Short Probabilistic Proof of the Brascamp-Lieb and Barthe Theorems

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Short Probabilistic Proof of the Brascamp-Lieb and Barthe Theorems

Joseph Lehec

To cite this version:

Joseph Lehec. Short Probabilistic Proof of the Brascamp-Lieb and Barthe Theorems. Bulletin cana-

dien de mathématiques, 2014, 57, pp.585 - 597. �10.4153/CMB-2013-040-x�. �hal-01100925�

AND BARTHE THEOREMS

JOSEPH LEHEC

Abstract. We give a short proof of the Brascamp-Lieb theorem, which asserts that a certain general form of Young’s convolution inequality is saturated by Gaussian functions. The argument is inspired by Borell’s stochastic proof of the Pr´ekopa-Leindler inequality and applies also to the reversed Brascamp- Lieb inequality, due to Barthe.

1. Introduction A Brascamp-Lieb datum onRⁿ is a finite sequence (1) (c1, B1), . . . ,(cm, Bm)

whereciis a positive number andBi: Rⁿ →Rⁿⁱ is linear and onto. The Brascamp- Lieb constant associated to this datum is the smallest real numberCsuch that the inequality

(2)

Z

Rⁿ m

Y

i=1

(fi◦Bi)^cidx≤C

m

Y

i=1

Z

Rⁿⁱ

fidxci

holds for every set of non-negative integrable functionsfi:Rⁿⁱ→R. The Brascamp- Lieb theorem [9, 13] asserts that (2) is saturated by Gaussian functions. In other words if (2) holds for every functionsf1, . . . , fm of the form

fi(x) = e^−hAix,xi/2

where Ai is a symmetric positive definite matrix on Rⁿⁱ then (2) holds for every set of functionsf1, . . . , fm.

The reversed Brascamp-Lieb constant associated to (1) is the smallest constant Crsuch that for every non-negative measurable functionsf1, . . . , fm, f satisfying (3)

m

Y

i=1

fi(xi)^ci ≤fX^m

i=1

ciB_i^∗xi

for every (x1, . . . , xm)∈Rⁿ¹× · · · ×R^nm we have (4)

m

Y

i=1

Z

Rⁿⁱ

fidxci

≤Cr

Z

Rⁿ

fdx.

It was shown by Barthe [1] that again Gaussian functions saturate the inequality.

The original paper of Brascamp and Lieb [9] rely on symmetrization techniques.

Barthe’s argument uses optimal transport and works for both the direct and the reversed inequality. More recent proofs of the direct inequality [5, 6, 10, 11] all

2010Mathematics Subject Classification. Primary 39B62, Secondary 60J65.

Key words and phrases. Functional inequalities, Brownian motion.

1

rely on semi-group techniques. There are also semi-group proofs of the reversed inequality, at least when the Brascamp-Lieb datum has the following property

BiB_i^∗= id_Rⁿⁱ, ∀i≤m,

m

X

i=1

ciB_i^∗Bi= id_Rⁿ, (5)

called frame condition in the sequel. This was achieved by Barthe and Cordero- Erausquin [2] in the rank 1 case (when all dimensionsni equal 1) and Barthe and Huet [3] in any dimension.

The purpose of the present article is to give a short probabilistic proof of the Brascamp-Lieb and Barthe theorems. Our main tool shall be a representation formula for the quantity

lnZ

e^g(x) γ(dx) ,

where γ is a Gaussian measure. Let us describe it briefly. Let (Ω,A,P) be a probability space, let (Ft)t∈[0,T] be a filtration and let

(Wt)t∈[0,T]

be a Brownian motion taking values inRⁿ(we fix a finite time horizonT). Assuming that the covariance matrix A of the random vectorW1 has full rank, we let Hbe the Cameron-Martin space associated toW; namely the Hilbert space of absolutely continuous paths u: [0, T]→Rⁿ starting from 0, equipped with the norm

kuk_H= Z T

0

hA⁻¹u˙s,u˙sids1/2

.

In the sequel we calldrift any adapted processU which belongs toHalmost surely.

The following formula is due to Bou´e and Dupuis [8] (see also [7, 12]).

Proposition 1. Let g:Rⁿ→Rbe measurable and bounded from below, then log

Ee^g(WT)

= supn E

g(WT+UT)−1

2kUk²_Ho where the supremum is taken over all drifts U.

