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We prove a global Li-Yau inequality for a general Markov semigroup under a curvature-dimension condition. This inequality is stronger than all classical Li-Yau type inequalities known to us. On a Riemannian manifold, it is equivalent to a new parabolic Harnack inequality, both in negative and positive curvature, giving new subsequents bounds on the heat kernel of the semigroup. Under positive curvature we moreover reach ultracontractive bounds by a direct and robust method.

We develop connections between Harnack inequalities for the heat flow of diffusion operators with curvature bounded from below and optimal transportation. Through heat kernel inequalities, a new isoperimetric-type Harnack inequality is emphasized. Commutation properties between the heat and Hopf-Lax semigroups are developed consequently, providing direct access to the heat flow contraction property along Wasserstein distances.

International audience; Using some harmonic extensions on the upper-half plane, and probabilistic representations, and curvature-dimension inequalities with some negative dimensions, we obtain some new opimal functional inequalities of the Beckner type for the Cauchy type distributions on the Euclidean space. These optimal inequalities appear to be equivalent to some non tight optimal Beckner inequalities on the sphere, and the family appear to be a new form of the Sobolev inequality.

We derive sharp, local and dimension dependent hypercontractive bounds on the Markov kernel of a large class of diffusion semigroups. Unlike the dimension free ones, they capture refined properties of Markov kernels, such as trace estimates. They imply classical bounds on the Ornstein-Uhlenbeck semigroup and a dimensional and refined (transportation) Talagrand inequality when applied to the Hamilton-Jacobi equation. Hypercontractive bounds on the Ornstein-Uhlenbeck semigroup driven by a non-diffusive L\'evy semigroup are also investigated. Curvature-dimension criteria are the main tool in the analysis., Comment: 24 pages

The aim of this short note is to explain how Nash inequalities lead to such estimates in a general setting and also to show simple techniques used to establish the required Nash inequalities.

We study the relationship between two classical approaches for quantitative ergodic properties: the first one based on Lyapunov type controls and popularized by Meyn and Tweedie, the second one based on functional inequalities (of Poincaré type). We show that they can be linked through new inequalities (Lyapunov–Poincaré inequalities). Explicit examples for diffusion processes are studied, improving some results in the literature. The example of the kinetic Fokker–Planck equation recently studied by Hérau and Nier, Helffer and Nier, and Villani is in particular discussed in the final section.

International audience; We present a finite dimensional version of the logarithmic Sobolev inequality for heat kernel measures of non-negatively curved diffusion operators that contains and improves upon the Li-Yau parabolic inequality. This new inequality is of interest already in Euclidean space for the standard Gaussian measure. The result may also be seen as an extended version of the semigroup commutation properties under curvature conditions. It may be applied to reach optimal Euclidean logarithmic Sobolev inequalities in this setting. Exponential Laplace differential inequalities through the Herbst argument furthermore yield diameter bounds and dimensional estimates on the heat kernel volume of balls.

On definit l'espace des formes bilineaires intrinseques associees a un operateur L differentiel du second ordre. Ces formes sont definies uniquement a partir de l'operateur L lui-meme et de la structure d'algebre sur les fonctions donnee par la multiplication. La positivite d'une de ces formes ne depend que de la structure locale de l'operateur. Pour des operateurs symetriques par rapport a une mesure positive, chaque forme bilineaire intrinseque positive donne des informations sur le spectre de L. Dans cet article, nous construisons une famille (R k ) de formes bilineaires intrinseques sur les spheres (dans ce cas, l'operateur L est l'operateur de Laplace-Beltrami) qui contient toute l'information sur le spectre. Plus precisement, si f est un vecteur propre de l'operateur de Laplace-Beltrami de valeur propre λ, et si g est une fonction lisse quelconque, alors pour la mesure Riemannienne μ, on a ∫R k (f,g)dμ = λ(λ-λ 1 )...(λ-λ k-1 ) ∫ fgdμ, ou 0, λ 1 ,...,λ k-1 sont les k premieres valeurs propres de l'operateur de Laplace-Beltrami. C'est une extension a tout le spectre de la formule de Bochner-Lichnerowicz, qui donne une minoration precise de la premiere valeur propre non nulle. On etend ensuite cette propriete a tout une famille d'operateurs du second ordre qui est une extension multidimensionnelle des operateurs ultraspheriques en dimension 1.

