Modified logarithmic Sobolev inequalities and transportation inequalities

Transcription

1 Probab. Theory Relat. Fields 133, Digital Object Identifier DOI /s Ivan Gentil Arnaud Guillin Laurent Miclo Modified logarithmic Sobolev inequalities and transportation inequalities Received: 16 May 004 / Revised version: 5 January 005 / Published online: 15 March 005 Springer-Verlag 005 Abstract. We present a class of modified logarithmic Sobolev inequality, interpolating between Poincaré and logarithmic Sobolev inequalities, suitable for measures of the type exp x α or exp x α log + x α ]1, [ and R which lead to new concentration inequalities. These modified inequalities share common properties with usual logarithmic Sobolev inequalities, as tensorisation or perturbation, and imply as well Poincaré inequality. We also study the link between these new modified logarithmic Sobolev inequalities and transportation inequalities. 1. Introduction A probability measure µ on R n satisfies a logarithmic Sobolev inequality if there exists C< such that, for every smooth enough functions f on R n, Ent µ f C f dµ, 1 where Ent µ f = f log f dµ f dµlog f dµ and where f is the Euclidean length of the gradient f of f. Gross in [18] defines this inequality and shows that the canonical Gaussian measure with density π n/ e x / with respect to the Lebesgue measure on R n is the basic example of measure µ satisfying 1 with the optimal constant C =. Since then, many results have presented measures satisfying such an inequality, among them the famous Bakry-Emery Ɣ criterion, we refer to Bakry [] and Ledoux [19] for further references and details on various applications of these inequalities. I. Gentil, A. Guillin: Ceremade UMR CNRS no. 7534, Université Paris-Dauphine, Place du Maréchal De Lattre De Tassigny, Paris Cédex 16, 75775, France. {gentil,guillin}@ceremade.dauphine.fr L. Miclo: Laboratoire d Analyse, Topologie, Probabilités UMR CNRS no. 663, Université de Provence, 39, rue F. Joliot Curie, Marseille cédex 13, 13453, France. miclo@cmi.univ-mrs.fr Send offprint requests to: Ivan Gentil

2 410 I. Gentil et al. Let α 1 and define the probability measure µ α on R by µ α dx = 1 Z α e x α dx, where Z α = e x α dx. It is well-known that the probability measure µ α satisfies a logarithmic Sobolev inequality 1 if and only if α. But for α [1, [, even if the measure µ α does not satisfy 1, it satisfies a Poincaré inequality or spectral gap inequality which is for every smooth enough function f, Var µα f C f dµ α, 3 where Var µα f = f dµ α fdµ α and C<. Recall, see for example Section 1..6 of [1], that if a probability measure on R n satisfies a logarithmic Sobolev inequality with constant C then it satisfies a Poincaré inequality with a constant no greater than C/. The problem is then to interpolate between logarithmic Sobolev and Poincaré inequalities, which will help us to study further properties, such as concentration, of measures µ n α for α [1, ] and n N. A first answer was brought by Latała-Oleszkiewicz in [1] and recently extended by Barthe-Roberto in [10]. Let µ be a probability measure on R n, µ satisfies inequality I µ a for a [0, 1] with constant 0 C< if for all p [1, [ and f be a measurable, square integrable non-negative function on R n, f dµ /p f p dµ C p a f dµ. 4 Inequality 4 was introduced by Beckner in [5], it interpolates Poincaré and logarithmic Sobolev inequalities for the Gaussian measure. In [1], the authors prove that the measure µ α for α [1, ] satisfies such an inequality for a constant C< and with a = α 1/α. And in [10] the authors present a simple proof of Latała-Oleszkiewicz s results and describe measures on the line which enjoy the same inequality. Our main purpose here will be to establish another type of interpolation between logarithmic Sobolev and Poincaré inequalities, more directly linked to the structure of the usual logarithmic Sobolev inequalities, i.e. an inequality entropy-energy where we will modify the energy to enable us to consider µ α measure. Note that this point of view was the one used by Bobkov-Ledoux in [8] when considering double sided exponential measure. Let us describe further these modified logarithmic Sobolev inequalities. Let α [1, ], a>0and satisfying 1/α + 1/ = 1, we note x if x a H a,α x = x a + a if x a and α 1 + if x a and α = 1.

3 Modified logarithmic Sobolev inequalities and transportation inequalities 411 In Section we give definition and general properties of the following inequality Ent µ f f C H a,α f dµ. LSI a,α C f In particular we prove that inequality LSI a,α satisfies some of the properties shared by Poincaré or Gross logarithmic Sobolev inequalities 1 or 3, namely tensorisation and perturbation. Note that in the case α = 1, we find inequalities used by Bobkov-Ledoux in [8] and for α = inequality LSI a,α C is exactly the Gross logarithmic Sobolev inequality. We present also a concentration property which is adapted to this inequality. More precisely, if a measure µ satisfies the inequality LSI a,α C, we have that if f is a Lipschitz function on R with f Lip 1 then, there is B>0such that for every λ>0one has µ α f fdµ α λ exp B min λ α,λ. 5 This inequality was proved by Talagrand in [6, 7] and also described by Maurey with the so called property τ in [3], Bobkov-Ledoux in [8] study the particular case α = 1. Let us note that the cases α are studied by Bobkov- Ledoux in [9], relying mainly on Brunn-Minkowski inequalities, and by Bobkov- Zegarlinski in [1] which refine the results presenting, via Hardy s inequality, some necessary and sufficient condition for measures on the real line. Let us note to finish that they use, for the case α, H x = x with 1/α + 1/ = 1. In Section., we extend Otto-Villani s theorem see [5] for the relation with logarithmic Sobolev inequality and transportation inequality. Let us define L a,α by L a,α = Ha,α, the Fenchel-Legendre transform of H a,α. We prove that if a probability measure µ on R n satisfies the inequality LSI a,α C then there are a > 0 and D>0such that it satisfies also a transportation inequality: for all function F on R n, density of probability with respect to µ, T La,α Fdµ,dµ DEnt µ F, T a,αd where { T La,α Fdµ,dµ = inf } L a,αx ydπx, y, where the infimum is taken over the set of probabilities measures π on R n R n such that π has two margins Fdµand dµ. This inequality was introduced by Talagrand in [8] for the case α = and α = 1. Let us note that the case α = 1 was also studied in [7] with exactly this form and the case α was studied in [16]. In Section 3 we prove, as in [1], that the measure µ α defined in satisfies the inequality LSI a,α C. More precisely we prove that there is A, B > 0 such that µ α satisfies for all smooth function such that f 0 and f dµ α = 1, Ent µα f AVar µα f + B f f f f dµ α. Due to the fact that µ α enjoys Poincaré inequality, µ α satisfies also inequality LSI a,α C for some constants C>0and a>0.

