Representation formula for the entropy and functional inequalities

www.imstat.org/aihp 2013, Vol. 49, No. 3, 885–899

DOI:10.1214/11-AIHP464

© Association des Publications de l’Institut Henri Poincaré, 2013

Representation formula for the entropy and functional inequalities

Joseph Lehec

CEREMADE, UMR CNRS 7534, Université Paris-Dauphine, place du maréchal de Lattre de Tassigny, 75016 Paris, France Received 11 June 2010; revised 9 September 2011; accepted 14 November 2011

Abstract. We prove a stochastic formula for the Gaussian relative entropy in the spirit of Borell’s formula for the Laplace trans- form. As an application, we give simple proofs of a number of functional inequalities.

Résumé. On démontre une formule stochastique pour l’entropie relative par rapport à la Gaussienne, dans le genre de la formule de Borell pour la transformée de Laplace. Cette formule donne des preuves simples d’un certain nombre d’inégalités fonctionnelles.

MSC:39B62; 60J65

Keywords:Gaussian measure; Entropy; Functional inequalities; Girsanov’s formula

1. Introduction: Borell’s formula

Letγ_dbe the standard Gaussian measure onR^d: γ_d(dx)=e^−|x|2/2

(2π)^d/2dx, where|x| =√

x·xdenotes the Euclidean norm ofx. In [5] Borell proves the following representation formula. Given a standardd-dimensional Brownian motionBand a bounded functionf:R^d→Rwe have

log

R^de^fdγd

=sup

u

E

f

B₁+

₁

0

u_sds

−1 2

₁

0 |u_s|²ds

, (1)

where the supremum is taken over all random processes u, say bounded and adapted to the Brownian filtration.

Among other applications, he derives easily the Prékopa–Leindler inequality. The name Borell’s formulamay be unfair to Boué and Dupuis who in an earlier paper [6] obtained a stronger result, allowing the functionf to depend on the whole path(B_t)_t∈[0,1](see Theorem9below for a precise statement). Anyway, Borell and Boué–Dupuis agree that representation formulas such as (1) arose much earlier in optimal control theory, particularly in Fleming and Soner’s work [13], and Borell should definitely be credited for bringing these techniques in the context of functional inequalities.

The present article deals with relative entropy. Let(Ω,A, m)be a measured space andμbe a probability measure.

The relative entropy ofμis defined by H(μ|m)=

Ω

dμ dmlog

dμ dm

dm ifμm

and H(μ|m)= +∞otherwise. It is well known that there is a Legendre duality between relative entropy and logarith- mic Laplace transform:

H(μ|m)=sup

f

fdμ−log

Ω

e^fdm

. (2)

The purpose of this article is to prove a representation formula for the Gaussian relative entropy, both inR^dand in the Wiener space, providing the entropy counterparts of the results mentioned above. All these formulas have a common feature: Girsanov’s theorem. However, our approach is somewhat different from that of Borell and Boué–Dupuis:

it draws a connection with the work of Föllmer [14,15] which makes the whole argument arguably simpler. As an application, we give new, unified and simple proofs of a number of Gaussian inequalities.

2. Representation formula for the entropy

This section contains the main results of the article. Let us recall a couple of classical facts about relative entropy, see for instance [23], Section 10, and the references therein. IfAis the Borelσ-field of a Polish topology onΩ then it is enough to take the supremum over bounded and continuous functions in (2). In particular the mapμ→H(μ|m)is lower semicontinuous with respect to the topology of weak convergence of measures. IfT:(Ω,A)→(Ω,A)is a measurable map then

H

μ◦T⁻¹|m◦T⁻¹

≤H(μ|m) (3)

and assuming that H(μ|m) <+∞, equality occurs if and only if the density dμ/dmis a function of T. We now describe the setting of the article. LetWbe the space of continuous paths

w∈C⁰

R₊,R^d

, w₀=0

equipped with the topology of uniform convergence on compact intervals. LetBbe the associated Borelσ-field and letγ be the Wiener measure on(W,B). Letx_t:w→w_tbe the coordinate process and(Gt)_t≥0be the natural filtration ofx. It is well known thatBcoincides with the smallestσ-field containing

t≥0Gt. LetHbe the Cameron–Martin space: a pathUbelongs toHif there existsu∈L²([0,+∞);R^d)such that

U_t= _t

0

u_sds, t≥0.