International audience; We investigate the hypergroup property for the family of orthogonal polynomials associated with the deltoid curve in the plane, related to the A 2 root system. This provides also the same property for another family of polynomials related to the G 2 root system.

International audience; We provide new examples of diffusion operators in dimension 2 and 3 which have orthogonal polynomials as eigenvectors. Their construction rely on the finite subgroups of O(3) and their invariant polynomials.

We describe some examples of classical and explicit h-transforms as particular cases of a general mechanism, which is related to the existence of symmetric diffusion operators having orthogonal polynomials as spectral decomposition.

We generalise for a general symmetric elliptic operator the different notions of dimension, diameter, and Ricci curvature, which coincide with the usual notions in the case of the Laplace-Beltrami operators on Riemannian manifolds. If λ 1 denotes the spectral gap, that is the first nonzero eigenvalue, we investigate in this paper the best lower bound on λ 1 one can obtain under an upper bound on the dimension, an upper bound on the diameter, and a lower bound of the Ricci curvature. Two cases are known: namely if the Ricci curvature is bounded below by a constant R>0, then λ 1≥nR/(n-1), and this estimate is sharp for the n-dimensional spheres (Lichnerowicz's bound). If the Ricci curvature is bounded below by zero, then Zhong-Yang's estimate asserts that λ 1≥π 2d 2, where d is an upper bound on the diameter. This estimate is sharp for the 1-dimensional torus. In the general case, many interesting estimates have been obtained. This paper provides a general optimal comparison result for λ 1 which unifies and sharpens Lichnerowicz and Zhong-Yang's estimates, together with other comparison results concerning the range of the associated eigenfunctions and their derivatives. © 2000 Academic Press.

We consider Brownian motions and other processes (Ornstein-Uhlenbeck processes, spherical Brownian motions) on various sets of symmetric matrices constructed from algebra structures, and look at their associated spectral measure processes. This leads to the identification of the multiplicity of the eigenvalues, together with the identification of the spectral measures. For Clifford algebras, we thus recover Bott’s periodicity.

This chapter is a brief investigation of the links between optimal transportation methods and functional inequalities in the Markov operator framework of this monograph. After a brief introduction to the basic material on optimal transportation, the main topic of transportation cost inequalities and first examples for Gaussian measures are presented. Interpolation along the geodesics of optimal transport is used towards logarithmic Sobolev inequalities and transportation cost inequalities comparing relative entropy and Wasserstein distances between probability measures. An alternate approach to sharp Sobolev or Gagliardo–Nirenberg inequalities in Euclidean space is provided next along these lines. Non-linear Hamilton–Jacobi equations and hypercontractivity properties of their solutions, analogous to the ones for linear heat equations, are investigated in the further sections towards the relationships between (quadratic) transportation cost inequalities and logarithmic Sobolev inequalities. Contraction properties in Wasserstein space along with the heat semigroup are investigated in the Markov operator setting. The last section is a very brief overview of recent developments towards a notion of Ricci curvature lower bounds based on optimal transportation and the connection with the Γ-calculus developed in this work.

After Poincare inequalities, logarithmic Sobolev inequalities are amongst the most studied functional inequalities for semigroups. They contain much more information than Poincare inequalities, and are at the same time sufficiently general to be available in numerous cases of interest, in particular in infinite dimension (as limits of Sobolev inequalities on finite-dimensional spaces). After the basic definition of a logarithmic Sobolev inequality together with its first properties, the first sections of this chapter present the exponential decay in entropy and the fundamental equivalence between the logarithmic Sobolev inequality and smoothing properties of the semigroup in the form of hypercontractivity. Next, integrability properties of eigenvectors and of Lipschitz functions under a logarithmic Sobolev inequality are discussed together with a criterion for measures on the real line to satisfy a logarithmic Sobolev inequality (for the usual gradient). The further sections deal with curvature conditions, first for the local logarithmic Sobolev inequalities for heat kernel measures, then for the invariant measure with an additional dimensional information. Local hypercontractivity and some applications of the local logarithmic Sobolev inequalities towards heat kernel bounds are further presented. Harnack-type inequalities under the infinite-dimensional curvature conditions, linked with reverse local logarithmic Sobolev inequalities complete the chapter.