4 41 I. Gentil et al. Our method relies crucially on Hardy s inequality we recall now: let µ, ν be two finite Borel measures on R +. Then the best constant A so that every smooth function f satisfies 0 fx f0 dµx A 0 f dν 6 is finite if and only if x dν ac 1 B = sup µ[x, [ dt 7 x 0 0 dt is finite, where ν ac is the absolutely continuous part of ν with respect to µ. Moreover, when A is finite we have B A 4B. This inequality was proved by Muckenhoupt [4], one can see also [1, 10] for interesting review and application of this result. Finally in Section 4 we will present some inequalities satisfied by other measures. More precisely, let ϕ be twice continuously differentiable and note the probability measure µ ϕ by, Among them is considered µ ϕ dx = 1 Z e ϕx dx. 8 ϕx = x α log + x, with α ]1, [, R, which exhibits a modified logarithmic Sobolev inequality of function H different in nature from H a,α, and which is not covered by Latała-Oleskiewickz inequality. We also present examples which are unbounded perturbation of µ α. We then derive new concentration inequalities in the spirit of Talagrand and Maurey or Bobkov- Ledoux Let us finally comment the case of general convex potential ϕ and µdx = e ϕx dx. The natural extension of our modified logarithmic Sobolev inequality would be to consider a function H behaving quadratically near the origin and like ϕ the Legendre transform of ϕ for large values. The extension is however by no way trivial and requires appropriate technique currently under study.. Modified logarithmic Sobolev inequalities: definition and general properties.1. Definitions and classical properties Let α [1, ] and satisfying 1/α + 1/ = 1 and let a>0. Let define the functions L a,α and H a,α.

5 Modified logarithmic Sobolev inequalities and transportation inequalities 413 and If α ]1, ] we note x L a,α x = a x H a,α x = a α x α x α + a α α + a if x a if x a if x a if x a If α = 1 we note x if x a x L a,1 x = and H a,1 x = if x a a x a if x a if x >a Let n N and x = x 1,,x n R n, we note L n a,α x = n i=1 L a,α x i and H n a,α x = n H a,α x i. Note that when there is no ambiguity we will drop the dependence in n and note L a,α instead of L n a,α. Let us define the logarithmic Sobolev inequality of function H a,α. Definition.1. Let µ be a probability measure on R n, µ satisfies a logarithmic Sobolev inequality of function H a,α with constant C, noted LSI a,α C, if for every C 1 function f>0such that every integrals exists one has where Ent µ f C H a,α f f H a,α f f = n i=1 f dµ, i=1 f 1 H a,α. x i f We detail some properties of L a,α and H a,α in the following lemma. LSI a,α C Lemma.. Functions L a,α and H a,α satisfies: i: If α ]1, ], L a,α and H a,α are C 1 on R. ii: L a,α = H a,α, where L is the Fenchel-Legendre transform of L a,α. Of course we have also Ha,α = L a,α. iii: For all t>0one has for all x R L a,α tx = t L a t,αx, H a,α tx = t H a t,αx.

6 414 I. Gentil et al. iv: Let 0 a a, one has for all x R L a,α x L a,αx, H a,αx H a,α x. v: If α ]1, ], L a,α and H a,α are strictly convex and satisfies H a,α x L a,α x lim = lim =. x x x x The assumptions given on α and are significant only for condition iv, and condition v is significant for Brenier-McCann-Gangbo s theorem, which is crucial for the study of the link between modified logarithmic Sobolev inequalities and transportation inequalities of the next section. Here are some properties of the inequality LSI a,α C. Proposition This property is known under the name of tensorisation. Let µ 1 and µ two probability measures on R n 1 and R n. Suppose that µ 1 resp. µ satisfies the inequality LSI a,α C 1 resp. LSI a,α C then the probability µ 1 µ on R n 1+n, satisfies inequality LSI a,α D, where D = max {C 1,C }.. This property is known under the name of perturbation. Let µ a measure on R n satisfying LSI a,α C. Let h a bounded function on R n and defined µ as d µ = eh Z dµ, where Z = e h dµ. Then the measure µ satisfies the inequality LSI a,α D with D = Ce osch, where osch = suph infh. 3. Link between LSI a,α C inequality with Poincaré inequality. Let µ a measure on R n.ifµ satisfies LSI a,α C, then µ satisfies a Poincaré inequality with the constant C/. Let us recall that µ satisfies a Poincaré inequality with constant C/ if Var µ f C f dµ, 9 for all smooth function f. Proof. One can find the details of the proof of the properties of tensorisation and perturbation and the implication of the Poincaré inequality in chapters 1 and 3 of [1] Section 1..6., Theorem 3..1 and Theorem Note that the tensorisation of the Entropy is well known property discussed by Lieb in [0]. Remark.4. We may of course define logarithmic Sobolev inequality of function H, where Hx is quadratic for small values of x and with convex, faster than quadratic, growth for large x. See Section 4 for such examples. Note that Proposition.3 is of course still valid for this kind of inequality. These inequalities are also studied in a general case in [19] in Proposition.9.

