R be the radius from [START_REF] Kozlov | The averaging of random operators[END_REF]. Then we have for any (2p-integrable) function

ζ : Ω → R and any 1 ≤ p < ∞ (ζ -ζ ) 2p z∈Z d ∩[-L 2 , L 2 ) d (osc B R (z) ζ) 2 p . ( 11
)
Here means up to a generic constant that only depends on p.

Together with the previous lemma, the following lemma gives an estimate of φ in terms of ∇φ + ξ.

Lemma 3. Suppose d > 2 and that • is stationary. Then we have for any d d-2 < p < ∞ and any R large enough

z∈Z d ∩[-L 2 , L 2 ) d (osc B R (z) φ(•; 0)) 2 p ˆB1 |∇φ + ξ| 2 p , (12)
where means up to a generic constant only depending on d, λ, p, and R.

The last lemma in turn gives an estimate of ∇φ + ξ in terms of φ.

Lemma 4. Suppose that • is stationary. Then we have for any

2 ≤ p < ∞ ˆB1 |∇φ + ξ| 2 p φ 2(p-1) + 1,
where means up to a generic constant only depending on d, λ, and p.

Proofs

Proof of Lemma 1.

Step 1. Generalization and reduction. The most natural form of the result of the lemma is the following: For any measurable partition

D 1 , • • • , D N of the torus we have (ζ -ζ ) 2 ≤ N n=1 (osc B 1 (Dn) ζ) 2 , (13)
where B 1 (D) is the set of all points on the torus that have distance less than one to D. In this step, we will derive this from the following similar estimate on the Poisson point process itself:

(ζ 0 -ζ 0 0 ) 2 0 ≤ N n=1 (osc 0,Dn ζ 0 ) 2 0 . (14)
Here • 0 denotes the expectation w. r. t. to the Poisson point process

X := {X n } n=1,••• ,N , ζ is a (square integrable
) function of the point configuration X, and the oscillation osc 0,D is defined in a similar way to (10):

(osc 0,D ζ 0 )(X) = sup{ζ 0 ( X)| X = Xoutside D} -inf{ζ 0 ( X)| X = Xoutside D}. (15)
Indeed, ( 13) is an immediate consequence of ( 14) because of the following two facts.

• We recall that (7) defines a mapping X → a from point configurations to coefficient fields. As such, it pulls back functions according to ζ 0 (X) = ζ(a(X)) and pushes forward the ensemble according to

ζ = ζ 0 0 . (16)
In particular we have for the variance

(ζ -ζ ) 2 = (ζ 0 -ζ 0 0 ) 2 0 .
• By definition [START_REF] Gloria | An optimal error estimate in stochastic homogenization of discrete elliptic equations[END_REF], if the point configurations X and X coincide outside D, then the corresponding coefficient fields a(X; •) and a( X; •) coincide outside B 1 (D). Hence for a given configuration X, the set {a

( X)| X = X outside D} is contained in the set {ã|ã = a(X) outside B 1 (D)} so that sup{ζ(a( X))| X = X outside D} ≤ sup{ζ(ã)|ã = a(X) outside B 1 (D)},
and the opposite inequality if we replace the supremum by the infimum.

From the definitions [START_REF] Marahrens | Annealed estimates on the Green's function[END_REF] and [START_REF] Rossignol | Noise-stability and central limit theorems for effective resistance of random electric networks[END_REF] of the oscillation we thus see

(osc 0,D ζ 0 )(X) ≤ (osc B 1 (D) ζ)(a(X)).
By [START_REF] Yurinskiȋ | Averaging of symmetric diffusion in random medium[END_REF], this implies as desired

(osc 0,D ζ 0 ) 2 0 ≤ (osc B 1 (D) ζ) 2 .
Step

ζ|D ∪ D ′ = ζ|D ′ . ( 17
)
Here comes the argument: By definition of conditional expectation, (17) follows if for any pair of (bounded and measurable) test functions u and u ′ which only depend on X |D and X |D ′ , respectively, we have

ζuu ′ = ζ|D ′ uu ′ .
Indeed, on the one hand, since ζ only depends on X |D c (where D c denotes the complement of D) and u ′ does only depend on X |D ′ (and thus a fortiori only on X |D c ) while u only depends on X |D , we have from ( 6):

ζuu ′ = ζu ′ u .
On the other hand, since ζ|D ′ u ′ only depends on X |D ′ (and in particular only on X |D c ) while u only depends on X |D , we have from [START_REF] Gloria | An optimal variance estimate in stochastic homogenization of discrete elliptic equations[END_REF]:

ζ|D ′ uu ′ = ζ|D ′ u ′ u = ζu ′ |D ′ u = ζu ′ u ,
where the middle identity holds since u ′ only depends on X |D ′ .

Step 3. Conditional expectation and oscillation. For any (Lebesgue measurable) disjoint subsets D and D ′ of the torus and any (square integrable) function, we have

| ζ|D ∪ D ′ -ζ|D ′ | ≤ osc D ζ|D ′ . (18)
By exchanging ζ with -ζ, we see that it is enough to show

ζ|D ∪ D ′ ≤ ζ|D ′ + osc D ζ|D ′ , (19)
We note that sup D ζ ≤ ζ + osc D ζ, where we've set for abbreviation

(sup D ζ)(X) := sup{ζ( X)| X = X outside D}. Hence (19) follows from ζ|D ∪ D ′ ≤ sup D ζ|D ′ .
The latter inequality can be seen as follows

ζ|D ∪ D ′ ≤ sup D ζ|D ∪ D ′ since ζ ≤ sup D ζ (17) = sup D ζ|D ′ since sup D ζ does not depend on X |D .
Step 4. Martingale decomposition. For conciseness, we only prove [START_REF] Papanicolaou | Boundary value problems with rapidly oscillating random coefficients[END_REF] for N = 3. So let {D 1 , D 2 , D 3 } be a partition of the torus, we claim

(ζ -ζ ) 2 (20) = (ζ -ζ|D 1 ∪ D 2 ) 2 + ( ζ|D 1 ∪ D 2 -ζ|D 1 ) 2 + ( ζ|D 1 -ζ ) 2 .
Indeed, this follows from the fact that

ζ -ζ|D 1 ∪ D 2 , ζ|D 1 ∪ D 2 -ζ|D 1 , ζ|D 1 -ζ are L 2 (Ω) -orthogonal.
The latter can be seen as follows: By definition of •|D as L 2 (Ω)-orthogonal projection, the two last functions ζ|D 1 ∪ D 2 -ζ|D 1 and ζ|D 1ζ do only depend on X |D 1 ∪D 2 , so that they are orthogonal to the first function ζ -ζ|D 1 ∪ D 2 . It remains to argue that the two last functions ζ|D 1 ∪ D 2 -ζ|D 1 and ζ|D 1ζ are orthogonal. To that purpose, we rewrite the middle function as

ζ|D 1 ∪ D 2 -ζ|D 1 = ζ ′ -ζ ′ |D 1 where ζ ′ := ζ|D 1 ∪ D 2 .
Since the last function only depends on X |D 1 , they are orthogonal.

Step 5. Conclusion, i. e. [START_REF] Nolen | Normal approximation for a random elliptic equation[END_REF] for N = 3. By Step 4, it remains to estimate the three r. h. s. terms of (20). For the first term, we use (18) with

D ′ = D 1 ∪ D 2 and D = D 3 and obtain because of ζ = ζ|D 1 ∪ D 2 ∪ D 3 (ζ -ζ|D 1 ∪ D 2 ) 2 ≤ osc D 3 ζ|D 1 ∪ D 2 2 Jensen ≤ (osc D 3 ζ) 2 |D 1 ∪ D 2 = (osc D 3 ζ) 2 .
The other two terms follow the same way.

Proof of Lemma 2. W. l. o. g. we may assume that ζ = 0.

Step 1. Application of the original Spectral Gap Estimate to ζ p . We claim that this yields

ζ 2p ζ p 2 + z∈Z d ∩[-L 2 , L 2 ) d (osc B R (z) ζ) 2 p . (21)
Indeed, (9) applied to ζ p at first gives

(ζ p -ζ p ) 2 z∈Z d ∩[-L 2 , L 2 ) d (osc B R (z) (ζ p )) 2 . ( 22
)
Using the triangle inequality in L 2 (Ω) on the l. h. s. of ( 22) in form of

(ζ p ) 2 1 2 ≤ (ζ p -ζ p ) 2 1 2 + | ζ p |,
we see that (21) follows from ( 22) by Young's inequality provided we can show

z∈Z d ∩[-L 2 , L 2 ) d (osc B R (z) (ζ p )) 2 ζ 2p 1-1 p z∈Z d ∩[-L 2 , L 2 ) d (osc B R (z) ζ) 2 p 1 p + z∈Z d ∩[-L 2 , L 2 ) d (osc B R (z) ζ) 2 p . (23)
The latter can be seen as follows: From the elementary real-variable estimate

| ζp -ζ p | |ζ| p-1 | ζ -ζ| + | ζ -ζ| p ,
we obtain by definition of osc that

osc B R (z) (ζ p ) |ζ| p-1 osc B R (z) ζ + (osc B R (z) ζ) p .
Using that the discrete ℓ 2p (Z d )-norm is estimated by the discrete ℓ 2 (Z d )norm, this implies

z∈Z d ∩[-L 2 , L 2 ) d (osc B R (z) (ζ p )) 2 ζ 2(p-1)
z∈Z d ∩[-L 2 , L 2 ) d (osc B R (z) ζ) 2 + z∈Z d ∩[-L 2 , L 2 ) d (osc B R (z) ζ) 2 p .
Hölder's inequality w. r.