This chapter focuses on inequalities comparing measure and capacity uniformly over a given class of sets as equivalent forms of functional inequalities. The chapter concentrates on the so-called 2-capacities which capture the relevant Dirichlet form information on sets and on the 1-capacities which are related to boundary or surface measures and isoperimetric-type inequalities. The first part of the chapter introduces the basic notions on capacities and co-area formulas to transfer (and back) functional inequalities into measure-capacity inequalities. Sobolev-type, Poincare and logarithmic Sobolev inequalities are analyzed in this respect. The second part is concerned with measure-capacity inequalities. Together with the heat kernel tools under curvature conditions, measure-capacity inequalities of isoperimetric-type are investigated, leading in particular to the Gaussian isoperimetric inequality as well as to comparison results under curvature conditions. Combination of the heat kernel isoperimetric inequality together with its reverse form further yields an isoperimetric-type Harnack inequality. The last section addresses the relationships between Poincare and logarithmic Sobolev inequalities and their associated concentration properties.

This chapter provides a general framework for the investigation of symmetric Markov diffusion semigroups and operators. Based on the early observations of the first chapter and on the investigation of the model examples, it develops all the fundamental properties which will justify the computations in the following chapters in the most convenient framework. The main setting consisting of a Markov Triple (E,μ,Γ) describes a convenient framework to develop the formalism of Markov semigroups and the Γ-calculus towards functional inequalities and convergence to equilibrium. The analysis of the concrete example of second order differential operators on smooth complete connected manifolds is the guide for the description, in this framework, of some main features such as connexity, completeness, weak hypo-ellipticity etc. With the main tool of essential self-adjointness at the center of the construction, the chapter emphasizes the relevant hypotheses and properties of Full Markov Triples, in force throughout the monograph, covering the main examples of illustrations.

The Poincare, logarithmic Sobolev and Sobolev inequalities capture different features of the associated semigroup or the invariant measure, in terms of convergence to equilibrium, estimates on the heat kernels or tail behaviors of the invariant measure. This chapter investigates intermediate or more general families of functional inequalities which are suited to a wide variety of regimes as well as to more precise, or different, features. The study is concerned with three main examples, entropy-energy, generalized Nash and weak Poincare inequalities. The first part of the chapter describes the family of functional entropy–energy inequalities well-suited to heat kernel bounds by means of the method developed for hypercontractivity under logarithmic Sobolev inequalities. Off-diagonal heat kernel estimates may be achieved in the same way. Further sections in this chapter investigate generalized Nash inequalities on the basis of the example of the classical Nash inequality in Euclidean space, as well as weighted Nash inequalities, with a focus on non-uniform heat kernel bounds and tail inequalities. Weak Poincare inequalities are studied as the main minimal tool to tighten families of standard functional inequalities. Weak Poincare inequalities may be further studied in their own from the viewpoint of heat kernel bounds and tail estimates. Further related families of functional inequalities of interest and illustrative examples complete the exposition of this chapter.

Resume L'objet de cette Note est d'etudier, sous des conditions simples, la normalite asymptotique locale du modele statistique associe a une chaine de Markov cachee homogene, a espace fini d'etats, de condition initiale deterministe, ainsi que la consistance et la normalite asymptotique de l'estimateur du maximum de vraisemblance.

We establish several new Harnack estimates for the nonnegative solutions of the heat equation on a complete Riemannian manifold with Ricci curvature bounded by a positive or negative constant. This extends to symmetric diffusions whose generator satisfies a “curvature-dimension” inequality.