The norm ofU inHis then defined by U =

_+∞

0

|u_s|²ds 1/2

.

The Cauchy–Schwarz inequality shows that the Hilbert spaceHembeds continuously inW. Given a probability space (Ω,A,P)equipped with a filtration(Ft)t≥0we calldriftany adapted processU which belongs toHalmost surely.

Lastly, our Brownian motions are alwaysd-dimensional, standard and always start from 0.

2.1. The upper bound

We shall use repeatedly Girsanov’s formula, see [18], Chapter 6.

Proposition 1. LetBbe a Brownian motion defined on some filtered probability space(Ω,A,P,F)and letU be a drift.Lettingμbe the law ofB+U,we have

H(μ|γ )≤1

2^EU². (4)

Proof. WriteU_t =_t

0u_sds and assume for the moment thatU²=_∞

0 |u_s|²ds is uniformly bounded. Then by Novikov’s criterion

M_t=exp

− _t

0

u_s·dBs−1 2

_t

0 |u_s|²ds

, t≥0,

is a uniformly integrable martingale and Girsanov’s formula applies. Under dQ=M_∞dP

the processX:=B+Uis a Brownian motion. ThereforeXhas lawμandγ underPandQ, respectively. Then by (3) H(μ|γ )≤H(P|^Q)= −^Elog(M_∞)=1

2^EU²,

which concludes the proof whenUis bounded. In the general case, define the stopping time T_n=inf

t≥0,

_t

0 |u_s|²ds≥n

,

letUnbe the stopped process(Un)t=Ut∧T_nandμnbe the law ofB+Un. With probability 1 we haveU²<+∞, thusTn→ +∞andUn→U inH, hence inW. Thereforeμn→μweakly. Also^EUn²→^EU² by monotone convergence. Thus, using the lower semicontinuity of the entropy (observe thatWis a Polish space)

H(μ|γ )≤lim inf

n H(μn|γ )

≤lim inf

n

1

2^EU_n²=1

2^EU².

Remark. It follows immediately that when^EU²<+∞,the law ofB+U is absolutely continuous with respect to the Wiener measureγ.Let us point out that this is actually true for all driftsU,even if^EU²= +∞,see[18], Chapter7.

2.2. Föllmer’s drift

Let us address the question whether, given a probability measureμonW, equality can be achieved in (4). Recall that (x_t)_t≥0is the coordinate process on Wiener space(W,B, γ )and that(Gt)_t≥0is its natural filtration. The following is due to Föllmer [14,15].

Theorem 2. Letμbe a measure on(W,B)having densityF with respect toγ.There exists an adapted processu such that underμthe following holds.

1. The processU_t=t

0u_sdsbelongs toHalmost surely. 2. The processy=x−Uis a Brownian motion.

3. The relative entropy ofμis H(μ|γ )=1

2^E

μU².

We sketch the proof for completeness.

Proof of Theorem2. ThroughoutE^γ andE^μdenote expectations with respect toγ andμ respectively. OnGt the measureμhas density

Ft:=^Eγ(F|Gt),

with respect toγ. A standard martingale argument shows that μ

tinf≥0F_t>0

=μ(F >0)=1. (5)

Since Brownian martingales can be represented as stochastic integrals there exists an adapted processvsatisfying γ

_+∞

0

|v_s|²ds <+∞

=1 (6)

and

F_t=1+ _t

0

v_s·dxs, t≥0.

Letube the process defined by u_t=1_{Ft>0}(F_t)⁻¹v_t. It is adapted and (5) and (6) yield

μ _∞

0

|u_s|²ds <+∞

=1, which is the first assertion of the theorem.

The assertion 2 follows from Girsanov’s formula, see [18], Theorem 6.2.

Underμ, we have F_t =1+

_t

0

F_su_s·dxs

=1+ _t

0

Fsus·dys+ _t

0

Fs|us|²ds.