7 Modified logarithmic Sobolev inequalities and transportation inequalities 415 As in [1, 8], by using Herbst s argument, one can give precise estimates about concentration. Proposition.5. Assume that the probability measure µ on R satisfies the inequality LSI a,α C. Let F be a function on R n such that i, i F ζ, then we get for λ 0, µ n F µ n F exp K α n α 1 ζ λ acnζ α αα a α if λ acnζ, α λ exp λ otherwise, ncζ where K α = α α 1 1 α a α αc α 1. Proof. Let us first present the proof when n = 1. Assume, without loss of generality, that Fdµ = 0. Let us recall briefly Herbst s argument see [1] for more details. Denote t = e tf dµ, and remark that LSI a,α C applied to f = e tf, using basic properties of H a,α, yields to t t t log t CH a,α tζ t 10 which, denoting Kt = 1/tlog t, entails K t C t H a,α tζ. Then, integrating, and using K0 = Fdµ = 0, we obtain t t exp Ct 0 1 s H a,α sζ ds. 11 The Laplace transform of F is then bounded by exp Ct ζ a t 1 + Ctaζ 1 Ca exp C ζ t 8 if t a ζ, if 0 t a ζ. For the n-dimensional extension, use the tensorisation property of LSI a,α and n i=1 t tζ H a,α if nh a,α. Then we can use the case of dimension 1 with the constant C replaced by Cn.

8 416 I. Gentil et al. Remark.6. Let us present a simple application of the preceding Proposition to deviation inequality of the empirical mean of a function. Consider the real valued function f, with f 1, and Fx 1,...,x n = 1 n nk=1 fx k, which inherits the property that i F 1/n, we thus get the following Hoeffding type inequality, X i 1 i n being independent and identically distributed according to µ α, 1 n P fx n k µ α f >λ k=1 exp exp nk α λ ac α α a α if λ ac α, n λ otherwise. C Remark.7. Note that the obtained form minλ α,λ is natural in regard to Gaussian approximation. Indeed, consider, for example, Fx 1,..., x n = 1 nk=1 n fx i where f 1wehave i F n 1/ which enables us to recover the Gaussian concentration 1 n P fx k µ α f n >λ e λ /C, for all n 4λ /ac. k=1 Remark.8. For general logarithmic Sobolev of function H, we may obtain crude estimation of the concentration, at least for large λ. Indeed, using inequality 11, we have directly that the concentration behavior is given by the Fenchel-Legendre transform of H for large values, see Section 4 for more details... Link between inequality LSI a,α C and transportation inequality Definition.9. Let µ be a probability measure on R n, µ satisfies a transportation inequality of function L a,α with constant C, noted T a,α C, if for every function F, density of probability with respect to µ, one has T La,α Fdµ,dµ CEnt µ F, T a,α C where { T La,α Fdµ,µ = inf } L a,α x ydπx, y, where the infimum is taken over the set of probabilities measures π on R n R n such that π has two margins Fdµand µ. Otto and Villani proved that a logarithmic Sobolev inequality implies a transportation inequality with a quadratic cost this is the case α = =, see [5, 7]. They prove that if µ satisfies the inequality LSI, C, when α = the constant a is not any more a parameter in this case, then µ satisfies the inequality T, 4C. In [7] another case is studied, when α = 1 and =. In this first theorem we give an extension for the other cases, where α [1, ].

9 Modified logarithmic Sobolev inequalities and transportation inequalities 417 Theorem.10. Let µ be a probability measure on R n and suppose that µ satisfies the inequality LSI a,α C. Then µ satisfies the transportation inequality T ac,α C/4. Proof. As in [7], we use Hamilton-Jacobi equations. Let f be a Lipschitz bounded function on R n, and set Q t fx= inf y R { fy+ tlac,α x y t },t>0, x R n, 1 and Q 0 f = f. The function Q t f is known as the Hopf-Lax solution of the Hamilton-Jacobi equation { v t t, x = H ac,α vt, x, t > 0, x Rn, v0,x = fx, x R n, see for example [3, 15]. For t 0, define the function ψ by ψt = e 4t C Q t f dµ. Since f is Lipschitz and bounded function one can prove that Q t f is also a Lipschitz and bounded function on t for almost every x R n, then ψ is a C 1 function on R +. One gets ψ t = 4 C Q tfe 4t C Q t f dµ = 1 t Ent µ e 4t 4t C H ac,α Q tf e 4t C Q t f dµ C Q t f + 1 4t t ψtlog ψt C H ac,α Q tf e 4t C Q t f dµ Let use inequality LSI a,α C to the function exp t C Q tf to get ψ t 1 t ψtlog ψt + C t H a,α t C Q tf e 4t C Q t f dµ 4t C H ac,α Q tf e 4t C Q t f dµ Due to the property of H a,α see Lemma., t H a,α C Q tf = 4t C H ac,α Q tf. t Then for all t [0, 1], one has t H a,α C Q tf 4t C H ac,α Q tf..