In [7], Borell rediscovers this formula and shows that it yields the Pr´ekopa- Leindler inequality (a reversed form of H¨older’s inequality) very easily. Later on Cordero and Maurey noticed that under the frame condition, both the direct and reversed Brascamp-Lieb inequalities could be recovered this way (this was not pub- lished but is explained in [12]). The purpose of this article is, following Borell, Cordero and Maurey, to show that the Brascamp-Lieb and Barthe theorems in full generality are direct consequences of Proposition 1.

2. The direct inequality

Replacefibyx7→fi(x/λ) in inequality (2). The left-hand side of the inequality is multiplied by λⁿ and the right-hand side by λ^Pmi=1cini. Therefore, a necessary condition forC to be finite is

m

X

i=1

cini=n.

This homogeneity condition will be assumed throughout the rest of the article.

Theorem 2. Assume that there exists a positive definite matrixA satisfying

(6) A⁻¹=

m

X

i=1

ciB_i^∗(BiAB_i^∗)⁻¹Bi. Then the Brascamp-Lieb constant is

C= det(A)

Qm

i=1det(BiAB_i^∗)^ci 1/2

,

and there is equality in (2) for the following Gaussian functions (7) fi:x∈Rⁿⁱ7→e^{−h(BiAB∗i)−1x,xi/2}, i≤m.

Remark. If the frame condition (5) holds then A = id_Rⁿ satisfies (6) and the Brascamp-Lieb constant is 1.

Proof. Because of (6), if the functionsfi are defined by (7) then

m

Y

i=1

fi(Bix)ci

= e^{−hA−1x,xi/2}.

The equality case follows easily (recall the homogeneity conditionP

cini=n).

Let us prove the inequality. Letf1, . . . , fmbe non-negative integrable functions on Rⁿ¹, . . . ,R^nm, respectively and let

f:x∈Rⁿ 7→

m

Y

i=1

fi(Bix)^ci. Fixδ >0, letgi= log(fi+δ) for everyi≤mand let

g(x) =

m

X

i=1

cigi(Bix).

The functions (gi)i≤m, gare bounded from below. Fix a time horizonT, let (Wt)t≥T

be a Brownian motion on Rⁿ, starting from 0 and having covarianceA; and letH be the associated Cameron-Martin space. By Proposition 1, given ǫ > 0, there exists a driftU such that

log

Ee^g(WT)

≤E

g(WT +UT)−1 2kUk²_H

+ǫ

=

m

X

i=1

ciEgi(BiWT+BiUT)−1

2EkUk²_H+ǫ.

(8)

The processBiW is a Brownian motion onRⁿⁱ with covarianceBiAB_i^∗. SetAi= BiAB_i^∗ and letHi be the Cameron-Martin space associated toBiW. Equality (6) gives

hA⁻¹x, xi=

m

X

i=1

cihA⁻¹_i Bix, Bixi

for everyx∈Rⁿ. This implies that kuk²_H=

m

X

i=1

cikBiuk²_Hi

for every absolutely continuous pathu: [0, T]→Rⁿ. So that (8) becomes log

Ee^g(WT)

≤

m

X

i=1

ciE

gi(BiWT+BiUT)−1

2kBiUk²_Hi +ǫ.

By Proposition 1 again we have E

gi(BiWT+BiUT)−1

2kBiUk²_Hi

≤log

Ee^gi(BiWT) for everyi≤m. We obtain (droppingǫ which is arbitrary)

(9) log

Ee^g(WT)

≤

m

X

i=1

cilog

Ee^gi(BiWT) .

Recall thatf ≤e^g and observe that

m

Y

i=1

E(e^gi(BiWT)ci

≤

m

Y

i=1

Efi(BiWT)ci

+O(δ^c),

for some positive constantc. Inequality (9) becomes (dropping theO(δ^c) term)

(10) Ef(WT)≤

m

Y

i=1

Efi(BiWT)ci

.

SinceWT is a centered Gaussian vector with covarianceT A Ef(WT) = 1

(2πT)^n/2det(A)^1/2 Z

Rⁿ

f(x)e^{−hA−1x,xi/2T}dx,

and there is a similar equality forEfi(BiWT). Then it is easy to see that lettingT tend to +∞in inequality (10) yields the result (recall thatP

cini =n).