Applying Itô’s formula (recall thatF is positive andyis a Brownian motion underμ) we obtain log(F )=

_+∞

0

us·dys+1 2

_+∞

0

|us|²ds.

IfE^μU²<+∞the local martingale part in the equation above is integrable and has mean 0 so that H(μ|γ )=^Eμlog(F )=1

2^E

μU².

Again, a localization argument shows that this equality remains valid whenE^μU²= +∞, see [14], Lemma 2.6.

To finish this subsection, we give a formula for Föllmer’s drift when the underlying density has a Malliavin deriva- tive, we refer to the first chapter of [19] for the (little amount of) Malliavin calculus we shall use. For suitable F:W→Rwe let DF:W→Hbe the Malliavin derivative ofF. The domain of D in the space L²(W,B, γ ) is denoted byD². IfF ∈D²then the Clark–Ocone formula asserts

E^γ(F|Gt)=1+ _t

0

E^γ(DsF|Gs)·dxs, t≥0.

We obtain the following result.

Lemma 3. WhenF∈D²the processu_t given by Theorem2is u_t=^Eγ(DtF|Gt)

E^γ(F|Gt) 1_{E^γ(F|Gt)>0}. This implies thatμ-almost surely

u_t=^Eμ DtF

F Gt

.

2.3. Optimal drift in a strong sense

According to Theorem2, the filtered probability space(W,B, μ,G)carries a Brownian motiony. The processx= y+Uhas lawμand the driftUsatisfies

H(μ|γ )=1 2^E

μU².

Still, it remains open whethergivena probability space, a filtration and a Brownian motion, there exists a drift achiev- ing equality in (4).

It this section, we show that this is indeed the case, under some restriction on the measureμ. The approach is taken from the article [4] in which Baudoin treats the case of Brownian bridges (see Section2.5below). We refer to [20] for the background on stochastic differential equations.

Theorem 4. LetBbe a Brownian motion defined on some filtered probability space(Ω,A,P,F).Letμbe a measure onW,absolutely continuous with respect toγ and letut:W→R^dbe the associated Föllmer process.If the stochastic differential equation

Xt=Bt+ t

0

us(X)ds, t≥0, (7)

has the pathwise uniqueness property,then it has a unique strong solution.This solutionXsatisfies the following.

1. The processUt=_t

0us(X)dsbelongs toHalmost surely. 2. The processXhas lawμ.

3. The relative entropy ofμis given by H(μ|γ )=1

2^EU².

Proof. According to Theorem2, on(W,B, μ)the coordinate processx satisfies x_t=y_t+

_t

0

u_s(x)ds,

wherey is a Brownian motion. Therefore (7) has a weak solution. By Yamada and Watanabe’s theorem, if pathwise uniqueness holds then (7) has a unique strong solution. Moreover, since pathwise uniqueness implies uniqueness in law, the solutionXhas lawμ. The rest of Theorem4concerns the law ofX, so it is contained in Theorem2.

We end this section by showing that for a reasonably large class of measuresμ, the stochastic differential equation (7) does satisfy the pathwise uniqueness property.

Definition 5. LetSbe the class of probability measures on(W,B, γ )having a density of the form

F (w)=Φ(w_t1, . . . , w_tn) (8)

for some integern,for some sample0≤t1< t2<· · ·< tnand for some functionΦ:(R^d)ⁿ→Rsatisfying

• Φis Lipschitz.

• ∇Φ is Lipschitz.

• There existsε >0such thatΦ≥ε.

Lemma 6. Ifμbelongs toSthen the equation(7)has the pathwise uniqueness property.

Proof. Letμhave densityF given by (8). ThenF∈D²and DF (w)=

n

i=1

∇iΦ(w_t1, . . . , w_tn)1_[0,ti],

where∇iΦis the gradient ofΦin theith variable. By Lemma3, the process associated toμis u_t(w)=^Eγ(DtF (w)|Gt)

E^γ(F (w)|Gt)

= n

i=1

E^γ(∇iΦ(wt₁, . . . , wt_n)|Gt)

E^γ(Φ(wt₁, . . . , wt_n)|Gt) 1_[0,ti](t).