ζ p ≤ ζ 2p p-2 2p-2 ζ 2 p 2p-2 . ( 24
)
Using ζ = 0 we obtain from the original Spectral Gap Estimate applied to ζ itself

ζ 2 z∈Z d ∩[-L 2 , L 2 ) d (osc B R (z) ζ) 2 Jensen ≤ z∈Z d ∩[-L 2 , L 2 ) d (osc B R (z) ζ) 2 p 1 p
.

(25) Inserting (25) into (24) we obtain

ζ p 2 ζ 2p p-2 p-1 z∈Z d ∩[-L 2 , L 2 ) d (osc B R (z) ζ) 2 p 1 p-1 .
Inserting this into (21) and using Young's inequality yields the claim of the lemma.

Proof of Lemma 3.

Step 1. Regularity theory for a-harmonic functions. We will use the following two ingredients from De Giorgi's theory for uniformly elliptic equations: For any a ∈ Ω and any a-harmonic function u in B 2 we have sup

B 1 |u| ˆB2 u 2 1 2 . ( 26
)
Moreover, there exists a Hölder exponent α > 0 only depending on d and λ such that sup

x 1 ,x 2 ∈B 1 |u(x 1 ) -u(x 2 )| |x 1 -x 2 | α ˆB2 |∇u| 2 1 2 . ( 27
)
Both ingredients follow from De Giorgi's theorem (see for instance [8, Theorem 4.11]):

sup

B 1 |u| + sup x 1 ,x 2 ∈B 1 |u(x 1 ) -u(x 2 )| |x 1 -x 2 | α ˆB2 u 2 1 2 . ( 28
)
Indeed, for (27), one may assume w. l. o. g. that ffl B 2 u = 0 so that (27) follows from (28) and Poincaré's inequality on B 2 for functions with mean value zero.

Here and in the sequel, we write B R = B R (0) for brevity. The crucial element of these estimates is that the constants depend on the coefficient field a only through the ellipticity ratio λ (as indicated by the use of ).

Step 2. In this step, we derive an auxiliary a priori estimate involving dyadic annuli. Let u be a function and g a vector field on the torus related by the elliptic equation

-∇ • a∇u = ∇ • g and normalized by ´[-L 2 , L 2 ) d u = 0. We claim that if g vanishes in B 1 we have |u(0)| ∞ n=1 (2 n ) 1-d 2 ˆB2 n \B 2 n-1 |g| 2 1 2 . ( 29
)
We note that this sum is actually finite since for 2 n ≫ L, the ball B 2 n-1 invades the entire torus so that the "annulus" B 2 n \ B 2 n-1 is actually void. Estimate (29) will be derived from (26) and an elementary scaling argument. Indeed, for n ∈ N, we introduce

g n := g on B 2 n \ B 2 n-1 0 else , so that g = ∞ n=1 g n . Let u n denote the solution of -∇ • a∇u n = ∇ • g n on the torus normalized by ´[-L 2 , L 2 ) d u n = 0, so that u = ∞ n=1 u n .
Hence by the triangle inequality, for (29) it is enough to show

|u n (0)| (2 n ) 1-d 2 ˆ[-L 2 , L 2 ) d |g n | 2 1 2 . ( 30
)
We now give the argument for (30). Testing -∇ • a∇u n = ∇ • g n with u n , using the uniform ellipticity, and Cauchy-Schwarz' inequality, we obtain

ˆ[-L 2 , L 2 ) d |∇u n | 2 1 2 ˆ[-L 2 , L 2 ) d |g n | 2 1 2 . Since d > 2, Sobolev's embedding together with ´[-L 2 , L 2 ) d u n = 0 yields ˆ[-L 2 , L 2 ) d |u n | 2d d-2 d-2 2d ˆ[-L 2 , L 2 ) d |g n | 2 1 2
.

By Hölder's inequality on B 2 n-1 with exponents ( d d-2 , d 2 ) we obtain

ˆB2 n-1 |u n | 2 1 2 2 n ˆ[-L 2 , L 2 ) d |g n | 2 1 2 .
We note that u n is a-harmonic on B 2 n-1 (since g n vanishes there). Hence by ( 26), which we rescale from B 2 to B 2 n-1 , we have

|u n (0)| ≤ sup B 2 n-2 |u n | (2 n ) -d ˆB2 n-1 |u n | 2 1 2 .
The combination of the two last estimates yields (30).

Step 3. As a preliminary, we study the local dependence of ∇φ + ξ on a: Let the two coefficient fields a and ã agree outside B R . Then we have

ˆBR |∇φ(ã; •) + ξ| 2 ˆBR |∇φ(a; •) + ξ| 2 . ( 31
)
Indeed, we note that the function φ(ã; •)φ(a; •) satisfies

-∇ • ã∇(φ(ã; •) -φ(a; •)) = ∇ • (ã -a)(∇φ(a; •) + ξ).
We test this equation with φ(ã; •)-φ(a; •) and obtain from uniform ellipticity and Cauchy-Schwarz' inequality

ˆ[-L 2 , L 2 ) d |∇(φ(ã; •) -φ(a; •))| 2 ˆ[-L 2 , L 2 ) d |(ã -a)(∇φ(a; •) + ξ)| 2 .
Since by assumption, ãa vanishes outside B R , the above yields

ˆBR |∇(φ(ã; •) -φ(a; •))| 2 ≤ ˆ[-L 2 , L 2 ) d |∇(φ(ã; •) -φ(a; •))| 2 ˆBR |∇φ(a; •) + ξ| 2 . ( 32
)
This implies (31) by the triangle inequality in L 2 (B R ).

Step 4. In this step, we derive the central deterministic estimate

z∈Z d ∩[-L 2 , L 2 ) d \B R+1 (osc B R (z) φ(•; 0)) 2 1 2 ∞ n=1 (2 n ) 1-d 2 z∈Z d ∩B 2 n +R ˆBR (z) |∇φ + ξ| 2 p 1 2p . ( 33
)
Given a coefficient field a on the torus and a point on the integer lattice

z ∈ Z d ∩ [-L 2 , L 2
) d , we denote by a z an arbitrary coefficient field on the torus that agrees with a outside B R (z). We note that the function φ(a z ; •)φ(a; •) satisfies

-∇ • a∇(φ(a z ; •) -φ(a; •)) = ∇ • (a z -a)(∇φ(a z ; •) + ξ). ( 34
) Given a discrete field {ω z } z∈Z d ∩[-L 2 , L
2 ) d we consider the function u and the vector field g on the torus defined through u(x) :=

z∈Z d ∩[-L 2 , L 2 ) d ω z (φ(a z ; x) -φ(a; x)), g(x) := z∈Z d ∩[-L 2 , L 2 ) d ω z (a z (x) -a(x))(φ(a z ; x) + ξ)
and note that (34) translates into -∇ • a∇u = ∇ • g. Provided ω z = 0 for z ∈ B R+1 , we have g(x) = 0 for x ∈ B 1 . Under this assumption, we may apply (29) from Step 2 and obtain

z∈Z d ∩[-L 2 , L 2 ) d ω z (φ(a z ; 0) -φ(a; 0)) ∞ n=1 (2 n ) 1-d 2 ˆB2 n \B 2 n-1 z∈Z d ∩[-L 2 , L 2 ) d ω z (a z -a)(∇φ(a z ; •) + ξ) 2 1 2 .
Since |a z -a| ≤ 1 is supported in B R (z) and since {B R (z)} z∈Z d locally have a finite overlap, this turns into

z∈Z d ∩[-L 2 , L 2 ) d ω z (φ(a z ; 0) -φ(a; 0)) ∞ n=1 (2 n ) 1-d 2 z∈Z d ∩[-L 2 , L 2 ) d ω 2 z ˆBR (z)∩B 2 n |∇φ(a z , •) + ξ| 2 1 2 ∞ n=1 (2 n ) 1-d 2 z∈Z d ∩B 2 n +R ω 2 z ˆBR (z) |∇φ(a z , •) + ξ| 2 1 2 (31) ∞ n=1 (2 n ) 1-d 2 z∈Z d ∩B 2 n +R ω 2 z ˆBR (z) |∇φ(a, •) + ξ| 2 1 2
Hölder in z

z∈Z d ∩[-L 2 , L 2 ) d ω 2q z 1 2q × ∞ n=1 (2 n ) 1-d 2 z∈Z d ∩B 2 n +R ˆBR (z) |∇φ(a, •) + ξ| 2 p 1 2p ,
where p and q are dual exponents, that is,

1 p + 1 q = 1. Since {ω z } z∈Z d ∩[-L 2 , L 2 ) d
was arbitrary under the constraint that ω z = 0 for z ∈ B R+1 this implies by the duality of ℓ 2q (Z d ) and

ℓ 2q 2q-1 (Z d ) z∈Z d ∩[-L 2 , L 2 ) d \B R+1 |φ(a z ; 0) -φ(a; 0)| 2q 2q-1 2q-1 2q ∞ n=1 (2 n ) 1-d 2 z∈Z d ∩B 2 n +R ˆBR (z) |∇φ(a, •) + ξ| 2 p 1 2p .
Since for any z ∈ Z d , a z was an arbitrary coefficient field that agrees with a outside B R (z), this implies by the definition of osc B R (z)

z∈Z d ∩[-L 2 , L 2 ) d \B R+1 (osc B R (z) φ(•; 0)) 2q 2q-1 2q-1 2q ∞ n=1 (2 n ) 1-d 2 z∈Z d ∩B 2 n +R ˆBR (z) |∇φ + ξ| 2 p 1 2p .
On the l. h. s. we use that since 2q 2q-1 ≤ 2, the discrete ℓ 2q 2q-1 (Z d )-norm dominates the discrete ℓ 2 (Z d )-norm to obtain (33).