10 418 I. Gentil et al. Then t [0, 1], tψ t ψtlog ψt 0 After integration on [0, 1], we have ψ1 exp ψ 0 ψ0, from where e C 4 Q1f 4C dµ e fdµ. 13 Since { Ent µ F = sup Fgdµ, } e g dµ 1, we have with g = C 4 Q 1f 4 C fdµ, F Q 1 f fdµ dµ C 4 Ent µf. Let take the supremum on the set of Lipschitz function f, the Kantorovich- Rubinstein s theorem applied to the distance T La,α F dµ, dµ, see [9], implies that T L ac,αf dµ, dµ C 4 Ent µf. As it is also the case in quadratic case, when the measure is log-concave one can prove that a transportation inequality implies a logarithmic Sobolev inequality. For the next theorem we suppose that the function of transport given by the theorem of Brenier-Gangbo-McCann is a C function. Such a regularity result is outside the scope of this paper and we refer to Villani [9] for further discussions around this problem. However we show here, that once this result assumed, the methodology presented in Bobkov-Gentil-Ledoux [7], for the exponential measure, still works. Theorem.11. Let µ be a probability measure on R n. Assume that µdx = e ϕx dx where ϕ is a convex function on R n. If µ satisfies the inequality T a,α C then for all λ>c, µ satisfies the logarith-. mic Sobolev inequality LSI a λ,α 4λ λ C Proof. Let note F density of probability with respect to µ. Assume that F is C, the general case can result by density. By the Brenier-Gangbo-McCann s theorem, see [11, 17], there exists a function such that

11 Modified logarithmic Sobolev inequalities and transportation inequalities 419 S = Id H a,α, transports Fdµto the measure µ, for every measurable bounded function g gsfdµ = gdµ. The function is a L a,α -concave function and if is C, a classical argument of convexity see chapter of [9], one has D [ H a,α x ] is diagonalizable with real eigenvalues, all less than 1. According to the assumption made on function, one can assume that S is sufficiently smooth and we obtain for x R n, Fxe ϕx = e ϕ Sx det Sx. 14 Moreover this function gives the optimal transport, i.e. T La,α Fdµ,dµ = L a,α Ha,α Fdµ. Then by 14, one has for x R n, log Fx=ϕx ϕx H a,α x + log det Id D [ H a,α x ]. Then since D [ H a,α x ] is diagonalizable with real eigenvalues, all less than 1, we get log det Id D [ H a,α x ] div H a,α x. Since ϕ is convex we have ϕx ϕx H a,α x H a,α x ϕx and we obtain { Ha,α Ent µ F x ϕx div H a,α x } F xdµx, after integration by parts Ent µ F F H a,α dµ. H a,α Thus Let λ>0 and let use Young inequality for the combined functions L a,α and λ F F Ent µ F 1 λ λ H a,α H a,α λ F + L a,α Ha,α. F H a,α λ F Fdµ+ 1 L a,α Ha,α Fdµ F λ F Fdµ+ 1 F λ T L a,α Fdµ,dµ. H a λ,α

12 40 I. Gentil et al. Thus if µ satisfies the inequality T a,α C we get for all λ>c Ent µ F λ F H a λ C λ,α Fdµ. F Let us note now f = F,weget Ent µ f λ H a λ C λ,α f f 4λ f H a λ C λ,α f Then µ satisfies, for all λ>cinequality LSI a 4λ λ,α λ C f dµ f dµ.. Remark.1. One can summarizes Theorem.10 and.11 by the following diagram under assumption of Theorem.11: LSI a,α C T ac,α C/4 { 4λ } T a,α C LSI a λ,α. λ C λ>c Notice, as it is the case for the traditional logarithmic Sobolev inequality, that there is a loss at the level of the constants in the direction transportation inequality implies logarithmic Sobolev inequality. When α = =, we get as in [5], T, C LSI, 16C. As in [5], Theorem.11 can be modified in the case Hessϕ λid, where λ R. Also let us notice that as in the quadratic case we do not know if these two inequalities are equivalent. As in Proposition.3, here are some properties of the inequality T La,α C. Proposition Concentration inequality. Assume that µ satisfies a transportation inequality T La,α C then µ satisfies the following concentration inequality A R n, with µa 1, µa r c e C 1 L a,αr where A r c = {x R n, da, x r}.. As in Proposition.3, the properties of tensorisation are also valid for transportation inequality T a,α C. Let µ 1 and µ be two probability measures on R n 1 and R n. Suppose that µ 1 resp. µ satisfies the inequality T a,α C 1 resp. T a,α C then the probability µ 1 µ on R n 1+n, satisfies inequality T a,α D, where D = max {C 1,C }. 3. If the measure µ verifies T a,α C, then µ satisfies a Poincaré inequality 9 with the constant C.,

13 Modified logarithmic Sobolev inequalities and transportation inequalities 41 Proof. The demonstration of 1, of these results is a simple adaptation of the traditional case introduce by Marton in [], We return to the references for proofs for example chapters 3, 7 and 8 of [1]. The proof of 3 is an adaptation of the quadratic case. Suppose that µ satisfies a T a,α C. By a classical argument of Bobkov-Götze, the measure µ satisfies the dual form of T a,α C which is the inequality 13, e 1 C Q 1f dµ e 1C fdµ, 15 where Q 1 f is defined as in 1 with the function L a,α. Let note f = ɛg with g, C 1, bounded and with g also bounded, we get { } Q 1 fx= Q 1 ɛgx = ɛ inf gx ɛz + ɛla z R n ɛ,αz = ɛgx ɛ g + oɛ Then we obtain by 15, 1 + ɛ gdµ ɛ g dµ + ɛ C C C g dµ + oɛ 1 + ɛ gdµ + gdµ ɛ C C + oɛ, imply that Var µ g C g dµ. Unfortunately, as in the traditional case of the transportation inequality, we do not know if this one has property of perturbation as for inequality LSI a,α C Proposition An important example on R, the measure µ α Let α 1 and define the probability measure µ α on R by µ α dx = 1 e x α dx, Z α where Z α = e x α dx. For Section 3 and 4 we will note by smooth function a locally absolutely continuous function on R. Theorem 3.1. Let α ]1, ]. There exists 0 A, B < such that the measure µ α satisfies the following modified logarithmic Sobolev inequality, for any smooth function f on R such that f 0 and f dµ α = 1 we have