Example(Optimal constant in Young’s inequality). Young’s convolution inequality asserts that ifp, q, r≥1 and are linked by the equation

(11) 1

p+1

q = 1 +1 r, then

kF∗Gkr≤ kFkpkGkq,

for allF ∈LpandG∈Lq. When eitherp,qorrequals 1 or +∞the inequality is a consequence of H¨older’s inequality and is easily seen to be sharp. On the other hand when p, q, r belong to the open interval (1,+∞) the best constant C in the inequality

kF∗Gkr≤CkFkpkGkq,

is actually smaller than 1. Let us compute it using the previous theorem. Observe that by dualityCis the best constant in the inequality

(12) Z

R²

f^c1(x+y)g^c2(y)h^c3(x) dxdy≤CZ

R

fc1Z

R

gc2Z

R

hc3

,

where

c1=1

p, c2= 1

q, c3= 1−1 r.

In other wordsCis the Brascamp-Lieb constant inR²associated to the data (c1, B1),(c2, B2),(c3, B3),

where B1 = (1,1), B2 = (0,1) andB3 = (1,0). According to the previous result, we have to find a positive definite matrixAsatisfying

A⁻¹=

3

X

i=1

ciB_i^∗(BiAB_i^∗)⁻¹Bi.

LettingA= x z

z y

, this equation turns out to be equivalent to (1−c2)xy+yz+c2z²= 0

(1−c3)xy+xz+c3z²= 0 c1+c2+c3= 2.

The third equation is just the Young constraint (11). The first two equations admit two families of solutions: either (x, y, z) is a multiple of (1,1,−1) or (x, y, z) is a multiple of

c3(1−c3), c2(1−c2),−(1−c2)(1−c3) .

The constraintxy−z²>0 rules out the first solution. The second solution is fine sincec1, c2andc3are assumed to belong to the open interval (0,1). By Theorem 2, the best constant in (12) is

C= det(A)

Q3

i=1det(BiAB_i^∗)^ci 1/2

=(1−c1)^1−c1(1−c2)^1−c2(1−c3)^1−c3 c^c₁¹c^c₂²c^c₃³

1/2

.

In terms ofp, q, rwe have

C= p^1/p q^1/q r^′1/r′ p^′1/p′ q^′1/q′ r^1/r

^1/2

where p^′, q^′, r^′ are the conjugate exponents of p, q, r, respectively. This is indeed the best constant in Young’s inequality, first obtained by Beckner [4].

3. The reversed inequality

Theorem 3. Again, assume that there is a matrix A satisfying (6). Then the reversed Brascamp-Lieb constant is

Cr= det(A) Qm

i=1det(BiAB_i^∗)^{ci 1/2}

.

There is equality in (4) for the following Gaussian functions fi:x∈Rⁿⁱ 7→e^{−hBiAB∗ix,xi/2}, i≤m.

f:x∈Rⁿ7→e^−hAx,xi/2.

Remark. Observe that under condition (6) the Brascamp-Lieb constant and the reversed constant are the same, but the extremizers differ.

We shall use the following elementary lemma.

Lemma 4. Let A1, . . . , Am be positive definite matrices on Rⁿ¹, . . . ,R^nm, respec- tively and let

A=X^m

i=1

ciB_i^∗A⁻¹_i Bi

−1

.

Then for allx∈Rⁿ

hAx, xi= infnX^m

i=1

cihAixi, xii,

m

X

i=1

ciB_i^∗xi=xo .

Proof. Letx1, . . . , xmand let

(13) x=

m

X

i=1

ciB_i^∗xi.

Then by the Cauchy-Schwarz inequality (recall that the matrices Ai are positive definite)

hAx, xi=

m

X

i=1

cihAx, B^∗_ixii=

m

X

i=1

cihBiAx, xii

≤X^m

i=1

cihA⁻¹_i BiAx, BiAxi1/2X^m

i=1

cihAixi, xii1/2

=hAx, xi^1/2 X^m

i=1

cihAixi, xii^1/2 .

Besides, givenx∈Rⁿ, setxi=A⁻¹_i BiAxfor alli≤m. Then (13) holds and there is equality in the above Cauchy-Schwarz inequality. This concludes the proof.

Proof of Theorem 3. The equality case is a straightforward consequence of the hy- pothesis (6) and Lemma 4, details are left to the reader.