Step 5. Using stationarity, we upgrade Step 4 to the stochastic estimate

z∈Z d ∩[-L 2 , L 2 ) d \B R+1 (osc B R (z) φ(•; 0)) 2 p 1 2p ˆBR |∇φ + ξ| 2 p 1 2p
.

(35) Indeed, we start from (33) in Step 4 and apply the triangle inequality to the sum over n w. r. t. the norm L 2p (Ω):

z∈Z d ∩[-L 2 , L 2 ) d \B R+1 osc B R (z) (φ(•; 0)) 2 p 1 2p ∞ n=1 (2 n ) 1-d 2 z∈Z d ∩B 2 n +R ˆBR (z) |∇φ(a, •) + ξ| 2 p 1 2p
.

We now note that the stationarity (4) of φ also yields

∇φ(a; x + z) = ∇φ(a(• + z); x)
and thus

ˆBR (z) |∇φ(a, x ′ ) + ξ| 2 dx ′ = ˆBR |∇φ(a(• + z), x) + ξ| 2 dx.
By stationarity of • , cf. (8) applied to ζ(a) = ´BR (z) |∇φ(a; x) + ξ| 2 dx, this implies

ˆBR (z) |∇φ(•, x ′ ) + ξ| 2 dx ′ p = ˆBR |∇φ(•, x) + ξ| 2 dx p . (37)
Inserting this into (36) yields (because of

z∈Z d ∩B 2 n +R 1 (2 n + R) d ) z∈Z d ∩[-L 2 , L 2 ) d \B R+1 osc B R (z) (φ(•; 0)) 2 p 1 2p ∞ n=1 (2 n ) 1-d 2 (2 n + R) d 2p ˆBR |∇φ(a, •) + ξ| 2 p 1 2p . ( 38
) Since for p > d d-2 the exponent 1 -d 2 + d 2p < 0 is negative we have ∞ n=1 (2 n ) 1-d 2 (2 n + R) d 2p
1.

Hence (38) turns into the desired (35).

Step 6. It remains to treat z ∈ Z d ∩ B R+1 in [START_REF] Naddaf | Estimates on the variance of some homogenization problems[END_REF]. By stationarity, it will be enough to consider z = 0, cf.

Step 7. In this step, we will derive from the Hölder continuity a priori estimate (27) the deterministic estimate

osc B R φ(a; 0) ˆB2R |∇φ(a; •) + ξ| 2 1 2 . ( 39
)
Let a ∈ Ω be given and ã ∈ Ω agree with a outside B R and otherwise be arbitrary. On the one hand, since d > 2 and

´[-L 2 , L 2 ) d (φ(ã; •) -φ(a; •)) (3)
= 0, we have by Sobolev's embedding

ˆBR |φ(ã; •) -φ(a; •)| 2d d-2 d-2 2d ≤ ˆ[-L 2 , L 2 ) d |φ(ã; •) -φ(a; •)| 2d d-2 d-2 2d ˆ[-L 2 , L 2 ) d |∇(φ(ã; •) -φ(a; •))| 2 1 2 (32) ˆBR |∇φ(a; •) + ξ| 2 1 2 . ( 40
)
On the other hand, we obtain from (27) applied to the a-harmonic function u(x) = φ(a; x) + ξ • x (rescaled from B 1 to B R ):

sup x 1 ,x 2 ∈B R |φ(a; x 1 ) -φ(a; x 2 ) + ξ • (x 1 -x 2 )| |x 1 -x 2 | α ˆB2R |∇φ(a; •) + ξ| 2 1 2 . ( 41
)
Replacing a by ã in the above and using (31) from Step 3 (with B R replaced by B 2R ) we likewise have sup

x 1 ,x 2 ∈B R |φ(ã; x 1 ) -φ(ã; x 2 ) + ξ • (x 1 -x 2 )| |x 1 -x 2 | α ˆB2R |∇φ(a; •) + ξ| 2 1 2 . ( 42
)
Combining ( 41) and (42), we obtain sup

x 1 ,x 2 ∈B R |(φ(ã; x 1 ) -φ(a; x 1 )) -(φ(ã; x 2 ) -φ(a; x 2 ))| |x 1 -x 2 | α ˆB2R |∇φ(a; •)+ξ| 2 1 2 . ( 43
)
By the following elementary interpolation estimate, valid for an arbitrary function u,

sup B R |u| ˆBR |u| 2d d-2 d-2 2d + sup x 1 ,x 2 ∈B R |u(x 1 ) -u(x 2 )| |x 1 -x 2 | α ,
we see that (40) and (43) combine to

|φ(ã; 0) -φ(a; 0)| ˆB2R |∇φ(a; •) + ξ| 2 1 2 .
Since ã was arbitrary besides agreeing with a outside B R , we obtain (39) by definition of osc.

Step 7. We upgrade Step 6 to the stochastic estimate

z∈Z d ∩B R+1 (osc B R (z) φ(•; 0)) 2 p ˆB3R+1 |∇φ + ξ| 2 p . (44)
Indeed, (39) from Step 6, with the origin replaced by z, implies after summation

z∈Z d ∩B R+1 (osc B R (z) φ(•; 0)) 2 ˆB3R+1 |∇φ + ξ| 2 .
Taking the p-th power and the expectation yields (44).

Step 8. From Steps 5 and 7 we learn that ( 12) is satisfied with B 1 replaced by B 3R+1 on the r. h. s. . We appeal once more to stationarity to get for a generic R 1

ˆBR |∇φ + ξ| 2 p ˆB1 |∇φ + ξ| 2 p . (45)
Indeed, there exist points z 1 , • • • , z N on the torus such that B R ⊂ N n=1 B 1 (z n ) and we can arrange for N 1 because of R 1. Thus we have

ˆBR |∇φ + ξ| 2 ≤ N n=1 ˆB1 (zn) |∇φ + ξ| 2 .
Taking the p-th power gives

ˆBR |∇φ + ξ| 2 p N n=1 ˆB1 (zn)
|∇φ + ξ| 2 p ;

taking the expectation yields

ˆBR |∇φ + ξ| 2 p ≤ max n=1,••• ,N ˆB1 (zn) |∇φ + ξ| 2 p .
By stationarity, cf. (37), this yields (45).

Proof of Lemma 4

Step 1. We start by establishing the deterministic estimate

ˆB1 |∇φ + ξ| 2 p ˆB2 (φ + ξ • x) 2(p-1) |∇φ + ξ| 2 , (46)
which we will use in form of

ˆB1 |∇φ + ξ| 2 p ˆB2 (φ 2(p-1) + 1)(|∇φ| 2 + 1). (47)
Estimate (46) relies on the fact that u(x)

:= φ(x) + ξ • x is a-harmonic, that is, -∇ • a∇u = 0.
We test this equation with η 2 u, where η is a cut-off function for B 1 in B 2 . By uniform ellipticity we obtain λ ˆ(η|∇u|) 2 ≤ 2 ˆ|∇η||u| η|∇u|.

We now use Young's inequality (and p ≥ 2 > 1) on the r. h. s. integrand in form of

2 λ |∇η||u| η|∇u| ≤ 1 2 (η|∇u|) 2 + C(|∇η|u) 2 p-1 p (η|∇u|) 2 p , which yields ˆ(η|∇u|) 2 ˆ(|∇η|u) 2 p-1 p (η|∇u|) 2 p .
By the choice of η, this implies

ˆB1 |∇u| 2 ˆB2 u 2 p-1 p |∇u| 2 p .
It remains to apply Jensen's inequality on the r. h. s. to obtain as desired

ˆB1 |∇u| 2 ˆB2 u 2(p-1) |∇u| 2 1 p .
Step 2. We continue with the deterministic estimate

ˆ[-L 2 , L 2 ) d φ 2(p-1) |∇φ| 2 ˆ[-L 2 , L 2 ) d φ 2(p-1) , (48)
which we will use in form of

ˆ[-L 2 , L 2 ) d (φ 2(p-1) + 1)|∇φ| 2 ˆ[-L 2 L 2 ) d (φ 2(p-1) + 1) (49)
that follows from the combination of (48) once with the generic exponent p and once with the exponent p = 2. Indeed, we test -∇ • a(∇φ + ξ) = 0 with the monotone-in-φ expression 1 2p-1 φ|φ|2(p-1) over the entire torus. Because of ∇ 1 2p-1 φ|φ| 2(p-1) = φ 2(p-1) ∇φ and by uniform ellipticity, we obtain

λ ˆ[-L 2 , L 2 ) d φ 2(p-1) |∇φ| 2 ≤ ˆ[-L 2 , L 2 ) d φ 2(p-1) |∇φ|.
Using Cauchy-Schwarz' inequality on the r. h. s. of that inequality yields (48).