14 4 I. Gentil et al. Ent µα f AVar µα f + B f f f f dµ α, 16 where 1/α + 1/ = 1. In the extreme case, α = 1, there exists λ>0 and 0 A < such that we obtain the following inequality: for all f smooth such that f 0, f dµ 1 = 1 and f λ, Ent µ1 f A Var µ1 f. 17 Corollary 3.. Let α ]1, ] and assume that f is a smooth function on R. Then we obtain the following estimation Ent µα f AVar µα f + B f f f dµ α, 18 where { } { } = f + f+ dµ α f f dµ α, f + = maxf, 0 and f = max f, 0. Proof. We have f = f+ + f. Then Ent µα f { } = sup f gdµ α with e g dµ α 1 { = sup f+ gdµ α + f gdµ α with Ent µα f + + Ent µα f. } e g dµ α 1 By Theorem 3.1 there exists A, B > 0 independent of f such that Ent µα f + AVar µα f + + B f + + f f+ dµ α, + Ent µα f AVar µα f + B { } where + = f + f + dµ α and = To conclude, it is enough to notice that Var µα f + + Var µα f = f f { f f dµ α, } f dµ α. f dµ α f + dµ α + Var µα f, f dµ α

15 Modified logarithmic Sobolev inequalities and transportation inequalities 43 and + f + f + f+ dµ α + f f f dµ α = f f f dµ α. It implies the existence of a α > 0 and 0 C α <, such that µ α satisfies a logarithmic Sobolev inequality of function H aα,α with constant C α. Indeed, this is clear that µ α satisfies a Poincaré inequality, see chapter 6 of [1], with constant 0 λ α <, Var µα f λ α f dµ α. Then, by inequality 16, we obtain for any smooth function f>0onr, Ent µα f Aλ α f f dµα + B f f dµ α. Let us give a few hint on the proof of the Theorem 3.1, which will enable us to present key auxiliary lemmas. We first use the following inequality for f 0 such that f dµ α = 1, f log f dµ α 5 f 1 dµ α + f + logf + dµ α 19 where it is obvious that truncation arguments are crucial. We will then need the following lemma: Lemma 3.3. Let µ be a probability measure on R and let f f dµ = 1 then we obtain 0 such that i: ii: iii: f 1 dµ Var µ f. f f f dµ 8Var µ f. f log f dµ log 4 log 4 1 Ent µ f < 4Ent µ f. Proof. i.wehave f 1 dµ = Var µ f + 1 fdµ. Since f dµ = 1 and f 0 we obtain that 0 fdµ 1, then fdµ fdµ. Then f 1 dµ Var µ f + Var µ f, but since f dµ = 1, Var µ f 1, then f 1 dµ Var µ f. ii. One verifies trivially that when x, x 4x 1 and apply i.

16 44 I. Gentil et al. iii. Let us give the proof given in [13]. If x>0wehavexlog x + 1 x 0 which yields f log f dµ + µf f dµ 0, f f hence Ent µ f f f log f dµ f f dµ. Since f dµ 1 f log f dµ, f log 4 f we obtain Ent µ f 1 log 4 1 f f log f dµ. Recall the Hardy s inequality presented in the introduction. Let µ, ν be Borel measures on R +, the best constant A so that every smooth function f such that integrals are well defined, satisfies 0 fx f0 dµx A 0 f dν, 0 is finite if and only if x dν ac 1 B = sup µ[x, [ dt 1 x 0 0 dµ is finite. And when A is finite we have this estimation B A 4B. A direct proof of this inequality with these properties of regularities for f can be found for example in Theorem 6..1 of [1]. We then present different proof of the desired inequality, starting from 19, according to the value of Ent µ f, in which Hardy s inequality plays a crucial role. First, when the entropy is large we will need Lemma 3.4. Let h defined as follow, { 1 if x 1, hx = x α if x 1. Then there exists 0 C h < such that for every smooth function g we have Ent µα g C h g hdµ α. Proof. We use Theorem 3 of [10] which is a refinement of the criterion of a Bobkov- Götze theorem see Theorem 5.3 of [6]. The constant C h satisfies maxb,b + C h maxb,b + where

17 Modified logarithmic Sobolev inequalities and transportation inequalities 45 b + = sup µ α [x,+ [ log 1 + x 0 b = sup µ α ],x] log x 0 B + = sup µ α [x,+ [ log x 0 B = sup µ α ],x] log x x µ α [x,+ [ 1 0 µ α ],x] x e x µ α [x,+ [ e µ α [,x[ An easy approximation prove that for large positive x µ α [x, [ = x x Z α e t α ht dt, Z α e t α ht dt, Z α e t α ht dt, Z α e t α ht dt. 1 1 e t α dt Z α Z α αx α 1 e xα 3 x 0 e t α Z α ht dt Z α αx exα, and one may prove same equivalent for negative x. A simple calculation then yields that constants b +, b, B + and B are finite and the lemma is proved. Note that the function h is the smallest function such that the constant C h in the inequality is finite. More precisely, if h satisfy hx lim x hx = then the constant C h =. In the case of small entropy, we will use so-called -Sobolev inequalities even if our context is less general, see Chafaï [14] for a comprehensive review, and Barthe-Cattiaux-Roberto [4] for a general approach in the case of measure µ α. Lemma 3.5. Let T 1 <T, T [T 1,T ] and g be a smooth function defined on [T, [. Assume that gt =, g and T g dµ α 13, then T g g dµ α C g [T, [ g dµα, 4 where x = log α 1 α x. The constant C g depend on the measure µ α but does not depend on the value of T [T 1,T ].