Let us prove the inequality. There is no loss of generality assuming that the func- tionsf1, . . . , fm are bounded from above (otherwise replacefi by max(fi, k), letk tend to +∞and use monotone convergence). Fix δ > 0 and let gi = log(fi+δ) for everyi≤m. By (3) and since the functionsfi are bounded from above, there exist positive constantsc, C such that the function

g:x∈Rⁿ7→log f(x) +Cδ^c ,

satisfies (14)

m

X

i=1

cigi(xi)≤gX^m

i=1

ciB_i^∗xi

for everyx1, . . . , xm. Observe that the functions (gi)i≤m, gare bounded from below.

Let (Wt)t≤T be a Brownian motion on Rⁿ having covariance matrix A. SetAi= BiAB_i^∗, thenA⁻¹_i BiW is a Brownian motion on Rⁿⁱ with covariance matrix

(A⁻¹_i Bi)A(A⁻¹_i Bi)^∗=A⁻¹_i (BiAB_i^∗)A⁻¹_i =A⁻¹_i .

LetHi be the associated Cameron-Martin space. By Proposition 1 there exists a (Rⁿⁱ-valued) drift Ui such that

(15) log

Ee^{gi(A−i1BiWT)}

≤E

gi(A⁻¹_i BiWT + (Ui)T)−1

2kUik²_Hi +ǫ.

By (14) and (6)

m

X

i=1

cigi(A⁻¹_i BiWT + (Ui)T)≤gX^m

i=1

ciB_i^∗(A⁻¹_i BiWT + (Ui)T)

=g

A⁻¹WT+

m

X

i=1

ciB_i^∗(Ui)T

.

The Brownian motion (A⁻¹W)t≤T has covariance matrixA⁻¹A(A⁻¹)^∗=A⁻¹. Let Hbe the associated Cameron-Martin space. Lemma 4 shows that

DAX^m

i=1

ciB^∗_ixi

,

m

X

i=1

ciB_i^∗xi

E≤

m

X

i=1

cihAixi, xii

for everyx1, . . . , xm inRⁿ¹, . . . ,R^nm, respectively. Therefore

m

X

i=1

ciB^∗iui

2 H≤

m

X

i=1

cikuik²_Hi.

for every sequence of absolutely continuous paths (ui: [0, T] → Rⁿⁱ)i≤m. Thus multiplying (15) byci and summing overiyields

m

X

i=1

cilog

Ee^{gi(A−i1BiWT)}

≤Eh

g A⁻¹WT +

m

X

i=1

ciB_i^∗(Ui)T

−1 2

m

X

i=1

ciB_i^∗Ui

2 H

i+

m

X

i=1

ciǫ.

Hence, using Proposition 1 again and droppingǫagain, (16)

m

X

i=1

cilog

Ee^{gi(A−i1BiWT)}ci

≤log

Ee^g(A−1WT) .

Recall that fi ≤e^gi for every i≤mand that e^g =f +Cδ^c. Since δ is arbitrary, inequality (16) becomes

m

Y

i=1

Efi(A⁻¹_i BiWT)^ci

≤Ef(A⁻¹WT).

Again, lettingT tend to +∞in this inequality yields the result.

4. The Brascamp-Lieb and Barthe theorems

So far we have seen that both the direct inequality and the reversed version are saturated by Gaussian functions when there exists a matrixAsuch that

(17) A⁻¹=

m

X

i=1

ciB_i^∗(BiAB_i^∗)⁻¹Bi.

In this section, we briefly explain why this yields the Brascamp-Lieb and Barthe theorems.

Applying (2) to Gaussian functions gives (18)

m

Y

i=1

det(Ai)^ci≤C²det

m

X

i=1

ciB^∗_iAiBi ,

for every sequenceA1, . . . , Amof positive definite matrices onRⁿ¹, . . . ,R^nm. LetCg

be the Gaussian Brascamp-Lieb constant; namely the best constant in the previous inequality. We haveCg≤Cand it turns out that applying (4) to Gaussian functions yieldsCg≤Cr (one has to apply Lemma 4 at some point).