Step 3. Conclusion using stationarity. We take the expectation of (47):

ˆB1 |∇φ + ξ| 2 p ˆB2 (φ 2(p-1) + 1)(|∇φ| 2 + 1)
.

By stationarity, we have

ˆB2 (φ 2(p-1) + 1)(|∇φ| 2 + 1) = |B 2 |L -d ˆ[-L 2 , L 2 ) d (φ 2(p-1) + 1)(|∇φ| 2 + 1) .
We now use the expectation of (49):

|B 2 |L -d ˆ[-L 2 , L 2 ) d (φ 2(p-1) + 1)|∇φ| 2 L -d ˆ[-L
We use once more stationarity in form of

L -d ˆ[-L 2 , L 2 ) d (φ 2(p-1) + 1) = φ 2(p-1) + 1.
Proof of Proposition 1. By Jensen's inequality, it is enough to prove the statement for p > d d-2 . We apply Lemma 2 to ζ(a) = φ(a; 0). We note that by stationarity of φ and • we have

φ = L -d ˆ[-L 2 , L 2 ) d φ (3) = 0.
Hence the statement of Lemma 2 assumes the form

φ 2p z∈Z d ∩[-L 2 , L 2 ) d (osc B R (z) φ(•; 0)) 2 p .
Estimating the r. h. s. by Lemmas 3 and 4, this turns into

φ 2p ≤ C φ 2(p-1) + 1 .
We conclude by using Jensen's and Young's inequalities in form of C φ2(p-1) ≤

C φ 2p p-1 p ≤ 1 2 φ 2p + C. Proof of Theorem 1. Step 1. Application of Lemma 1 to ζ = ξ ′ • a L hom ξ yields (ξ ′ • a L hom ξ -ξ ′ • a L hom ξ ) 2 z∈Z d ∩[-L 2 , L 2 ) d (osc B R (z) ξ ′ • a L hom ξ) 2 . ( 50
)
Step 2. Deterministic estimate of the oscillation. We first rewrite ξ ′ • a L hom ξ as

ξ ′ • a L hom ξ = L -d ˆ[-L
where φ ′ (a; x) is the corrector associated with the pointwise transpose field a t of a and direction ξ ′ . Indeed, (51) holds by (3) since φ ′ is periodic. We claim

osc B R (z) ξ ′ • a L hom ξ L -d ˆBR (z) |∇φ ′ + ξ ′ | 2 1 2 ˆBR (z) |∇φ + ξ| 2 1 2 . ( 52
)
Indeed, consider two arbitrary coefficient fields a 0 , a 1 ∈ Ω that agree outside B R (z). We write for abbreviation φ i (x) = φ(a i ; x), φ ′ i (x) = φ ′ (a i , x), and a L hom,i = a L hom (a i ) for i = 0, 1. By definition of osc it is enough to show

L d |ξ ′ •a L hom,1 ξ-ξ ′ •a L hom,0 ξ| ˆBR (z) |∇φ ′ 0 +ξ ′ | 2 1 2 ˆBR (z) |∇φ 0 +ξ| 2 1 2 . ( 53
)
Indeed, we have by definition of a L hom and of φ, φ ′

L d (ξ ′ • a L hom,1 ξ -ξ ′ • a L hom,0 ξ) (51) = ˆ[-L 2 , L 2 ) d (∇φ ′ 1 + ξ ′ ) • a 1 (∇φ 1 + ξ) - ˆ[-L 2 , L 2 ) d (∇φ ′ 0 + ξ ′ ) • a 0 (∇φ 0 + ξ) = ˆ[-L 2 , L 2 ) d ∇(φ ′ 1 -φ ′ 0 ) • a 1 (∇φ 1 + ξ) + ˆ[-L 2 , L 2 ) d (∇φ ′ 0 + ξ ′ ) • a 0 ∇(φ 1 -φ 0 ) + ˆ[-L 2 , L 2 ) d (∇φ ′ 0 + ξ ′ ) • (a 1 -a 0 )(∇φ 1 + ξ) = ˆ[-L 2 , L 2 ) d ∇(φ ′ 1 -φ ′ 0 ) • a 1 (∇φ 1 + ξ) + ˆ[-L 2 , L 2 ) d ∇(φ 1 -φ 0 ) • a t 0 (∇φ ′ 0 + ξ ′ ) + ˆ[-L 2 , L 2 ) d (∇φ ′ 0 + ξ ′ ) • (a 1 -a 0 )(∇φ 1 + ξ).
Using the equation (3) for φ 1 and for φ ′ 0 , the first two r. h. s. terms vanish and this identity turns into

L d (ξ ′ • a L hom,1 ξ -ξ ′ • a L hom,0 ξ) = ˆ[-L 2 , L 2 ) d (∇φ ′ 0 + ξ ′ ) • (a 1 -a 0 )(∇φ 1 + ξ),
so that we obtain Step 3. Stochastic estimate based on Proposition 1. We claim

L d |ξ ′ • a L hom,1 ξ -ξ ′ • a L hom,0 ξ| ≤ ˆBR (z) |∇φ ′ 0 + ξ ′ | 2 ˆBR (z) |∇φ 1 + ξ| 2
ˆBR (z) |∇φ + ξ| 2 2 1. (54)
Indeed, by Step 8 from the proof of Lemma 3 (and stationarity to replace z by 0) we have

ˆBR (z) |∇φ + ξ| 2 2 ˆB1 |∇φ + ξ| 2 2 .
An application of Lemma 4 with p = 2 and of Proposition 1 with p = 1 yields (54). Since the ensemble • ′ that is obtained from • as pushforward under a → a t satisfies our assumptions, we have

ˆBR (z) |∇φ ′ + ξ ′ | 2 2 = ˆBR (z) |∇φ + ξ ′ | 2 2 ′ 1. (55)
Step 4. Conclusion:

(ξ ′ • a L hom ξ -ξ ′ • a L hom ξ ) 2
(50)

z∈Z d ∩[-L 2 , L 2 ) d (osc B R (z) ξ ′ • a L hom ξ) 2 (52)
L -2d

z∈Z d ∩[-L 2 , L 2 ) d ˆBR (z) |∇φ ′ + ξ ′ | 2 ˆBR (z) |∇φ + ξ| 2 ≤ L -2d z∈Z d ∩[-L 2 , L 2 ) d ˆBR (z) |∇φ ′ + ξ ′ | 2 2 1 2 ˆBR (z) |∇φ + ξ| 2 2 1 2
(54),(55)

L -d .

)

  It measures how sensitively ζ(a) depends on a |D . Note that (osc D ζ)(a) does not depend on a |D .

1 2 .

 12 Now (53) follows from this and Step 3 in the proof of Lemma 3 in form of ´BR(z) |∇φ 1 + ξ| 2 ´BR (z) |∇φ 0 + ξ| 2 .

  t. this ensemble with • . The Poisson ensemble is stationary which means that for any shift vector z ∈ Z d the random field a and its shifted version a(•+z) : x → a(x + z) have the same distribution. In other words, for any (integrable) function ζ : Ω → R (which we think of as a random variable) we have that a → ζ(a(• + z)) and ζ have the same expectation:

	Remark 1. The coefficient field a associated with the Poisson point process
	is a stationary ergodic coefficient field to which the theory by Papanicolaou
	and Varadhan applies. In particular, as a direct consequence of the homoge-
	nization result, we have almost surely
	lim L↑∞	a L hom (a) = a hom ,
	where a hom is as in (1) and a L hom (a) is as in Definition 1. The random vari-able a L hom (a) is called the approximation by periodization of the homogenized
	coefficient a hom .
	For our result, we only need the following two properties of the Poisson
	ensemble.
	Lemma 1.
	Stationarity.

  2. Conditional expectations and independence. From now on, we prove statement[START_REF] Papanicolaou | Boundary value problems with rapidly oscillating random coefficients[END_REF] for the Poisson point process. For brevity, we drop the subscript 0. For a given (Lebesgue measurable) subset D of the torus, we denote by •|D the expectation conditioned on the restriction X |D of the (random) point configuration X on D. We note that for a function ζ : Ω → R which is square integrable, ζ|D is the L 2 (Ω)-orthogonal projection of ζ onto the space of square integrable functions ζ : Ω → R that only depend on X via X |D . With help of these conditional expectations the independence assumption (6) can be rephrased as follows: For any two (Lebesgue measurable) subsets D, D ′ that are disjoint and any (square integrable) function ζ : Ω → R that does not depend on X |D we have

  t. to • applied to the first r. h. s. term with exponents ( p p-1 , p) yields (23). Step 2. Conclusion in case of p ≥ 2 (the other case is easier and not needed later). It remains to treat the first r. h. s. term of (21). By Hölder's inequality w. r. t. • we have

, L 2 ) d (φ 2(p-1) + 1) .

, L 2 ) d (∇φ ′ + ξ ′ ) • a(∇φ + ξ),(51)


Francois Bonnet

Sociology and political science in the patrimonial society: implications of Piketty's Capital

Keywords: inequality, capitalism, corporate governance, education, social structure, power JEL classification P10 Capitalists system N30 Labor and consumers, demography, education, health, welfare, income, wealth, religion, and philanthropy I24 Education and inequality

What are the implications of Piketty's Capital for sociology and political science? Capital's argument focuses on the evolution of the r/g ratio (capital returns over growth rate) and outlines two modes of economic inequalities. One is characteristic of affluent (g > r) societies and the other is characteristic of patrimonial (r > g) societies. With the current return to a patrimonial society, corporations become political actors; occupational status and education's relevance are declining; the meaning of poverty is transformed, and welfare and punishment become interdependent means to social order; in politics, elitist theories gain traction; immigration is less about assimilation, and more about transnationalism and nationalist politics. We show that some theories are more relevant in an affluent society, and others are more adequate to a patrimonial society.

