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International audience; Besides fractional Brownian motion most non-Gaussian fractional fields are obtained by integration of deterministic kernels with respect to a random infinitely divisible measure. In this paper, generalized shot noise series are used to obtain approximations of most of these fractional fields, including linear and harmonizable fractional stable fields. Almost sure and $L^r$-norm rates of convergence, relying on asymptotic developments of the deterministic kernels, are presented as a consequence of an approximation result concerning series of symmetric random variables. When the control measure is infinite, normal approximation has to be used as a complement. The general framework is illustrated by simulations of classical fractional fields.

In a recent work, E. Cinti and F. Otto established some new interpolation inequalities in the study of pattern formation, bounding the Lr(μ)-norm of a probability density with respect to the reference measure μ by its Sobolev norm and the Kantorovich-Wasserstein distance to μ. This article emphasizes this family of interpolation inequalities, called Sobolev-Kantorovich inequalities, which may be established in the rather large setting of non-negatively curved (weighted) Riemannian manifolds by means of heat flows and Harnack inequalities.

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.

We study Lipschitz contraction properties of general Markov kernels seen as operators on spaces of probability measures equipped with entropy-like ``distances''. Universal quantitative bounds on the associated ergodic constants are deduced from Dobrushin's ergodic coefficient. Strong contraction properties in Orlicz spaces for relative densities are proved under more restrictive mixing assumptions. We also describe contraction bounds in the entropy sense around arbitrary probability measures by introducing a suitable Dirichlet form and the corresponding modified logarithmic Sobolev constants. The interest in these bounds is illustrated on the example of inhomogeneous Gaussian chains. In particular, the existence of an invariant measure is not required in general.

We develop connections between Stein's approximation method, logarithmic Sobolev and transport inequalities by introducing a new class of functional inequalities involving the relative entropy, the Stein kernel, the relative Fisher information and the Wasserstein distance with respect to a given reference distribution on $\mathbb{R}^d$. For the Gaussian model, the results improve upon the classical logarithmic Sobolev inequality and the Talagrand quadratic transportation cost inequality. Further examples of illustrations include multidimensional gamma distributions, beta distributions, as well as families of log-concave densities. As a by-product, the new inequalities are shown to be relevant towards convergence to equilibrium, concentration inequalities and entropic convergence expressed in terms of the Stein kernel. The tools rely on semigroup interpolation and bounds, in particular by means of the iterated gradients of the Markov generator with invariant measure the distribution under consideration. In a second part, motivated by the recent investigation by Nourdin, Peccati and Swan on Wiener chaoses, we address the issue of entropic bounds on multidimensional functionals $F$ with the Stein kernel via a set of data on $F$ and its gradients rather than on the Fisher information of the density. A natural framework for this investigation is given by the Markov Triple structure $(E, \mu, \Gamma)$ in which abstract Malliavin-type arguments may be developed and extend the Wiener chaos setting., Comment: 52 pages

This paper is concerned with a general class of self-interacting diffusions {X t } t ≥0 living on a compact Riemannian manifold M. These are solutions to stochastic differential equations of the form : dX t = Brownian increments + drift term depending on X t and μ t , the normalized occupation measure of the process. It is proved that the asymptotic behavior of {μ t } can be precisely related to the asymptotic behavior of a deterministic dynamical semi-flow Φ = {Φ t } t ≥0 defined on the space of the Borel probability measures on M. In particular, the limit sets of {μ t } are proved to be almost surely attractor free sets for Φ. These results are applied to several examples of self-attracting/repelling diffusions on the n-sphere. For instance, in the case of self-attracting diffusions, our results apply to prove that {μ t } can either converge toward the normalized Riemannian measure, or to a gaussian measure, depending on the value of a parameter measuring the strength of the attraction.

Following the equivalence between logarithmic Sobolev inequalities and hypercontractivity showed by L. Gross, we prove that logarithmic Sobolev inequalities are related similarly to hypercontractivity of solutions of Hamilton–Jacobi equations. By the infimum-convolution description of the Hamilton–Jacobi solutions, this approach provides a clear view of the connection between logarithmic Sobolev inequalities and transportation cost inequalities investigated recently by F. Otto and C. Villani. In particular, we recover in this way transportation from Brunn–Minkowski inequalities and for the exponential measure.

We develop several applications of the Brunn—Minkowski inequality in the Prekopa—Leindler form. In particular, we show that an argument of B. Maurey may be adapted to deduce from the Prekopa—Leindler theorem the Brascamp—Lieb inequality for strictly convex potentials. We deduce similarly the logarithmic Sobolev inequality for uniformly convex potentials for which we deal more generally with arbitrary norms and obtain some new results in this context. Applications to transportation cost and to concentration on uniformly convex bodies complete the exposition.

We present a simple proof, based on modified logarithmic Sobolev inequalities, of Talagrand’s concentration inequality for the exponential distribution. We actually observe that every measure satisfying a Poincare inequality shares the same concentration phenomenon. We also discuss exponential integrability under Poincare inequalities and its consequence to sharp diameter upper bounds on spectral gaps.

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.