18 46 I. Gentil et al. Proof. Let use Hardy s inequality as explained in the introduction. We have gt =. We apply inequality 0 on [T, [ and with the function f = g note that ft= 0 and the following measures α 1 dµ = log g α dµ α and ν = µ α. Then the constant C in inequality 4 is finite if and only if x α 1 B = sup Z α e t α dt log g α dµ α, x T T x is finite. Since α 1/α < 1 the function x log x α 1 α is concave on [4, [. By Jensen inequality we obtain for all x T, α 1 log g α dµ α log α 1 α x g dµ α µ α [x, [. µ α [x, [ x Then by the property of g we have B sup x T sup x T 1 x T x Using the approximation x Z α e t α dt log α 1 α Z α e t α α 1 dt log α T 1 0 Z α e t α dt 13 µ α [x, [ 13 µ α [x, [ Z α αx α 1 exα, µ α [x, [ µ α [x, [. and that given in equality 3 we prove that B is finite, bounded by a constant C g which does not depend on T. We divide the proof of Theorem 3.1 in two parts: large and small entropy, both in the case of positive function. Let us now present the proof in the case of large entropy. Large entropy case: Proposition 3.6. Suppose that α ]1, ]. There exists 0 A L,B L < such that for any smooth functions f 0 satisfying f dµ α = 1 and Ent µα f 1, 5 we have Ent µα f A L Var µα f + B L f f f f dµ α. 6

19 Modified logarithmic Sobolev inequalities and transportation inequalities 47 If α = 1, there exists 0 A L < and λ>0such that for every smooth function f 0 with f dµ 1 = 1, when f λ, then we get Ent µ1 f A L Var µ 1 f. Proof of Proposition 3.6 Let f be a smooth function satisfying f 0, f dµ α = 1 and f 4 dµ α <. A careful study on R + of this function x x log x + 5x 1 + x 1 + x + logx + proves that for every x 0 x log x 5x 1 + x 1 + x + logx +. Then we obtain by Lemma 3.3.i, recalling that f dµ α = 1 and f 0, f log f dµ α 5 f 1 dµ α + f 1dµ α + f + logf + dµ α 10Var µα f + f + logf + dµ α which is the announced starting point inequality 19. Since f dµ α = 1, one can easily prove that f + dµ α 1, then f + logf + dµ α Ent µα f +, and Ent µα f 10Var µα f + Ent µα f +. 7 Hardy s inequality of Lemma 3.4 with g = f + gives Ent µα f + C h f + hdµ α = C h f f hdµ α. 8 For p, q > 1 such that and 1/p + 1/q = 1 we have for every x,y > 0by Young inequality, xy xp p + yq q. 9 Consider then α ]1, ] and = α/α 1. Let p = / and q = /. Let ε>0and let apply inequality 9 to the right term of 8, we obtain on {f }, 1 ε / f ε / h f ε / f f + εh/,

20 48 I. Gentil et al. then Ent µα f + C h ε / + C hε f f f f dµ α f h / f dµ α Let µ a probability measure, then we have for every function f such that f dµ = 1 and for every measurable function g such that f gdµ exists we get f gdµ Ent µ f + log e g dµ. This inequality is also true for all function g 0 even integrals are infinite. Let η>0and we apply the previous inequality with g = ηh / g 0, we obtain then Ent µα f + C h f f dµ α + C hε η ε / f Ent µα f + log f exp ηh / dµ α. 30 Since = α/α 1, hx / = x α if x 1, then we fix η = 1/. And note 1 = log exp h/ dµ α <. Fix now ε = inf {/ 4C h, / 4C h } and note κ = C h /ε / We obtain Ent µα f + κ f f f f dµ α Ent µ α f As Ent µα f 1, inequality 7 implies Ent µα f f 0Var µα f + κ f dµ α, f f which proves inequality 6 with A L = 0 and B L = κ. Assume now that α = 1 and take f such that f λ. We apply the limit case of Young inequality to get on {f }, f λ hf hf. f Then, with the same computation as in the case α ]1, ], on can found λ>0and 0 A L < such that for all smooth function f satisfying hypothesis on 5 and f λ, Ent µ1 f A L Var µ 1 f.

21 Modified logarithmic Sobolev inequalities and transportation inequalities 49 Remark 3.7. With the same method as developed in Proposition 3.6 we can prove the inequality 6 without Var µα f. Suppose that α ]1, ]. There exists A>0 such that for any functions f>0 satisfying f dµ α = 1 and Ent µα f 1 we have Ent µα f f A f f dµ α. Small entropy case: Proposition 3.8. Let α ]1, ]. There exists A S,B S > 0 such that for any functions f 0 satisfying f dµ α = 1 and Ent µα f 1, we have Ent µα f A S Var µα f + B S f f f f dµ α. 31 If α = 1, there exists 0 A S < such that, for all f such that f 1. Ent µ1 f A S Var µ 1 f. Proof of Proposition 3.8 Let f 0 satisfying f dµ α = 1. As in Proposition 3.6, we start with inequality 19, which readily implies Ent µα f = f log f dµ α 10Var µα f + f + log f dµ α. 3 We will now control the second term of the right hand side of this last inequality via the use of -Sobolev inequalities, namely Lemma 3.5. Therefore we have to construct a function g defined on [T, [ for a well chosen T with g and gt = and which satisfies, g1 g g3 T T T g dµ α 13; g g dµ α C g dµ α C [T, [ {f } f + log f dµ α ; T f f f dµ α + D Ent µα f,