Sociology and political science in the patrimonial society: implications of Piketty's Capital

Francois Bonnet and Clement Thery

August 2014

Partners // Introduction This is not a review of Thomas Piketty's Capital in the Twenty-First Century, but a commentary on the implications that the book holds for sociology and political science.

Capital is a history of capital returns over two centuries. The main finding of the book is that capital returns (r) have exceeded growth rates (g) (r > g) over most of the past two centuries. When r > g, economic inequalities increase indefinitely. In the few decades when g > r, inequalities decreased significantly.

Reading Capital suggests a simple yet useful image of two ideal-types of modern societies (Piketty himself does not make this claim). The first is the society of g > r: a society with high economic growth, low unemployment and relative equality. In this society, the wealthiest 10% "only" owns 50% of the capital, workers receive a growing share of the national income and social mobility redistributes positions in the social space. In the United States, this is the postwar prosperity of the Affluent Society [START_REF] Galbraith | The Affluent Society[END_REF]). In the regulation school parlance, this is the Fordist society; it refers to the French "Trente Glorieuses" between 1945 and 1975, the "Italian miracle", the "German miracle", and so on. But, as Piketty shows, this is a historical anomaly. High growth is a temporary bump caused by catching-up effects. Long-term growth rates over centuries are stable at 1.5%. In the long term, capital returns are higher than growth rates: r > g. r > g societies are extremely unequal: the wealthiest 10% of the population own 90% of capital, and the wealthiest 1% own 60% of capital. Most of the national income rewards capital owners, not workers. r > g societies are patrimonial societies, where inheritance trumps work, exemplified by Europe pre-World War I and the Gilded Age in the United States. Piketty's data show that Western societies have ended their spell with high growth, and are coming back to being patrimonial societies-barring other major events, such as global taxation, another world war, or catastrophic climate change.

Our point is that key concepts and theories in sociology and political sciences have been developed both in an era of affluence, when g > r, and in a period of patrimonialization of societies, when r > g. But these two societies are different, and they each require a specific set of analytical tools. In this article, we discuss the social and political structure of both societies, and their corresponding sociology and political science. We look at the social groups, and economic and political actors of r and g. Piketty's approach in this regard is abstract: r and g are conceptual constructs, not empirical entities. In Capital, the r/g ratio depicts the mathematical logic of inequality generation.

With the relevant social scientific literature, we bring back social groups and actors in.

Our objective is twofold: to look at the social, political and economic reality of the affluent and patrimonial societies, and at the theories and concepts of these realities. We do not aim at conceptualizing correspondences between the two; instead, we show that some theories, concepts and debates are more relevant in a g > r (or affluent) society, and others are more adequate to a r > g (or patrimonial) society. Capital's strength is to offer the possibility of constructing such perspective on the history of postwar social sciences. The article does not aim at providing a new history of ideas. Our goal is to stimulate novel questions and hypotheses about the current return to the patrimonial society, to push for original connections between fields, and to allow for inventive casing [START_REF] Ragin | Casing' and the process of social inquiry[END_REF]. In this paper, we move from one field to another, from the United States to Western Europe, and from one historical context to the next in just a few pages. We hope the resulting broad span will stimulate new objects of studies and theories.

The article assumes that the r/g ratio drives the organization of society. We plead guilty of economic determinism, with mitigating circumstances. We do not claim to demonstrate causal relationships between the r/g ratio and other macro-social trends.

We give substance to the ratio by showing the social actors that exist behind r and g. To do so, we mobilize canonic scholarly works for which we explain how their inner reasoning is (unbeknownst to them) conditioned by the processes that r/g captures. We are therefore not determinist in the strictest sense. Moreover, not all social facts and scholarly debates need to be related to the dynamics of economic inequalities captured by r and g. Our argument is that a number of key debates in social sciences should take into account the return to a patrimonial society-in particular, questions around the distribution of economic rewards. These debates have in common to discuss processes which overlap and are closely related to the formation of economic inequalities as described by Piketty. We have selected four broad questions for this article. ( 1) What is the nature of market competition, the nature of the firm, and the role of corporations in the allocation of economic rewards? (2) How do non-economic dimensions of social stratification-education, cultural capital-relate to economic inequalities? (3) How to think about poverty and its regulation? (4) How do the possession of economic resources and political power relate to each other? For each question, we study how the passage from an affluent society to a patrimonial society transforms scholars' posture and conceptualizations, and we suggest issues for future research.

The large corporation: from institution to political player

The r/g ratio is a mathematical abstraction. In the material world, the main economic actors who are busy maximizing r and indirectly producing interdecile inequality are corporations. The evolution of r and g coincides with the transformations of corporate governance and of the nature of the corporation as an economic actor.

In a g > r society, a significant social innovation has been the rise of managers running large corporations with a scientific outlook toward productivity gains and mass markets (Piore and Sabel 1984). This represented a departure from the Gilded Age, when financiers and industrialists ran the firms they owned and were looking for unfair competitive advantages to create monopolistic situations-the "robber barons". Leading sociologists such as Ralf Dahrendorf, Daniel Bell and David Riesman lauded the demise of owner control as an improvement of democracy [START_REF] Mizruchi | Berle and Means revisited: The governance and power of large US corporations[END_REF]). The scientific manager was the epitome of a meritocratic system that rewards education and competence. The large corporation provided jobs, including many unskilled positions in manufacturing, healthcare benefits, retirement plans, and opportunities for social mobility by climbing the organizational ladder [START_REF] Davis | After the corporation[END_REF]). In the United States, lowest levels of income inequality have coincided with highest levels of employment in large corporations. This was a situation where the strength of unions, the "scientific" organization of production and the legitimacy of both enabled workers and capitalists to work out compromises, and when the ideas of "people's capitalism", "soulful corporation", "post-capitalist society" or "what's good for General Motors is good for the country" made sense (Mizruchi 2004: 583, 607).

Because it had become such a fundamental organizing force of postwar American society, economists and sociologists questioned the corporation as a site of production and work. Alfred Chandler in The Visible Hand (1977) emphasized that the corporation exists because it makes possible productivity gains. Neo-institutionalists uncovered the hidden logic of integrating production [START_REF] Williamson | Markets and Hierarchies[END_REF]. Sociologists analyzed the work process as a series of dialectic oppositions, between integration and conflict, formal and informal organization, alienation and motivation.

The return to a r > g society at the end of the 1970s transforms this setting entirely. Since 1997, the number of public corporations in the US has declined by 55% [START_REF] Davis | After the corporation[END_REF]. Corporations are increasingly owned by investment management firms such as BlackRock (which owned in 2011 at least 5% of the shares of more than 40% of the companies listed on American stock markets) and Fidelity (the largest shareholder of one in ten American corporations) (Davis 2013: 289). "Even at the height of "finance capitalism" in the early twentieth century, the United States has never before seen corporate ownership this concentrated in the hands of a small number of financial institutions" (Davis 2013: 289). This shift toward the maximization of the "shareholder value" leads to the decline of the large, stable corporation. It is enormously detrimental to workers. Flat hierarchies in small firms make internal social mobility less likely; short-lived companies have no reason "to build strong corporate cultures with generous employee benefits intended to ensure commitment" (Davis 2013: 290); employment becomes more uncertain. Globalization, cheap foreign labor and low shipping costs enable corporations to function without a large workforce. Nike or Apple's production lines are entirely outsourced and only retain "core competence" (design, marketing, etc.). Manufacturing jobs only remain in military industries, and 9 out of 12 largest US employers are retailers: "To oversimplify only slightly, big firms prefer investing in machines to people, and small firms rely on outside vendors for much of the heavy lifting of production and distribution" (Davis 2013: 295).

As a consequence, corporations are no longer to be studied as sites of production and work. With the crisis of the affluent society, the scholarship's focus has shifted toward the growing networks of cooperation, sub-contracting and outsourcing between corporations of various sizes. This is the rise of the literature on "flexible accumulation" depicted in the Second Industrial Divide (Piore and Sabel 1984), the "network form of governance" [START_REF] Powell | Neither market nor hierarchy: Network forms of organization[END_REF]), or regional ecosystems fostering innovations by opposition to the stifling effects of large corporations [START_REF] Saxenian | Regional advantage[END_REF]. More importantly, the blurring of the firms' boundaries and the gutting of their production process go beyond the restructuring of a value chain. Scholars are now documenting how corporations have become active towards lobbying the state in order to manipulate legal norms and public subsidies to their advantage. In a society of r > g, the corporation is more and more an political actor targeting the state for private profit in an economic game that has less and less to do with market competition and production (King and Pearce 2010) (one recent legal translation of the new status of corporations is the Citizens United decision).

Such shift raises questions and problems. How did the bargaining structure between the state, the corporations and social movements tilt in favor of corporations? In the affluent society, many worried about the military-industrial complex, because they perceived the overlap of economic and political power as a democratic anomaly. With the reversal to a patrimonial society, such intermingling has become the norm. Corporations started to heavily invest in lobbying in the 1970s; the driving force behind the strategy of lobbying the state was Procter and Gamble, not a weapons manufacturer (Hacker and Pierson 2010). What is the content of the lobbying?