It is enough to prove that there is a constantCsuch that u_t(w)−u_t(w)˜ ≤C sup

0≤s≤t

|w_s− ˜w_s| (9)

for allt≥0 and for allw,w˜ ∈W. Fixt≥0 and assume thatt_k≤t < t_k+1for somek∈ {0, . . . , n−1}. By the Markov property of the Brownian motion

E

Φ(w_t1, . . . , w_tn)|Gt

=Ψ (w_t1, . . . , w_tk, w_t), whereΨ (x₁, . . . , x_k, x)equals

WΦ(x₁, . . . , x_k, x+w_tk+1−t, . . . , x+w_tn−t)γ (dw).

Then observe that Ψlip≤ Φlip. We have a similar property when 0≤t < t1 and when tn≤t. The argument

applies also to∇iΦ. The inequality (9) follows easily.

To sum up, we have the following representation formula.

Theorem 7. Let(Ω,A,P,F)be a filtered probability space and letB:Ω→Wbe a Brownian motion.For allμ∈S we have

H(μ|γ )=min

U

1 2^EU²

,

where the minimum is on all driftsUsuch thatB+Uhas lawμ.

2.4. The Boué and Dupuis formula

In this subsection the previous results are translated in terms of log-Laplace using the following lemma.

Lemma 8. Letf:W→Rbounded from below.For every positiveεthere existsμ∈Ssuch that log

We^fdγ

≤

Wfdμ−H(μ|γ )+ε. (10)

Proof. By monotone convergence we can assume that f is also bounded from above, and that

e^fdγ =1. Set F =e^f and letμbe a probability measure onWhaving densityGwith respect toγ. Then

H(μ|γ )−

fdμ= G

F log G

F

Fdγ . Usingtlog(t)≤ |t−1| + |t−1|²/2 we get

H(μ|γ )−

fdμ≤ G F −1

Fdγ+1

2 G

F −1 ²Fdγ

≤ F −GL¹(γ )+CF−G²_L2(γ )

for some constantC(recall thatf is bounded below). Therefore, it is enough to show that there existsμ∈S whose

densityGis arbitrarily close toF in L²(γ ). This is a standard fact.

Here is the Boué and Dupuis formula.

Theorem 9. For every functionf:W→Rmeasurable and bounded from below,we have log

We^fdγ

=sup

U

E

f (B+U )−1 2U²

,

where the supremum is taken over all driftsU.

This is actually slightly more general than the result in [6], which concerns the spaceC([0, T],R^d)for some finite time horizonT.

Proof of Theorem9. LetU be a drift and μbe the law of B+U. By Proposition1 and the entropy/log-Lapace duality

E

f (B+U )−1 2U²

≤

fdμ−H(μ|γ )≤log

We^fdγ

.

On the other hand, givenε >0, there exists a probability measureμ∈S satisfying (10). Sinceμ∈S, Theorem7 asserts that there exists a driftUsuch thatB+U has lawμand satisfying

H(μ|γ )=1 2^EU². Then (10) becomes

log

We^fdγ

≤^E

f (B+U )−1 2U²

+ε,

which concludes the proof.

2.5. Brownian bridges A measureμonWsatisfying

μ(dw)=ρ(w₁)γ (dw), (11)

whereρis some density on(R^d, γd)is said to be a Brownian bridge. It can be seen as the law of a Brownian motion conditioned to have lawρ(x)γd(dx)at time 1.

Lemma 10. Letνhave densityρwith respect toγ_d,we have H(ν|γd)=inf

μ

H(μ|γ ) ,

where the infimum is on all probability measures satisfyingμ◦(x₁)⁻¹=ν.The infimum is attained whenμis the bridge(11).

In other words, among all processes having lawν at time 1, the bridge minimizes the relative entropy. This is essentially a particular case of (3), see also [4] and [16], p. 161.

Assume that ρ is differentiable and that ∇ρ ∈L²(γ_d). Then F (w)=ρ(w₁) belongs toD² and has Malliavin derivative

DF (w)= ∇ρ(w1)1_[0,1].

By Lemma3the Föllmer process of the bridgeμis such that u_t=^Eμ

∇log(ρ)(w1)|Gt

1_[0,1](t), μ-a.s.