22 430 I. Gentil et al. with x = log α 1 α x,0<d 1/ and x = x. Let now define T 1 < 0 and T > 0 such that µ α ],T 1 ] = 3 8,µ α[t 1,T ] = 1 4 and µ α[t, + [ = 3 8. Since f dµ α = 1 there exists T [T 1,T ] such that ft. Introduce now g on [T, ] as follow g = + f + log γ f, where γ = α/α. Due to the fact that f is a smooth function locally absolutely continuous function then g is also a smooth function. Moreover g satisfies gt = and gx for all x T. Let now compute T g dµ α. We get g dµ α 4dµ α + f + logγ f dµ α T T 1 T f log γ f dµ α. [T 1, [ {f } Since γ [0, 1] we have log γ f log f on {f }. Then we obtain by Lemma 3.3.iii g dµ α 5 + f log γ f dµ α T f 5 + 8Ent µα f 13, since Ent µα f 1. Assumptions on Lemma 3.5 are satisfied, we obtain by inequality 4 g α 1 + log α g dµ α C g g dµα. T Let us compare the various terms now. First, denote u = α 1/α, we thus obtain [T, [ {g } g + logu g = f + logγ f log u + f + log γ f. On {f },wehave+ f + log γ f + f + K, where K = log γ 4. Since K 1 and u + γ = 1, one has Then we obtain g + logu g f + logγ +u f = f + log f. T f + log f dµ α T g + log α 1 α g dµ α. 33

23 Modified logarithmic Sobolev inequalities and transportation inequalities 431 Secondly one has on {f }, g = f log γ f 1 + γ γ f f log f, then using log f log 4 on {f } one obtain g f log γ f 1 + γ γ. łog4 Denoting D = 1 + γ γ /log 4, one has g dµ α D f log γ f dµ α, 34 [T, [ {f } [T, [ {f } on [T, [ {f }. Then, using inequalities 33 and 34, there exists C 0 independent of T [T 1,T ], such that f + log f dµ α C f log γ f dµ α. 35 T [T, [ {f } When α ]1, ], we apply Inequality 9 with q = α/ α and p = α/α 1. We obtain for every ε>0, f log γ f f α 1 dµ α f [T, [ {f } f αε α 1 α [T, [ {f } f f dµ α +ε α f log f dµ α. α [T, [ {f } Fix ε such that εc α α < 1/16, then there exists A>0such that f + log f dµ α A f T [T, [ {f } f f dµ α + 1 f log f dµ α. 16 Using Lemma 3.3.iii we have, [T, [ {f } f + log f dµ α T A f [T, [ {f } f f dµ α Ent µ α f. The same method can be used on ],T] and then we get T f + log f dµ α A [,T ] {f } f f f dµ α Ent µ α f.

24 43 I. Gentil et al. And then, we get f + log f dµ α A f f f f dµ α + 1 Ent µ α f. Note that the constant A does not depend on T. Then, by inequality 3, inequality 31 is proved for α ]1, ], with A S = 34 and B S = 4A. Assume now that α = 1. In this case γ = 1, then using inequality 35 we obtain f + log f dµ 1 D log f dµ 1 8D f dµ 1, [T, [ {f } [T, [ {f } T for some constant C. Then by Lemma 3.3.ii, we obtain the result which concludes the proof in this case with A S = D. Let us give now a proof of the theorem. Proof of Theorem 3.1 The proof of the theorem is a simple consequence of Propositions 3.6 and 3.8. For α ]1, ], we get inequality 16 with A = max{a L,A S } and B = max{b L,B S } and for α = 1 one find λ>0and 0 A = max{a S,A L } < such that inequality 17 is true. 4. Extension to other measures We will present in this section modified logarithmic Sobolev inequality of function H for more general measure than µ α which can be derived using the proof carried on in Section 3: the large entropy case where the optimal Hardy function h is identified and used to derive the optimal H, and the small entropy case where and g used on the proof of Proposition 3.7 have to be identified leading to the same H function. Let us first consider the following probability measure µ α, for α [1, ] and R defined by µ α, dx = 1 Z e ϕx dx where ϕx = x α log x for x 1 and ϕ twice continuously differentiable. Theorem 4.1. There exists 0 A, B < such that the measure µ α, satisfies the following logarithmic Sobolev inequality: for any smooth f on R such that f dµ α = 1 and f 0, we have Ent µα, f f AVar µα, f + B H f f f dµ α,, 36

25 Modified logarithmic Sobolev inequalities and transportation inequalities 433 where H is positive smooth and given for x by Hx = x α α 1 if α ]1, [, R, log α 1 x Hx = x e x1/ if α = 1, R +, Hx = x log x if α =, R. Proof. We will mimic closely the proof given in the µ α case, considering large and small entropy case. We will not present all the calculus but give the essential arguments. Let now treat the case α ]1, [. Large entropy. We will first apply Lemma 3.4 to measure µ α,, one has then that b +, b, B +, B are finite if one take h positive smooth hx = x α log x x. One has then to determine H to construct ψ such that there exists η>0with ηψh exponentially integrable with respect to µ α, and H = ψ x where ψ is the Fenchel-Legendre transform of ψ. Considering the exponential integrability condition leads us to consider ψx behaving asymptotically as x α α log α x. One may thus derive the asymptotic behavior of ψ and finally H. Small entropy. One desires here to apply Lemma 3.5, evaluating and then build the function g satisfying conditions g1, g and g3. By Hardy s inequality and arguments in the proof of Lemma 3.5, one may choose for x large enough as Setting then x = log α 1 α x log log x α. g = + f + log α α f log log f α, one may then verify g1, g and g3 with = H defined in the large entropy step. Now if α = 1 and 0, then the same arguments gives that for large enough x ψx = x log x, ψ x = xe x1/ and Hx = x e x1/. If α = and 0, we have for large enough x. ψx = 1 x ex 1/, ψ x = x log x and Hx = x log x.