We give a simple proof of the Levy concentration of measure phenomenon on the sphere and on a compact Riemannian manifold with strictly positive Ricci curvature using the heat semigroup and Bochner's formula.

In the line of investigation of the works by D. Bakry and M. Emery (Lecture Notes in Math., Vol. 1123, pp. 175–206, Springer-Verlag, New York/Berlin 1985) and O. S. Rothaus ( J. Funct. Anal. 42 (1981) 102–109; J. Funct. Anal. 65 (1986) 358–367), we study an integral inequality behind the “Γ2 criterion” of D. Bakry and M. Emery (see previous reference) and its applications to hypercontractivity of diffusion semigroups. With, in particular, a short proof of the hypercontractivity property of the Ornstein-Uhlenbeck semigroup, our exposition unifies in a simple way several previous results, interpolating smoothly from the spectral gap inequalities to logarithmic Sobolev inequalities and even true Sobolev inequalities. We examine simultaneously the extremal functions for hypercontractivity and logarithmic Sobolev inequalities of the Ornstein-Uhlenbeck semigroup and heat semigroup on spheres.

We present a simple direct proof of the classical Sobolev inequality in \(\mathbb{R}^n\) with best constant from the geometric Brunn–Minkowski–Lusternik inequality.

Perturbations of functional inequalities are studied by using merely growth conditions in terms of a distance-like reference function. As a result, optimal sufficient conditions are obtained for perturbations to reach a class of functional inequalities interpolating between the Poincaré inequality and the logarithmic Sobolev inequality.

Processus de Levy: R.A. Doney: Tanaka's construction for random walks and Levy processes.- R.A. Doney: Some excursion calculations for spectrally one-sided Levy processes.- A.E. Kyprianou, Z. Palmowski: A martingale review of some fluctuation theory for spectrally megative Levy processes.- M.R. Pistorius: A potential-theoretical review of some exit problems of spectrally negative Levy processes.- L. Nguyen-Ngoc, M. Yor: Some martingales associated to reflected Levy processes.- K.B. Erickson, R.A. Maller: Generalised Ornstein-Uhlenbeck processes and the convergence of Levy integrals.- Autres Exposes: P. Fougeres: Spectral gap for log-concave probability measures on the real line.- L. Godefroy: Propriete de Choquet-Deny et fonctions harmoniques sur les hypergroupes commutatifs.- M. Buiculescu: Exponential decay parameters associated with excessive measures.- V. Grecca: Positive bilinear mappings associated with stochastic processes.- A. Jakubowski: An almost sure approximation for the predictable process in the Doob-Meyer decomposition theorem.- A. Cherny, A. Shiryaev: On stochastic integrals up to infinity and predictable criteria for integrability.- Y. Kabanov, C. Stricker: Remarks on the true no-arbitrage property.- H. Buhler: Information-equivalence: On filtrations created by independent increments.- M. Zakai: Rotations and tangent processes on Wiener space.- I. Shigekawa: Lp multiplier theorem for the Hodge-Kodaira operator.- G. Peccati, C.A. Tudor: Gaussian limits for vector-valued multiple stochastic integrals.- J. Rosen: Derivatives of self-intersection local times.- N. Eisenbaum, C. A. Tudor: On squared fractional Brownian motions.- A. Ayache et al: Regularity and identification of generalised miltifractional Gaussian processes.- F. Benaych-Georges: Failure of the Raikov theorem for free random variables.- G. Aubrun: Aharp small deviation inequality for the largest eigenvalue of a randommatrix.- F. Baudoin: The tangent space to a hypoelliptic diffusion and applications.- A. Bencherif-Madani, E. Pardoux: Homogenization of a diffusion with locally periodic coefficients.

We examine the Charlier, Meixner, Krawtchouk and Hahn discrete orthogonal polynomial ensembles, deeply investigated by K. Johansson, using integration by parts for the underlying Markov operators, differential equations on Laplace transforms and moment equations. As for the matrix ensembles, equilibrium measures are described as limits of empirical spectral distributions. In particular, a new description of the equilibrium measures as adapted mixtures of the universal arcsine law with an independent uniform distribution is emphasized. Factorial moment identities on mean spectral measures may be used towards small deviation inequalities on the rightmost charges at the rate given by the Tracy-Widom asymptotics.

Following the investigation by U. Haagerup and S. Thorbjornsen, we present a simple differential approach to the limit theorems for empirical spectral distributions of complex random matrices from the Gaussian, Laguerre and Jacobi Unitary Ensembles. In the framework of abstract Markov diffusion operators, we derive by the integration by parts formula differential equations for Laplace transforms and recurrence equations for moments of eigenfunction measures. In particular, a new description of the equilibrium measures as adapted mixtures of the universal arcsine law with an independent uniform distribution is emphasized. The moment recurrence relations are used to describe sharp, non asymptotic, small deviation inequalities on the largest eigenvalues at the rate given by the Tracy-Widom asymptotics.