Piketty makes an implicit critique of the mainstream explanation for growing inequalities, that is, technological progress empower a few workers to be highly productive (and therefore highly paid), while the rest are made redundant. Capital argues that inequality depends on r and g, not technology. For economists defending the "skill-biased technical change hypothesis of income inequality", the only contribution of corporate lobbying to inequality is tax cuts; the bulk of inequality generation happens before tax, because of technology. Political scientists suggest instead that corporations actively seek to maximize their economic activity through corporate lobbying before tax; they lobby for lawmaking that undermines consumer protection and creates direct business opportunities. Recent work suggests that corporate lobbies have successfully fought against the Consumer Protection Agency for thirty years and the regulation of top executives' remuneration (Hacker and Pierson 2010) and obtain the privatization of public prerogatives (in education, energy, security, healthcare, and so on) at the expense of consumers and citizens and for private profit, not to mention criminal banking practices at the expense of the taxpayer [START_REF] Galbraith | The predator state: How conservatives abandoned the free market and why liberals should too[END_REF].

Finally, what enabled corporations to become opportunistic political? Davis argues that corporations owned by hedge funds have less employees, are more likely to be restructured, sold, renamed, and are less embedded in communities. They do not have to "bear the costs of being a social institution" anymore (Davis, quoted in King 2014). The key point is that "The larger an organization's employee base, the more obligations it likely has to those employees and the communities in which they live, and so the more constraints there are on its behavior. [These organizations] are also those most likely to violate societal expectations of what it means to be a good, responsible company" [START_REF] King | where is power elite theory when you need it ?, orgtheory.net[END_REF]. Further research needs to investigate how corporate power expresses itself in other countries where rules are different; for instance [START_REF] Culpepper | Quiet politics and business power: Corporate control in Europe and Japan[END_REF] has studied the cases of Europe and Japan where campaign donations play little role.

Economic and non-economic inequalities: the declining significance of education and cultural capital?

How do non-economic dimensions of social stratification, such as education, human capital, or cultural capital, relate to economic inequalities? The answer to this question is markedly different in g > r and r > g societies.

The social structure and stratification of the g > r (affluent) society is difficult to read. High GDP growth associated with lower income inequality and full employment equalize social conditions, lifting the working classes to the consumerist lifestyle of the growing middle-class. In affluent societies, occupation is more salient than capital ownership in the determination of social position. This era produced two major intellectual innovations. First, the reduction of the social hierarchy to occupational status opens up the possibility for individuals' mobility in society through their career, their job history. This is [START_REF] Blau | The American Occupational Structure[END_REF]'s point when they recommended studying occupational attainment from father to son, rather than class belonging. Blau and Duncan's Socio-Economic Index of occupations is continuous and captures both the economic and the prestige dimensions of social position. In g > r societies, such a stratificationist perspectives, where social hierarchies are gradual and open, make more sense than classist perspectives, where social positions are discrete and ascribed. Second, Gary [START_REF] Becker | Investment in human capital: A theoretical analysis[END_REF] redefines work as the exercise of human capital: something that requires an investment (education and job training) and yields rewards (wage and productivity). The number of years of schooling is for the first time linked to income. In the g > r society, education trumps inherited economic capital in the making of one's social position: workers with the right degrees may earn as much, and command more respect, than many business owners. The image of a fluid and meritocratic society, where individuals who invest in the right skill sets are rewarded, has taken roots. This is the society in which it is meaningful for Bourdieu to write Distinction (1984) and to introduce the notion of "cultural capital".

The theoretical project of Distinction is to salvage a class analysis of social structure from the blurring of social groups in g > r societies. In interviews and conferences, Bourdieu has repeatedly stated that he had become impatient with the then-prevailing stratificationist wisdom, according to which diminishing profits had not happen and had not led to society's polarization. Indeed, many sociologists were abandoning the classist perspective and were embracing an integration (versus conflict) perspective on social structure. Bourdieu went for the contrarian argument that (a) social mobility is an illusion (The Inheritors, Reproduction) and (b) the mechanics of social reproduction has just become more insidious and involves the unequal distribution in society of "cultural capital" (Distinction). Cultural capital is the subtle competences in the consumption of cultural goods, from table manners to musical tastes, inherited from one's family that school implicitly demands from its students, and sanctions, without ever teaching these competences. Schooling is not about investing in one's human capital with the expectation of future upward mobility. It is about the conversion of inherited cultural capital into formal education and diploma that can be valorized on the labor market -a process of social "certification" rather than accumulation of valuable skills. In Distinction, Bourdieu develops an understanding of macro-social differentiation and reproduction where cultural and economic capitals are complementary forces.

It is striking how both Bourdieu's position and the stratificationism of Blau and Duncan are similar in their fundamental diagnosis about society. In the g > r society, the social structure becomes so economically homogenous through education and labor markets that we need Distinction to explain its inner workings. Social positions are objectively defined by occupational status and subjectively lived through everyday consumption and cultural practices. Consumption becomes a key status marker: the foods, clothes, music and movies that people consume are signals of their group belonging, relative group position and can be understood as boundary-making. This is the key insight of Distinction: we enact our social class (or, more tellingly, our "class fraction") with our myriad particular consumptions. We are (were) fascinated with Distinction because it deciphers the "thin differences" (Die feine Unterschieden, Distinction's German title) that make up social structure.

Such complicated social structure is radically simplified in r > g societies, which rest on the ownership of material capital. The happy few who own capital enjoy comfortable lives, regardless of their occupational status; the rest who does not own capital has to work for those who do, and only top earners manage to reach the ease and comfort of capital owners. Piketty notes that today in France, a lifetime of earning minimum wage and the associated retirement equals 750,000 euros-the price of a three-bedroom apartment in Paris. Thus, a first consequence of the return to r > g is the questioning of whether educational and occupational attainment are good indicators of social position. Piketty's Capital is a virulent indictment of the notion of human capital and more generally of occupation as a significant variable for understanding economic stratification. According to Piketty, workers' skills and competences do not have the key property that defines capital. They are factors of production, as material capital is, but they cannot store economic value and they cannot be owned and sold. Therefore, they cannot produce income by themselves, unlike (material) capitalists, who can live off rent. This difference would explain why technological advancements and the increasing demand for high level human capital has not produced the convergence of economic standings that economists were expecting. Following Piketty, using education and occupation as explaining variables for economic inequalities in r > g societies is a quaint anachronism: the true measure of economic reproduction is the inheritance of capital. This is controversial. Documenting an increasing polarization of the labor market after 1980, Kalleberg in Good Jobs, Bad Jobs (2011) insists that the critical divide in society is between highly skilled workers who can reap the benefit of the modern economy with its ever-shifting labor market, and low-skill workers whose life is worsened by the new economic behaviors of corporations. Kalleberg keeps alive the idea of human capital.

To what extent, then, education and occupational attainment really matter for understanding economic inequalities in a patrimonial society? Piketty's critique of the concept of human capital may be extended to Distinction. In this view, Distinction rests on a historical anomaly, g > r, where cultural capital and education mattered. In the return to normalcy of r > g societies. r > g suggests, therefore, the declining significance of Bourdieu's Distinction, sociology's most cited work.

The diminishing relevance of education and occupation in a patrimonial society raises concern over meritocracy. Meritocracy is the idea that individual talent, hard work, entrepreneurial spirit and above all school-sanctioned intelligence find its just reward on the marketplace, and is the legitimizing ideology of economic inequality in affluent societies. The empirical assessment of meritocracy (social mobility studies) is an important scientific endeavor in g > r scholarship. This stands in stark contrast with previous experiences of r > g societies. In Social Darwinism in American Thought (1955), Hofstadter depicts the peculiar ideology of holders of great fortunes during the American Gilded Age. The automatic transmission of social position through inheritance of immense fortunes during the Gilded Age was not justified by meritocracy but by social Darwinism, imported from Belle Epoque Europe. To survive and dominate the economic struggles was the mark of an exceptional genetic quality. The legitimacy of inherited wealth went without saying: it rewarded one's superior innate genetic character. Transmission of name, wealth and genetic make-up was going hand-in-hand, inseparably.

In r > g societies, education and occupation have less explaining power on economic inequalities, which questions the function of the meritocratic discourse. With the return to a patrimonial society, will the class of capital owners develop a new legitimizing discourse about the fairness of economic disparities? Or will they cling to the old meritocratic story? In Privilege, Shamus [START_REF] Khan | Privilege: The Making of an Adolescent Elite at St. Paul's School[END_REF] documents the attachment of the current elites to the discourse of hard work and individual achievement in full contradiction with what their privileged environment should indicate to them. Can this disconnection be sustained in the long-term? Will the current elites mimic their ancestors and develop a modernized social Darwinism, where a difference in quality is created between capital owners and non-capital owners?

From 'the poor and the middle-class' to 'non-capital owners'

Piketty's Capital focuses on the upper strata of income and wealth's hierarchy, but it can also inform how to study the rest of society. The transition towards a patrimonial society has consequences for the study of the middle class and the poor. Prolonging Piketty's approach, we analyze the poor and the middle class not as qualitatively distinct, but through the hypothesis of their common lack of ownership of capital.