26 434 I. Gentil et al. Remark Using once again Herbst s argument, we may derive concentration properties for the measure µ α, of desired order, for every function F with F 1, there exists C>0 such that, for all λ>0, µ α, F µ α, F λ e C minλα log λ,λ. The extension to greater dimension being handled as in the previous case. Note that the Latała-Oleszkiewicz inequalities Ir see [1] are not well adapted for the family of measures µ α,. Indeed, using Hardy s characterization of this inequalities obtained by Barthe-Roberto [10, Th. 13 and Prop. 15], one may show that µ α, satisfies an Iα/ inequality if 0 and an Iα/ ɛ ɛ being arbitrary small for < 0, which entails consequently not optimal concentration properties. 3. By the characterization of the spectral gap property on R, one obtains that each measure µ α, satisfies a Poincaré inequality and thus a modified logarithmic Sobolev inequality. Following the previous proof, we may generalize the family µ α, adding an explicit multiplicative term to the potential x α log x, as for example log log γ x which will give us new modified logarithmic Sobolev inequality, but each of this new measure has to be considered one-by-one we hope some general results for ϕ convex. We may now state a result enabling us to get the stability of these modified logarithmic Sobolev inequality by addition of an unbounded perturbation: consider the measures dx dτ α x = exp x α x α 1 cosx, α ]1, ], Z α dγ α,b x = 1 + x b e dx xα 1 x 0, α ]1, ],b R. Z α,b Proposition 4.3. There exists a>0 such that the measures τ α and γ α,b satisfy a logarithmic Sobolev inequality of function H a,α. Proof. Following the proof given in Section 3, one sees that the result hold true once one may verify that the Hardy s inequalities of Lemma 3.4 and Lemma 3.5 hold with the h and obtained for the case of µ α. It is easily checked once remarked that log dτ αx dx x α and the same for γ α,b. and log dτ αx α 1 x α 1 dx Acknoledgements. The authors sincerely thank referee for helpful and interesting remarks.

27 Modified logarithmic Sobolev inequalities and transportation inequalities 435 References 1. Ané, C., Blachère, S., Chafaï, D., Fougères, P., Gentil, I., Malrieu, F., Roberto, C., Scheffer, G.: Sur les inégalités de Sobolev logarithmiques, volume 10 of Panoramas et Synthèses. Société Mathématique de France, Paris, 000. Bakry, D.: L hypercontractivité et son utilisation en théorie des semigroupes. In: Lectures on probability theory. École d été de probabilités de St-Flour 199, volumn 1581 of Lecture Notes in Math. Springer, Berlin, 1994, pp Barles, G.: Solutions de viscosité des équations de Hamilton-Jacobi. Springer-Verlag, Paris, Barthe, F., Cattiaux, P., Roberto, C.: Interpolated inequalities between exponential and Gaussian, Orlicz hypercontractivity and application to isoperimetry. Preprint, Beckner, W.: A generalized Poincaré inequality for Gaussian measures. Proc. Amer. Math. Soc. 105, Bobkov, S.G., Götze, F.: Exponential integrability and transportation cost related to logarithmic Sobolev inequalities. J. Funct. Anal , Bobkov, S.G., Gentil, I., Ledoux, M.: Hypercontractivity of Hamilton-Jacobi equations. J. Math. Pu. Appli. 80 7, Bobkov, S.G., Ledoux, M.: Poincaré s inequalities and Talagrand s concentration phenomenon for the exponential distribution. Probab. Theor. Related Fields 107 3, Bobkov, S.G., Ledoux, M.: From Brunn-Minkowski to Brascamp-Lieb and to logarithmic Sobolev inequalities. Geom. Funct. Anal. 10 5, Barthe, F., Roberto, C.: Sobolev inequalities for probability measures on the real line. Studia Math. 159, Brenier, Y.: Polar factorization and monotone rearrangement of vector-valued functions. Comm. Pure Appl. Math. 44 4, Bobkov, S.G., Zegarlinski, B.: Entropy bounds and isoperimetry. To appear in Memoirs AMS, Cattiaux, P., Guillin, A.: Talagrand s like quadratic transportation cost inequalities. Preprint, Chafaï, D.: Entropies convexity and functional inequalities. J. Math. Kyoto Univ. 44, Evans, L.C.: Partial differential equations. American Mathematical Society, Providence, RI, Gentil, I.: Inégalités de Sobolev logarithmiques et hypercontractivité en mécanique statistique et en EDP. Thèse de l Université Paul Sabatier, Gangbo, W., McCann, R.J.: The geometry of optimal transportation. Acta Math. 177, Gross, L.: Logarithmic Sobolev inequalities. Amer. J. Math. 97 4, Ledoux, M.: Concentration of measure and logarithmic Sobolev inequalities. In: Séminaire de Probabilités XXXIII, vol 1709 of Lecture Notes in Math, Springer, Berlin, 1999, pp Lieb, E.H.: Some convexity and subadditivity properties of entropy. Bull. Amer. Math. Soc. 81, Latała, R., Oleszkiewicz, K.: Between Sobolev and Poincaré. In: Geometric aspects of functional analysis, vol 1745 of Lecture Notes in Math, Springer, Berlin, 000, pp Marton, K.: Bounding d-distance by informational divergence: a method to prove measure concentration. Ann. Probab. 4,

28 436 I. Gentil et al. 3. Maurey, B.: Some deviation inequalities. Geom. Funct. Anal. 1, Muckenhoupt, B.: Hardy s inequality with weights. collection of articles honoring the completion by Antoni Zygmund of 50 years of scientific activity 5. Otto, F., Villani, C.: Generalization of an inequality by Talagrand and links with the logarithmic Sobolev inequality. J. Funct. Anal. 173, Talagrand, M.: A new isoperimetric inequality and the concentration of measure phenomenon. Geometric aspects of functional analysis Lecture Notes in Math Springer, Berlin, 1991, pp Talagrand, M.: Concentration of measure and isoperimetric inequalities in product spaces. Inst. Hautes Études Sci. Publ. Math. 81, Talagrand, M.: Transportation cost for Gaussian and other product measures. Geom. Funct. Anal. 6 3, Villani, C.: Topics in optimal transportation, volumn 58 of Graduate Studies in Mathematics. American Mathematical Society, Providence, RI, 003