It is known since the work of Carlen and Cordero [10] that there is a dual formulation of (2) in terms of relative entropy. In the same way, there is a dual formulation of (18). For every positive matrixAonRⁿ, one has

log det(A) = inf

B>0

tr(AB)−n−log(det(B)) , with equality when B = A⁻¹. Using this and the equality Pm

i=1cini = n, it is easily seen thatCg is also the best constant such that the inequality

(19) det(A)≤C_g²

m

Y

i=1

det(BiAB^∗_i)^ci holds for every positive definite matrixAonRⁿ.

Example. Assume that m = n, that c1 =· · · =cn = 1 and that Bi(x) =xi for i ∈ [n]. Inequality (18) trivially holds with constant 1 (and there is equality for everyA1, . . . , An). On the other hand (19) becomes

det(A)≤

n

Y

i=1

aii,

for every positive definiteA, with equality whenAis diagonal. This is Hadamard’s inequality.

Lemma 5. IfA is extremal in (19)thenA satisfies (17).

Proof. Just compute the gradient of the map A >07→log det(A)−

m

X

i=1

cilog det(BiAB^∗_i).

Therefore, if the constantCg is finite and if there is an extremizerAin (19) then A satisfies (17) and together with the results of the previous sections we get the Brascamp-Lieb and Barthe equalities

(20) C=Cr=Cg.

Although it may happen thatCg<+∞and no Gaussian extremizer exists, there is a way to bypass this issue. For the Brascamp-Lieb theorem, there is an abstract argument showing that is it is enough to prove the equality C=Cg when there is a Gaussian extremizer. This argument relies on:

(1) A criterion for having a Gaussian extremizer, due to Barthe [1] in the rank 1 case (namely when the dimensions ni are all equal to 1) and Bennett, Carbery, Christ and Tao [6] in the general case.

(2) A multiplicativity property of CandCg due to Carlen, Lieb and Loss [11]

in the rank 1 case and obtained in full generality in [6] again.

There is no point repeating this argument here, and we refer to [11, 6] instead. This settles the case of theC=Cgequality. As for theC=Crequality, we observe that the above argument can be carried out verbatim once the mutliplicativity property of the reversed Brascamp-Lieb constant is established. This is the purpose of the rest of the article.

Definition 6. Given a proper subspaceE ofRⁿ, we let fori≤m Bi,E:E→BiE

x7→Bix,

Bi,E^⊥:E^⊥→(BiE)^⊥ x7→qi(Bix),

where qi is the orthogonal projection onto (BiE)^⊥. Observe that both Bi,E and Bi,E^⊥ are onto. Now we let Cr,E be the reversed Brascamp-Lieb constant on E associated to the datum

(c1, B1,E), . . . ,(cm, Bm,E)

andC_r,E⊥ be the reversed Brascamp-Lieb constant onE^⊥associated to the datum (c1, B1,E^⊥), . . . ,(cm, Bm,E^⊥)

Proposition 7. Let E be a proper subspace of Rⁿ, and assume that E is critical, in the sense that

dim(E) =

m

X

i=1

cidim(BiE).

Then Cr=Cr,E×C_r,E⊥.

Bennett, Carbery, Christ and Tao proved the corresponding property ofC and Cg, we adapt their argument to prove the multiplicativity ofCr.

Let us prove the inequalityCr≤Cr,E×Cr,E^⊥ first. This does not requireEto be critical. Let f1, . . . , fm, f be functions onRⁿ¹, . . . ,R^nm,Rrespectively, satisfying

m

Y

i=1

fi(zi)^ci ≤fX^m

i=1

ciB_i^∗zi

for all z1, . . . , zm. Fix (x1, . . . , xn)∈B1E× · · · ×BmE. Since (Bi,E^⊥)^∗yi=B_i^∗yi

for everyyi ∈(BiE)^⊥, applying the reversed Brascamp-Lieb inequality onE^⊥ to the functions

y∈(BiE)^⊥7→fi(xi+y), i≤m yields

m

Y

i=1

Z

(BiE)^⊥

fi(xi+yi) dyi

ci

≤C_r,E⊥

Z

E^⊥

fX^m

i=1

ciB^∗_ixi+y dy

=C_r,E⊥

Z

E^⊥

fX^m

i=1

ci(Bi,E)^∗xi+y dy.

For the latter equality, observe that (Bi,E)^∗xi =p(B_i^∗xi) wherepis the orthogonal projection with rangeEand use the translation invariance of the Lebesgue measure.