We obtain the following result.

Lemma 11. Underμ,the process(u_t)_t∈[0,1]is a martingale.In particular

E^μ(u_t)=^Eμ∇log(ρ)(w1)=^Eγ∇ρ(w₁)=

R^dxν(dx).

Now assume thatρand∇ρare Lipschitz and thatρ≥ε, so that the bridgeμbelongs toS. It is easily seen thatu_t can also be written as

ut(w)= ∇logP1−tρ(wt)1_[0,1](t), whereP_tdenotes the heat semigroup onR^d:

∂_tP_t=1 2P_t.

The stochastic differential equation (7) becomes Xt=Bt+

t∧1 0

∇log(P1−sρ)(Xs)ds, t≥0. (12)

By Lemma6, there is a unique strong solution. Combining Lemma10with Theorem4we obtain the following dual formulation of Borell’s result (1).

Theorem 12. Letνandρbe as above.Then H(ν|γd)=inf

U

1 2^EU²

,

where the infimum is taken on all driftsUsatisfyingB₁+U₁=νin law.The infimum is attained by the drift Ut=

t∧1 0

∇log(P1−sρ)(Xs)ds, whereXis the unique solution of(12).

3. Applications

Following Borell, we now derive functional inequalities from the representation formula. Let us point out that in all but one applications we use Proposition1and Theorem2rather than Theorem7.

3.1. Transportation cost inequality

Let T2be the transportation cost for the Euclidean distance squared: given two probability measuresμandνonR^d T2(μ, ν)=inf

R^d×R^d|x−y|²dπ(x, y)

, (13)

where the infimum is taken over all couplings π ofμandν, namely all probability measures on the product space R^d×R^dhaving marginalsμandν. There is a huge literature about this optimization problem, usually referred to as Monge–Kantorovitch problem, see Villani’s book [24]. Talagrand’s inequality [22] asserts that

T2(ν, γd)≤2H(ν|γd)

for every probability measure ν on R^d. The purpose of this subsection is to prove a Wiener space version of this inequality.

On Wiener space the natural definition of T2involves the norm of the Cameron–Martin spaceH: given two proba- bility measuresμ, νon(W,B)

T2(μ, ν)=inf

W×W

w−w²π

dw,dw ,

where the infimum is taken over all couplingsπ ofμandνsuch thatw−w∈Hforπ-almost all(w, w).

Theorem 13. Letμbe a probability measure on(W,B).Then T2(μ, γ )≤2H(μ|γ ).

Here is a short proof based of Theorem2. Fair enough, Feyel and Üstünel [12] have a very similar argument.

Proof of Theorem13. Assume thatμis absolutely continuous with respect toγ (otherwise H(μ|γ )= +∞). Ac- cording to Theorem2there exists a Brownian motionBand a driftU such thatB+Uhas lawμand

H(μ|γ )=1 2^EU².

Then(B, B+U )is a coupling of(γ , μ)and by definition of T2

T2(μ, γ )≤^EU²=2H(μ|γ ).

Let us point out that Talagrand’s inequality can be recovered easily from this theorem, applying it to a Brownian bridge. Details are left to the reader.

3.2. Logarithmic Sobolev inequality

In this section we prove the logarithmic Sobolev inequality for the Wiener measure, which extends the classical log- Sobolev inequality for the Gaussian measure, due to Gross [17]. Whenμis a measure on(W,B, γ )with densityF such that DF is well defined, the Fisher information ofμis

I(μ|γ )=

W

DF² F dγ=

W

DF F

²dμ.

Theorem 14. Letμhave densityF with respect toγ and assume thatF ∈D².Then H(μ|γ )≤1

2I(μ|γ ). (14)

Proof. We consider the probability space(W,B, μ). Recall that(Gt)_t≥0is the filtration of the coordinate process. By Theorem2and Lemma3, letting

u_t=^Eμ DtF

F Gt

we have

H(μ|γ )=1 2^E

μ

_∞

0

|ut|²dt.

By Jensen’s inequality E^μ|u_t|²≤^Eμ

DtF F

²

so that

H(μ|γ )≤1 2^E

μ DF

F ²

which is the result.