Cours specialises et exposes thematiques: A. Guionnet, B. Zegarlinski: Lectures on Logarithmic Sobolev inequalities.- A. Lejay, L. Pastur: Matrices aleatoires : statistique asymptotique des valeurs propres.- N. O'Connell: Random matrices, non-colliding particle systems and queues.- Exposes: A. Dermoune, O. Moutsinga: Generalized variational principles.- D. Chafai: Gaussian maximum of entropy and reversed log-Sobolev inequality.- L. Miclo: About projections of logarithmic Sobolev inequalities.- L. Miclo: Sur l'inegalite de Sobolev logarithmique des operateurs de Laguerre a petit parametre.- A. Bentaleb: Sur les fonctions extremales des inegalites de Sobolev des operateurs de diffusion.- C. Donati-Martin, Y. Hu: Penalization of the Wiener measure and principal values.- C. Leuridan: Theoreme de Ray-Knight dans un arbre : Une approche algebrique.- R. Bass: Stochastic differential equations driven by symmetric stable processes.- T. Simon: Support d'une equation d' Ito avec sauts en dimension 1.- N. Eisenbaum: A Gaussian sheet connected to symmetric Markov chains.- C. Leuridan: Filtration d'une marche aleatoire stationnaire sur le cercle.- S. Beghdadi-Sakrani: Une martingale non pure, dont la filtration est brownienne.- J. Hannig: On filtrations related to purely discontinuous martingales.- S. Beghdadi-Sakrani: Calcul stochastique pour des mesures signees.- J. Jacod: On processes with conditional independent increments and stable convergence in law.- V. Grecea: Duality and quasi-continuity for supermartingales.- Y. Kabanov, C. Stricker: On the true submartingale property, d'apres Schachermayer.- C. Stricker: Simple strategies in exponential utility maximization.- M. Arnaudon, A. Thalmaier: Horizontal martingales in vector bundles.- D. Kurtz: Representation nucleaire des martingales d'Azema.- S. Attal: Approximating the Fock space with the toy Fock space.- Corrections auxvolumes anterieurs.

We analyze recent proofs of decay of correlations and logarithmic Sobolev inequalities for unbounded spin systems in the perturbative regime developed by B. Zegarlinski, N. Yoshida, B. Helffer, Th. Bodineau. We investigate to this task a simple analytic model. Proofs are short and self-contained.

International audience; Some Gaussian functional inequalities have simple generalizations to some Gaussianlike cases. They allow us to establish Gaussian-like Large Deviations Principles and bounds via Gaussian concentration and shift inequalities for certain families of Boltzmann measures and laws of diffusion semigroups in short time. Beyond the results themselves, we would like to emphasize here the method and the symmetry of the arguments used for upper and lower bounds by means of the functional inequalities.

We show how the Clark-Ocone-Haussmann formula for Brownian motion on a compact Riemannian manifold put forward by S. Fang in his proof of the spectral gap inequality for the Ornstein-Uhlenbeck operator on the path space can yield in a very simple way the logarithmic Sobolev inequality on the same space. By an appropriate integration by parts formula the proof also yields in the same way a logarithmic Sobolev inequality for the path space equipped with a general diffusion measure as long as the torsion of the corresponding Riemannian connection satisfies Driver's total antisymmetry condition.

In this first chapter, we present the isoperimetric inequalities which now appear as the crucial concept in the understanding of various concentration inequalities, tail behaviors and integrability theorems in Probability in Banach spaces. These inequalities often arise as the final and most elaborate forms of previous, weaker (but already efficient) inequalities which will be mentioned in their framework throughout the book. In these final forms however, the isoperimetric inequalities and associated concentration of measure phenomena provide the appropriate ideas for an in depth comprehension of some of the most important theorems of the theory.

This chapter collects in rather an informal way some basic facts about processes and infinite dimensional random variables. The material that we present actually only appears as the necessary background for the subsequent analysis developed in the next chapters. Only a few proofs are given and many important results are only just mentioned or even omitted. It is therefore recommended to complement, if necessary, these partial bases with the classical references, some of which are given at the end of the chapter.

This paper studies an inequality of H. P. Rosenthal for vector valued random variables, its relations with some geometric properties of Banach spaces and its applications to the study of the central limit theorem in Banach spaces.

We use the continuity theorem of Lyons for rough paths in the p-variation topology to produce an elementary approach to the large deviation principle and the support theorem for di$usion processes. The proofs reduce to establish the corresponding results for Brownian motion itself as a rough path in the p-variation topology, 2 ipi 3, and the technical step is to handle the L4 evy area in this respect. Some extensions and applications are discussed. c � 2002 Published by Elsevier Science B.V.

We show that for any positive functionfon the discrete cube {0,1}n,Entμnp(f)⩽pqEμnp1f|Df|2whereμnpis the product measure of the Bernoulli measure with probability of successp, as well as related inequalities, which may be shown to imply in the limit the classical Gaussian logarithmic Sobolev inequality as well as a logarithmic Sobolev inequality for Poisson measure. We further investigate modified logarithmic Sobolev inequalities to analyze integrability properties of Lipschitz functions on discrete spaces. In particular, we obtain, under modified logarithmic Sobolev inequalities, some concentration results for product measures that extend the classical exponential inequalities for sums of independent random variables.