We establish optimal uniform upper estimates on heat kernels whose generators satisfy a logarithmic Sobolev inequality (or entropy-energy inequality) with the optimal constant of the Euclidean space. Off-diagonals estimates may also be obtained with however a smaller d istance involving harmonic functions. In the last part, we apply these methods to study some heat kernel decays for diffusion operators of the type Laplacian minus the gradient of a smooth potential with a given growth at infinity.

We establish, by simple semigroup arguments, a Levy–Gromov isoperimetric inequality for the invariant measure of an infinite dimensional diffusion generator of positive curvature with isoperimetric model the Gaussian measure. This produces in particular a new proof of the Gaussian isoperimetric inequality. This isoperimetric inequality strengthens the classical logarithmic Sobolev inequality in this context. A local version for the heat kernel measures is also proved, which may then be extended into an isoperimetric inequality for the Wiener measure on the paths of a Riemannian manifold with bounded Ricci curvature.

For a given orthonormal basis (f n ) on a probability measure space, we want to describe all Markov operators which have the f n as eigenvectors. We introduce for that what we call the hypergroup property. We study this property in three different cases.

We give a simple and direct proof of the existence of a spectral gap under some Lyapunov type condition which is satisfied in particular by log-concave probability measures on $\mathbb{R}^n$. The proof is based on arguments introduced in Bakry and al, but for the sake of completeness, all details are provided.

We give a characterization of the eigenvalues of Markov operators which admit an orthogonal polynomial basis as eigenfunctions, in the Hermite and the Laguerre cases, as well as for the sequences of orthogonal polynomials associated to some probability measures on ℕ. In the Hermite case, we also give a description of the path of the associated Markov processes, as well as a geometric interpretation.

Nous decrivons des bases orthonormees sur l’espace L2 d’un ensemble fini dans lequel des inegalites de la forme GKS1 prennent une forme simple. Ceci s’applique en particulier aux classes de conjugaison d’un groupe fini quelconque. Dans tous les exemples que nous considerons, l’inegalite GKS2 en est une consequence, sans que nous ayons une interpretation generale de ce phenomene.

A weak Sobolev inequality (WSI) is a weakened form of a Sobolev inequality associated to a Dirichlet form. It turns out that it is in fact equivalent to a Sobolev inequality. If there is a spectral gap and a (WSI) holds, then we can get a tight weake Sobolev quality (TWSI). Starting with a TWSI. we get upper and lower bounds on the density of the heat semigroup associated with the Dirichlet form, when t → 0 as well as when t → ∞. In the last chapter, we give a Γ2 criterium to get a TWSI on a manifold.

This paper is a brief survey of some properties of the “iterated squared gradient” associated to some diffusion semigroup. In the first part, we use this notion to give an “intrinsic” definition of the Ricci curvature and of the dimension of the semigroup: in the case of the heat semigroup on a Riemannian manifold, we recover the usual notions. In the second part, we describe some properties of diffusions with Ricci curvature bounded from below. In the third part, we show how to improve these properties in the case of diffusions with finite dimension. The fourth part is devoted to examples, and was worked out with the help of M.Emery.

This is an expository paper about some recent works on the Riesz transforms associated with a general symmetric, elliptic second order differential operator. Except in the third part, there are no new results and the proofs are often ommited or just sketched.

Lorsqu'on a un semigroupe markovien symetrique Pt, le theoreme d'interpolation de Stein permet de voir que, si z est un complexe de partie reelle positive, l'operateur Pz, defini a partir de la decomposition spectrale de Pt, est borne sur Lp(μ), pourvu que l'exposant p soit dans un intervalle contenant 2 et qui depend de l'angle que fait z avec avec l'axe reel. Pour un semigroupe de diffusion, nous ameliorons ce resultat, c'est a dire que nous obtenons un intervalle plus grand que celui donne par le theoreme d'interpolation. La methode que nous utilisons n'ayant que peu a voir avec la structure complexe, nous donnons quelques exemples de generalisation: par exemple, apres les avoir definis, nous donnons des estimations sur les operateurs Ph, ou h est un quaternion de partie reelle positive, et plus generalement sur les operateurs PM, ou M est une matrice normale de partie symetrique positive.