This may not be the most straightforward proof, see [8]. Let us emphasize that applying (14) to a Brownian bridge yields the usual log-Sobolev inequality. More precisely, letνbe a probability measure onR^dhaving a smooth density ρwith respect toγdand letμbe the measure onWgiven by

μ(dw)=ρ(w₁)γ (dw).

Then H(ν|γ_d)=H(μ|γ ). On the other hand lettingF (w)=ρ(w₁)we have DF (w)= ∇ρ(w1)1_[0,1],

which implies easily that I(ν|γ_d)=I(μ|γ ). Thus (14) becomes H(ν|γd)≤1

2I(ν|γd).

3.3. Shannon’s inequality

Given a random vectorηonR^dhaving densityρwith respect to the Lebesgue measure, Shannon’s entropy is defined as

S(η)= −

R^dρlog(ρ)dx.

In other words S(η)= −H(ν|λ_d)whereνis the law ofηandλ_dis the Lebesgue measure onR^d. Theorem 15. Letη, ξ be independent random vectors onR^dandθ∈ [0,π/2]

S

cos(θ )η+sin(θ )ξ

≥cos(θ )²S(η)+sin(θ )²S(ξ ). (15)

This inequality plays a central role in information theory, see [11] for an overview on the topic.

Proof of Theorem15. Let ν_θ be the law of cos(θ )η+sin(θ )ξ. By Theorem 2, Lemmas 10and11there exists a Brownian motionXand a driftUsuch that

• X1+U1has lawν0.

• H(ν0|γ_d)=^EU²/2.

• ^E(U )=^E(η)1_[0,1].

Similarly, there exists a Brownian motionY and a driftV satisfying the corresponding properties forν_π/2. Besides, we can clearly assume thatY is independent ofX. Then cos(θ )X+sin(θ )Y is a Brownian motion and

cos(θ )X1+sin(θ )Y1+cos(θ )U1+sin(θ )V1

has lawν_θ. By Proposition1and Lemma10 H(νθ|γ_d)≤1

2^Ecos(θ )U+sin(θ )V². Denoting the inner product inHby·,·we have

EU, V = ^EU,EV =(Eη)·(Eξ ), so that

H(νθ|γ_d)≤cos(θ )²H(ν0|γ_d)+sin(θ )²H(ν_π/2|γ_d) +cos(θ )sin(θ )(Eη)·(Eξ ).

This is easily seen to be equivalent to (15).

3.4. Brascamp–Lieb inequality

Let us focus on a family of inequalities dating back to Brascamp and Lieb’s article [7] on optimal constants in Young’s inequality. Since then a number of nice alternate proofs have been discovered, see [3,9,10] and the survey article [1].

This subsection is inspired by the (unpublished) proof of Maurey relying on Borell’s formula.

LetE be a Euclidean space, letE₁, . . . , E_m be subspaces and for alli letP_i be the orthogonal projection with rangeE_i. The crucial hypothesis is the so-called frame condition: there existc₁, . . . , c_minR₊such that

m

i=1

c_iP_i=idE. (16)

Letx∈E, we then have|x|²=(

ciPix)·xand sincePi is an orthogonal projection

|x|²= m

i=1

c_i|P_ix|². (17)

From now on Wdenotes the space of continuous paths taking values inE and starting from 0 and γ denotes the Wiener measure onW. The spacesWi and measuresγ_i are defined similarly.

Theorem 16. Under the frame condition,for every probability measureμonWwe have H(μ|γ )≥

m

i=1

c_iH(μi|γ_i),

whereμi =μ◦P_i⁻¹is the push-forward ofμby the projectionPi.

Proof. According to Theorem2there exists a standard Brownian motionBonEand a driftUsuch thatB+Uhas lawμand

H(μ|γ )=1 2^EU².

SinceP_i is an orthogonal projection, the processP_iBis a standard Brownian motion onE_i. AlsoP_iB+P_iUhas law μ◦P_i⁻¹=μ_i. By Proposition1

H(μi|γ_i)≤1

2^EP_iU², i=1, . . . , m.