The modern sociology of poverty is born in the US in the 1960s, when g > r, around the debates about the culture of poverty. It was a society of middle-class affluence and general optimism and scholars asked the following questions: why do poor people not benefit from the general prosperity of modern capitalism? Why don't poor people move up to a middle-class status? Culture of poverty theorists argued that poverty makes the poor act in ways that perpetuate poverty [START_REF] Lewis | La vida: a Puerto Rican Family in the Culture of Poverty-San Juan and New York[END_REF]); structuralists analyzed this as an instance of "blaming the victim" and focused on explanations based on cumulative disadvantage on the labor market, discrimination and racism, etc. Doing so culturalists and structuralists defined poverty in relative terms, not by the income below which the most basic needs (food and shelter) are met, but by the lack of participation to mainstream rituals and middle-class lifestyle: drug use, teen pregnancies, hostile attitudes towards the police, absentee fathers and crime. Culturalists and structuralists also emphasized the barriers that the poor face to become middle-class: culturalists focused on childhood socialization, the inception of inadequate values, and the generous welfare policies that prevent the poor from seizing the ever present economic opportunities of an affluent society; and structuralists focused on spatial and social isolation that preclude the poor from having access to the pool of jobs of the modern capitalist economy [START_REF] Wilson | The Truly Disadvantaged. The Inner City, the Underclass and Public Policy[END_REF][START_REF] Wilson | The World of the New Urban Poor[END_REF]. For both culturalists and structuralists, the poor were middle-class in the making. The general optimism of the era and the influence of the economic context on the study of poverty are captured in The Affluent Society: "The poverty-stricken are further forgotten because it is assumed that with increasing output poverty must disappear. Increased output eliminated the general poverty of all who worked. Accordingly, it must, sooner or later, eliminate the special poverty that still remains" (Galbraith 1958: 255).

Such a conceptualization of poverty is obsolete in a society where r > g. In a patrimonial society, the poor cannot be considered middle-class in the making anymore. The changing economic behaviors of corporations has made obsolete the classic portrait of the economically secure middle-class of Mills' White Collar (1951) Capital owners can put to work their growing resources in reformatting the economic life of non-capital owners so that, in spite of sluggish economic output, "r" is maintained at the highest level possible. Technological progress plays an important role in this story; the examples of Uber for car services and taxis, AirBnB for the hotel industry, and Kindle-Amazon for publishing show how these corporations rely on new technologies, in association with heavy legal resources, to challenge economic circuits in which regulations and unions previously had a major role. The sociology of noncapital owners would be then a sociology of an emerging class replacing the slowly fading divisions between the poor, the working-class and the middle-class. The hypothesis of a rising new class of non-capital owners would be tested and amended by analysis of patterns of inter-marriage and socio-political movements that cut across previously existing boundaries between the poor, the working-class and the middleclass.

The expansion of the logic of economic recombination by capital owners ("disruption" in business lingo) to previously untouched corners of the economy is of particular interest. The literature suggests that work instability and limited economic prospects begin to homogenize various social profiles. Low-income minority individuals are constantly in and out low-paying job, often without benefits, alternating period of relative economic ease with spells of poverty. Educated artists and creative workers accumulate transient work experience, filling the gaps with alimentary jobs in order to support themselves. Employees at technology firms face a risk/reward profile that put them outside the common categories of employee and equity partner. There is a growing sense of a shared work experience between these non-capital owners. A key illustration of how previously upper-middle jobs come to gradually look-like low-income jobs is close to home: the college professor. Galbraith concludes the Affluent Society with a prophetic call of a rising new class, with leisure, autonomy, education, economic stability, and benefits. The vanguard of this new class is the tenured college professor, he says. But the changes of the past thirty years in the academic labor market have falsified Galbraith's prophecy. The tenured professor is being replaced by the adjunct and the clinical visiting position. MOOCs, the flipped classroom and recent legal challenges on tenure suggest that the dynamics will amplify in the next years. For all these often-young people, the challenge is to make it from one job to another without falling into the cracks of unemployment for too long, always trying to be ahead of the curve of economic restructuration.

The shared economic fate of non-capital owners suggests a renewed attention to the concept of informal economy. We tend to associate the informal economy with the Third World or the urban poor. In fact, Venkatesh already documents in Off the Books (2006) the surprisingly long ramifications of the underground economy in the Chicago's South Side, in which the lower rungs of the African-American ghetto and the black bourgeoisie are tied together. The normalization of work instability and low economic prospects for an ever greater share of the active population would suggest the extension of the underground economy to populations that we previously assumed were living their whole life on the formal side of the economy. It is not just that many may have to engage in informal economic activities to make ends meet; it is also that "informalization" may become to mean something slightly different. In the new economic world of non-capital owners, employment is precarious, labor regulations are undone and rules are rewritten. In short, capital owners have the power to informalize the economic life of non-capital owners [START_REF] Portes | Making it underground: Comparative material on the informal sector in Western market economies[END_REF].

The difference between today's class structure and the Belle Epoque's or the Gilded Age's is the contemporary presence of the "patrimonial middle class". With this expression, Piketty refers to the households whose income is constituted by both rent from capital and wage from labor, for comparable amounts. Among the 10% highest income, the patrimonial middle class is the first 9%, while for the 1%, rent largely dominates households' income. These are households that cannot be passive rentiers; they need to work, but they combine work with revenues generated by capital ownership. To be middle class today is to own capital, which in itself is a vast redefinition of the term. But one central sociological problem for this group remains the same: what are their strategies to avoid downward mobility and to integrate the 1%? By contrast with the affluent society's middle class, upward mobility for the patrimonial middle class cannot be achieved by education alone, and capital can be used to prevent downward mobility. An authentic patrimonial strategy becomes necessary. Scholars should therefore study, among others, intergenerational strategies aiming at preserving capital's unity, matrimonial strategies with the 1%, and entrepreneurial strategies coupled with educational trajectories. For members of the patrimonial middle class, long-term and especially intergenerational stability of their social position is the critical concern.

Finally, the capacity of capital owners to recombine economic circuits through technology makes human work less necessary, raising the issue of the useless poor. Technological progress, Piketty writes, is a good thing: humans do less of the repetitive, dirty, dangerous and unhealthy work, robots do more, and humans have more time for other endeavors. The problem is: who own the robots? Robots are capital. The productivity gains they enable benefit capitalists in the first place, and a few highly skilled workers; meanwhile, a lot of unskilled workers are made less employable.

Combined with slow growth, this runs the risk of creating a class of permanently unemployed humans, who live off informal economic activities and government benefits, while productivity gains make capitalists wealthier, increasing overall inequality. In the 2010s, in the United States, 16 million children grow up in poverty; their father often are cycled in and out of prison; it is unlikely that they will be part of the "knowledge economy". In the Euro zone, 19 million people are unemployed-many more do not even appear in official statistics, because they have long ceased to look for a job. What is to be made of them? A logical consequence of the creation of a large reserve army of labor is the lowering of the minimum wage.

All this creates threats of unrest and disorder from the poorest. Here again we may need to update our concepts. In the g > r scholarship, welfare and punishment have become decoupled. Affluence did so much to solve the social question that welfare has been increasingly conceptualized as poverty reduction and as a component of advanced citizenship (Esping-Andersen 1990), while punishment has become concerned with the technical concern of crime control and rehabilitation. But this way of thinking misses the fundamental problem of lesser eligibility, which cannot be escaped in a r > g society. The principle of lesser eligibility is the iron law of social engineering: punishment has to be worse than welfare, and welfare has to be worse than minimum wage (Rusche and [START_REF] Kirchheimer | Punishment and Social Structure[END_REF]. Otherwise, rational individuals will prefer welfare over work, and crime over welfare. Welfare has always been a means to maintain the social order by distributing relief in ways that keep the poor quiet and incentivize low-wage work [START_REF] Piven | Regulating the Poor: The Functions of Public Welfare[END_REF]; welfare and punishment are part of the same project of social engineering. Welfare and punishment are both tools to regulate the labor markets; declining living standards necessarily implies less welfare and harsher punishment. Welfare reform (towards "workfare") and the rise of mass incarceration in the United States exemplify our point (Pierson 2001, Western and[START_REF] Western | How Unregulated Is the US Labor Market? The Penal System as a Labor Market Institution[END_REF].

Patrimonial society and the limits of democratic pluralism

In the society of g > r, capital versus labor is the central political conflict. Politically, this means that there is electoral competition between the left and right, representing workers and capital owners [START_REF] Lipset | Cleavage Structures, Party Systems, and Voter Alignments: An Introduction[END_REF]Rokkan 1967, Korpi 1983). In each society, this conflict has been complicated by regional particularities, ethnic strife, one-party dominance, and so on, but the basic insight is this: g > r societies experience genuine political competition for power. This led political scientists of the g > r era to formulate influential theories in political science. The pluralist paradigm holds that power is not concentrated in the hands of the wealthy few, but fragmented, equilibrated, and above all contested between competitive alternative options. Partisan politics theory perceives political parties as "representative of social constituencies, mostly defined in terms of industrial classes, and as bearers of clear ideological stances for social-democratic or conservative welfare policies" (Häusermann et al. 2012: 222). It assumed that left-wing parties pursued policies favorable to the working class, and that left-wing parties have deep ties with unions. From the point of view of political theory, the affluent society has an affinity with social democracy.