Applying the reversed Brascamp-Lieb inequality (this time onE) and using Fubini’s theorem we get

m

Y

i=1

Z

Rⁿⁱ

fidxci

≤Cr,ECr,E^⊥

Z

Rⁿ

fdx, which is the result.

We start the proof of the inequalityCr,EC_rE⊥ ≤Cr with a couple of simple obser- vations.

Lemma 8. Upper semi-continuous functions having compact support saturate the reversed Brascamp-Lieb inequality.

Proof. The regularity of the Lebesgue measure implies that given a non-negative in- tegrable functionfionRⁿⁱ andǫ >0 there exists a non-negative linear combination of indicators of compact setsgi satisfying

gi≤fi and Z

Rⁿⁱ

fidx≤(1 +ǫ) Z

Rⁿⁱ

gidx.

The lemma follows easily.

The proof of the following lemma is left to the reader.

Lemma 9. If f1, . . . , fm are compactly supported and upper semi-continuous on Rⁿ¹, . . . ,R^nm respectively, then the function f defined onRⁿ by

f(x) = supnY^m

i=1

fi(xi)^ci,

m

X

i=1

ciB_i^∗xi =xo ,

is compactly supported and upper semi-continuous as well.

Remark. If the Brascamp-Lieb datum happens to bedegenerate, in the sense that the map (x1, . . . , xm) 7→ Pm

i=1B_i^∗xi is not onto, then Brascamp-Lieb constants are easily seen to be +∞. Still the previous lemma remains valid, provided the convention sup∅= 0 is adopted.

Let us prove thatCr,E×Cr,E^⊥ ≤Cr. By Lemma 8, it is enough to prove that the inequality

m

Y

i=1

Z

BiE

fidxci

×

m

Y

i=1

Z

(BiE)^⊥

gidxci

≤Cr

Z

E

fdxZ

E^⊥

gdx .

holds for every compactly supported upper semi-continuous functions (fi)i≤m and (gi)i≤m, wheref andg are defined by

f: x∈E 7→supnY^m

i=1

fi(xi)^ci,

m

X

i=1

ci(Bi,E)^∗xi=xo

g: y∈E^⊥7→supnY^m

i=1

gi(yi)^ci,

m

X

i=1

ci(B_i,E⊥)^∗yi=yo .

Letǫ >0. Fori≤mdefine a functionhi onRⁿⁱ by

hi(x+y) =fi(x/ǫ)gi(y), ∀x∈BiE, ∀y∈(BiE)^⊥, and let

h:z∈Rⁿ 7→supnY^m

i=1

hi(zi)^ci,

m

X

i=1

ciB_i^∗zi=zo .

By definition of the reversed Brascamp-Lieb constantCr

(21)

m

Y

i=1

Z

Rⁿⁱ

hidxci

≤Cr

Z

Rⁿ

hdx.

Using the equalityPm

i=1cidim(BiE) = dim(E) we get ǫ^−dim(E)

m

Y

i=1

Z

Rⁿⁱ

hidxci

=

m

Y

i=1

Z

BiE

fidxci

×

m

Y

i=1

Z

(BiE)^⊥

gidxci

.

On the other hand, we let the reader check that for everyx∈E, y∈E^⊥ h(ǫx+y)≤f(x)gǫ(y),

where

gǫ(y) = sup

g(y^′), |y−y^′| ≤Kǫ

and K is a constant depending on the diameters of the supports of the functions fi. Therefore

ǫ^−dimE Z

Rⁿ

hdx= Z

E×E^⊥

h(ǫx+y) dxdy≤Z

E

fdxZ

E^⊥

gǫdx .

Inequality (21) becomes

m

Y

i=1

Z

BiE

fidxci

×

m

Y

i=1

Z

(BiE)^⊥

gidxci

≤Cr

Z

E

fdxZ

E^⊥

gǫdx .

By Lemma 9, the function g has compact support and is upper semi-continuous.

This implies easily that

ǫ→0lim Z

E^⊥

gǫdx= Z

E^⊥

gdx, which concludes the proof.

Acknowledgment

I would like to thank the anonymous referee for his careful reading of the paper and his accurate remarks.

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Joseph Lehec Ceremade (UMR CNRS 7534), Universit´e Paris-Dauphine, Place de Lat- tre de Tassigny, 75016 Paris, France.

E-mail address: lehec@ceremade.dauphine.fr