On the other hand, the frame condition (17) implies easily that U²=

n

i=1

ciPiU²

pointwise. Taking expectation yields the result.

As observed by Carlen and Cordero [9], this super-additivity property of the relative entropy is equivalent to the following Brascamp–Lieb inequality.

Corollary 17. Under the frame condition,givenmfunctionsF_i:Wi→R₊,we have

W

m

i=1

(F_i◦P_i)^cidγ≤ m

i=1

Wi

F_idγi

c_i

.

When the functions Fi depend only on the point w1 rather than on the whole path w we recover the usual Brascamp–Lieb inequality for the Gaussian measure.

3.5. Reversed Brascamp–Lieb inequality

Again E is a Euclidean space and E₁, . . . , E_m are subspaces satisfying the frame condition (16). Observe that if x₁, . . . , x_mbelong toE₁, . . . , E_mrespectively, then for anyy∈E, the Cauchy–Schwarz inequality and (17) yield

_m

i=1

cixi

·y= m

i=1

ci(xi·Piy)

≤ _m

i=1

c_i|x_i|²

1/2 _m

i=1

c_i|P_iy|² 1/2

= _m

i=1

c_i|x_i|² 1/2

|y|. Hence

m

i=1

c_ix_i

2

≤ m

i=1

c_i|x_i|². (18)

LetSi be the class of probability measures onEi which satisfy the conditions of Definition5, replacingR^d byEi. Here is the reversed version of Theorem16.

Theorem 18. Givenmprobability measuresμ₁, . . . , μ_mbelonging toS1, . . . ,Smrespectively,there existmprocesses X₁, . . . , X_m(defined on the same probability space)such that

1. X_ihas lawμ_i for alli=1, . . . , m.

2. Lettingμbe the law of

c_iX_iwe have

H(μ|γ )≤ m

i=1

ciH(μi|γi).

Proof. Again letB be a standard Brownian motion onE. Fori=1, . . . , m, the processP_iB is a standard Brownian motion onE_i. Sinceμ_i∈Sithere exists a driftU_i such that the processX_i=P_iB+U_i has lawμ_iand

H(μi|γ_i)=1

2^EU_i². LetX=

c_iX_i and letμbe the law ofX. Since

c_iP_i is the identity ofE X=B+

m

i=1

ciUi.

By Proposition1, we get

H(μ|γ )≤1 2^E

m

i=1

c_iU_i

2

.

On the other hand (18) easily implies that

m

i=1

c_iU_i

2

≤ m

i=1

c_iU_i²,

pointwise. Taking expectation we get the result.

This sub-additivity property of the entropy is a multi-marginal version of thedisplacement convexityproperty put forward by Sturm [21]. By duality, we obtain the following reversed Brascamp–Lieb inequality.

Corollary 19. Assuming the frame condition,givenmfunctionsF_i:Wi→R₊bounded away from0,and a function G:W→R₊satisfying

m

i=1

Fi(wi)^ci≤G _m

i=1

ciwi

(19) for all(w₁, . . . , w_m)∈W1× · · · ×Wm,we have

m

i=1

Wi

F_idγi

c_i

≤

WGdγ .

Proof. By Lemma8, for everyi, there exists a measureμi ∈Si such that log

Wi

Fidγi

≤

Wi

log(Fi)dμi−H(μi|γi)+ε.

LetX₁, . . . , X_mbe the random processes given by the previous theorem, letX=

c_iX_i and letμbe the law ofX.

Then by duality and the hypothesis (19) we get log

WGdγ

≥^Elog(G)(X)−H(μ|γ )

≥^E _m

i=1

cilog(Fi)(Xi)

−H(μ|γ ).

Since H(μ|γ )≤

c_iH(μi|γ_i), this is at least c_i

log

Wi

F_idγi

−ε

.

Lettingεtend to 0 yields the result.

Again when the functions depend only on the value of the path at time 1, we recover the reversed Brascamp–Lieb inequality for the Gaussian measure, which is due to Barthe [2].

Acknowledgements

The author is grateful to Patrick Cattiaux and Massimiliano Gubinelli for communicating references and to Christian Léonard, Bernard Maurey and Patrick Cattiaux again for valuable discussions.

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