In the return to patrimonial society, extreme inequalities subvert electoral competition. Sociologists now theorize that capitalism and democracy were only compatible in times of high growth (in Piketty's terms, when g > r): "To work properly, capitalism requires a rule-bound economic policy, with protection of markets and property rights constitutionally enshrined against discretionary political interference" (Streeck 2011: 7)-unelected bodies such as courts, central banks or the European Commission. German chancellor Merkel recently spoke of the necessity of a "market-conforming" democracy. In the United States, Gilens and Page (forthcoming) have shown that "economic elites and organized groups representing business interests have substantial independent impact on U.S. government policy, while average citizens and mass-based interest groups have little or no independent influence". Such a statement would have sounded like unsophisticated Marxism in the early 1970s, but it now reflects the political condition of the r > g society. Partisan politics theory also becomes obsolete: political parties are more autonomous from their base (Katz and Mair 1994) and leftwing parties pursue similar economic policies as right-wing parties (Häusermann et al. 2012).

In the r > g society, political disenchantment and apathy become important topics of study, as elections become increasingly meaningless and governments are perceived as working for private foreign interests. In the developing world, middle classes increasingly perceive democracy as favoring populist demands from the poor, instead of fostering the rule of law and economic development, and protest against elected leaders and calls for military coups (Kurlantzick 2013). In the Western world, elites increasingly believe that electoral populism threatens "responsible" governance and praise authoritarian systems (such as China's) for their efficacy [START_REF] Streeck | How will capitalism end?[END_REF].

In short, in the r > g society, the pluralist paradigm becomes less relevant, and is taken over by the elitist paradigm [START_REF] Mills | The Power Elite[END_REF]). The financial crisis of 2008 has played an important role in making this once-controversial statement obvious. As sociologist Brayden King observes in a blog post, "When I was in grad school, power elite theory seemed antiquated, an explanation founded on paranoid underpinnings. It was an undergrad-ish view of the world. (…) It turns out that the power elite has been really busy while sociologists have been off studying other things." [START_REF] King | where is power elite theory when you need it ?, orgtheory.net[END_REF]. In this society, the ideology of legitimation is Margaret Thatcher's "there is no alternative". Future research needs to investigate the political behavior of disenfranchised citizens, emerging ideologies of contestation and new ways of co-opting social movements.

In Capital, Piketty mentions in passing the "nationalist responses" (p. 27), probably referring to far-right European populist parties which have had electoral success in France, Greece, Netherlands, Hungary and Denmark. Piketty's implicit stance here is that electors vote under false consciousness: their objective problem is the concentration of wealth at the top, and their subjective perception of the problem is ethnic minorities and poor migrants. Capital brings up a point that complicates conventional postnationalist wisdom. Free trade means that the ownership of capital will become ever less national. The concentration of capital at the top-the crushing logic of r > gmeans that most capital will be owned by a few wealthy entities. Piketty notes that his reader may soon have to pay her rent to the emir of Qatar; sovereign wealth funds will be major capital owners at the global scale by 2050.

To whom rent is due may be irrelevant; but people over the world value ethnic or national sovereignty, which they often define as not being owned by foreign capitalists. What Piketty suggests here-probably unwillingly-is that nationhood matters. r > g not only produces extreme inequalities, it also goes with the transnationalization of capital ownership. Foreign-owned capital hurts national feelings. As Piketty writes, "no one knows the precise location of the psychological and political boundaries that must not be crossed when it comes to the ownership of one country by another" (p. 460). Implied here is a theory of sovereignty: the tolerance threshold for asset inequality is greater when assets remain nationally owned. The theory has implications for political stability. Piketty shows that there is no spontaneous economic mechanism that corrects the dynamic of capital accumulation; a venue for research is how nationalism may become a means to reclaim the ownership of capital. For instance, Streeck analyzes "the drama of democratic states being turned into debt-collecting agencies on behalf of a global oligarchy of investors" as translating "class conflicts into international conflicts" (2011: 28), as creditor countries (like Germany) impose austerity on debtor countries (like Greece).

In the g > r society, nationalism is traditionally analyzed as xenophobia, ignorance and false consciousness. In the r > g society, nationalism becomes intelligible: it is one of the ways in which societies can reclaim a sense of ownership. Historical fascism was a product of the First World war. But it was also "a reaction against globalization" and the "corrosive effects of transnational capitalism" (Mann 2012:315 et sq), and a solution to the corruption of parliamentary democracy under extreme inequality. Fascism was an "extreme form" of nation-statism; it sought to transcend class conflict and to restore order by way of state despotism. These elements help understand why most countries in Europe, South America and Asia became nationalist, rightist dictatorships between the two world wars, often with electoral support [START_REF] Mann | The Sources of Social Power: Volume 3, Global Empires and Revolution[END_REF].

Another implication relates to migration. In the affluent society, high growth allowed for the integration of migrants. This has led g > r social scientists to formulate the influential assimilation theory [START_REF] Gordon | Assimilation In American Life[END_REF]. Migrants, like the poor, were middle class in the making, and "the major preoccupation of postwar migration studies was to measure and scrutinize the cultural differences between immigrants and nationals and to describe pathways of assimilation (…), in short, to deliver a description of the mechanics of successful nation making process" (Wimmer and Glick Schiller 2002: 310). Nation and society were synonymous: the national society was a discrete totality, as exemplified by Bourdieu's Distinction, theorized to be incomparable to other nations/societies. After industrialization, prosperity was supposed to make nationalism a non-problem [START_REF] Gellner | Nations and nationalism[END_REF]).

These conceptions are challenged in the patrimonial society. Maximizing r means making capital more mobile, but also labor: in American corporate lobbying, immigration reform is one of the best funded causes (in particular from high tech industries). In a context of globalization, the r > g society is less likely to follow a national path of development. From Detroit to the Philippines, transnational capital flows force workers to migration. Migration in the r > g society becomes less an issue of assimilation and more an issue of transnationalism [START_REF] Wimmer | Methodological nationalism and beyond: nation-state building, migration and the social sciences[END_REF]. Low growth makes non-capital owning migrants unlikely candidates for upward mobility. Growing competition for work fosters conflict with nationals. Moreover, sending states increasingly develops diaspora policies, to maximize revenues from remittances and implement counter-brain drain measures [START_REF] Varadarajan | The Domestic Abroad: Diasporas in International Relations[END_REF]. Under r > g, the concept of national society as discrete totality dissolves.

Conclusion

In short, the evolution of the r/g ratio seems tremendously influential in structuring societies' social and political structures. Piketty shows how capital returns shape societies, regardless of their polity. In the 19 th century, France experienced five revolutions and a civil war to end up being a republic, England remained a stable monarchy, and the United States consistently remained a republic: same r, same g, and same social structure. In the 20 th century, when world wars, genocides, destructions and so on tempered with r and g, allowing for g > r, these three societies evolved in similar directions. And the return to the normalcy of r > g is about to bring back the fundamental social dynamics that we have experienced in the past. Capital strongly supports the idea that ownership is the organizing principle of capitalist society. Many have made this claim in the past and present, but Piketty brings particularly convincing supporting data.

We have no data to offer, only conjectures. In this paper, we have tried to flesh out the mechanisms which produce inequality when r exceeds g. We have looked into the transformation of the role of corporations, which are key actors in the process of maximizing r. We have explored the main implications of the return to the r > g society: the new economic behavior of corporations, the transformation of the class structure, the demise of the relevance of human and cultural capital, the informalization of workers, the new arrangements of welfare and punishment, the end of democratic pluralism and the return of nationalist politics.

All these transformations underline the historical situatedness of the concepts we routinely use to analyze social and political processes: concepts borne out of the affluent society and made obsolete by the return to the r > g society. It may have become more relevant to study corporations as political actors than sites of work and production. Occupational status and education are losing their relevance in providing an accurate account of one's social position, as inheritance trumps work. Poor people are not middle class in the making any more, and welfare and punishment become conceptually interdependent means to social order. It is increasingly obvious that once-polemical elitist perspective are more useful to account for contemporary politics than pluralist ones. Immigration is less an issue of assimilation, and more one of transnationalist and nationalist politics. Thanks to the wealth of hard results about the functioning of capitalist societies, Capital has streamlined entire fields for social research; this is a first sketch in exploring implications.

To conclude: Capital's trends about r and g have been calculated over the past two centuries. We do not know how r and g will be affected by climate change, "peak everything" and the resulting rising prices, declining fossil fuel reserves, losses in biodiversity, and so on. Social and political implications of the interpolation of the patrimonial society with environmental change certainly is one of the most important venue for future research.

  The context of r > g creates therefore a new population of study: the people who do not own capital, and who cannot realistically aspire to become capital owners in ways that would significantly alter their economic standing in society. This group is much larger than the minority officially living in poverty and includes most of the middle-class that face a new economic environment. In a patrimonial society, what defines non-capital owners' common social position is their dependence on capital owners' resources and lobbying capacity of the state. Through technology, regulation and globalization, capital owners recombine the economic circuits in which non-capital owners make a living.

	and Whyte's The
	Organization Man (1956). Countless studies chronicle the increasing precariouness of
	the American middle-class, from Sennett's The Corrosion of Character (1998) to
	Hacker, who writes in 2006 that job insecurity "isn't just a problem of the poor and
	uneducated … Increasingly it affects … educated, upper-middle class Americans" (cited
	in Kalleberg 2011: 15).

Acknowledgments

We are grateful for helpful feedback from Bernard Augier, Quentin Batréau, Marc Bonnet, Christophe Bouillaud, Sébastien Chauvin, Claire Dupuy, Herb Gans, Patrick Le Galès, Raul Magni-Berton, Pierre Martin and Francesco Ragazzi.

August 15 th , 2014