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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 establish modified logarithmic Sobolev inequalities for the path distributions of some continuous time random walks on graphs, including the simple examples of the discrete cube and the lattice ZZ d . Our approach is based on the Malliavin calculus on Poisson spaces developed by J. Picard and stochastic calculus. The inequalities we prove are well adapted to describe the tail behaviour of various functionals such as the graph distance in this setting.

This paper describes some of the analytic tools developed recently by Ghirlanda and Guerra in the investigation of the distribution of overlaps in the Sherrington–Kirkpatrick spin glass model and of Parisi's ultrametricity. In particular, we introduce to this task a simplified (but also generalized) model on which the Gaussian analysis is made easier. Moments of the Hamiltonian and derivatives of the free energy are expressed as polynomials of the overlaps. Under the essential tool of self-averaging, we describe with full rigour, various overlap identities and replica independence that actually hold in a rather large generality. The results are presented in a language accessible to probabilists and analysts.

— Let M be a complete n-dimensional Riemanian manifold with non-negative Ricci curvature in which one of the Sobolev inequalities (∫ |f |dv )1/p ≤ C(∫ |∇f |qdv)1/q, f ∈ C∞ 0 (M), 1 ≤ q 1. Moreover, for q > 1, the equality in (1) is attained by the functions (λ + |x|q/(q−1))1−(n/q), λ > 0, where |x| is the Euclidean length of the vector x in IR. We are actually interested here in the geometry of those manifolds M for which one of the Sobolev inequalities (1) is satisfied with the best constant C = K(n, q) of IR. The result of this note is the following theorem. Theorem. Let M be a complete n-dimensional Riemannian manifold with nonnegative Ricci curvature. If one of the Sobolev inequalities (1) is satisfied with C = K(n, q), then M is isometric to IR. The particular case q = 1 (p = n/(n − 1)) is of course well-known. In this case indeed, the Sobolev inequality is equivalent to the isoperimetric inequality ( voln(Ω) )(n−1)/n ≤ K(n, 1)voln−1(∂Ω) where ∂Ω is the boundary of a smooth bounded open set Ω in M . If we let V (x0, s) = V (s) be the volume of the geodesic ball B(x0, s) = B(s) with center x0 and radius s in M , we have d ds voln ( B(s) ) = voln−1 ( ∂B(s) ) . Hence, setting Ω = B(s) in the isoperimetric inequality, we get V (s)(n−1)/n ≤ K(n, 1)V ′(s) for all s. Integrating yields V (s) ≥ (nK(n, 1))−nsn, and since K(n, 1) = n−1ω n , for every s, (2) V (s) ≥ V0(s) where V0(s) = ωns is the volume of the Euclidean ball of radius s in IR. IfM has nonnegative Ricci curvature, by Bishop’s comparison theorem (cf. e.g. [Ch]) V (s) ≤ V0(s) for every s, and by (2) and the case of equality,M is isometric to IR. The main interest of the Theorem therefore lies in the case q > 1. As usual, the classical value q = 2 (and p = 2n/(n − 2)) is of particular interest (see below). It should be noticed that known results already imply that the scalar curvature of M is zero in this case (cf. [He], Prop. 4.10). Proof of the Theorem. It is inspired by the technique developed in the recent work [B-L] where a sharp bound on the diameter of a compact Riemannian manifold satisfying a Sobolev inequality is obtained, extending the classical Myers theorem. We thus assume that the Sobolev inequality (1) is satisfied with C = K(n, q) for some q > 1. Recall first that the extremal functions of this inequality in IR are the functions (λ + |x|q)1−(n/q), λ > 0, where q′ = q/(q − 1). Let now x0 be a fixed point in M and let θ > 1. Set f = θ−1d(·, x0) where d is the distance function on M . The idea is then to apply the Sobolev inequality (1), with C = K(n, q), to (λ+ f ′ )1−(n/q), for every λ > 0 to deduce a differential inequality whose solutions may be compared to the extremal Euclidean case. Set, for every λ > 0, F (λ) = 1 n− 1 ∫ 1 (λ+ fq)n−1 dv. Note first that F is well defined and continuously differentiable in λ. Indeed, by Fubini’s theorem, for every λ > 0

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.

International audience; We establish an improved form of the classical logarithmic Sobolev inequality for the Gaussian measure restricted to probability densities which satisfy a Poincar\'e inequality. The result implies a lower bound on the deficit in terms of the quadratic Kantorovich-Wasserstein distance. We similarly investigate the deficit in the Talagrand quadratic transportation cost inequality this time by means of an ${\rm L}^1$-Kantorovich-Wasserstein distance, optimal for product measures, and deduce a lower bound on the deficit in the logarithmic Sobolev inequality in terms of this metric. Applications are given in the context of the Bakry-\'Emery theory and the coherent state transform. The proofs combine tools from semigroup and heat kernel theory and optimal mass transportation.

E. Mossel and J. Neeman recently provided a heat flow monotonicity proof of Borell’s noise stability theorem. In this note, we develop the argument to include in a common framework noise stability, Brascamp-Lieb inequalities (including hypercontractivity), and even a weak form of Slepian inequalities. The scheme applies furthermore to families of measures with are more log-concave than the Gaussian measure.

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.

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 present a new and simple approach to some of the deviation inequalities for product measures deeply investigated by M. Talagrand in the recent years. Our method is based on functional inequalities of Poincare and logarithmic Sobolev type and iteration of these inequalities. In particular, we establish with theses tools sharp deviation inequalities from the mean on norms of sums of independent random vectors and empirical processes. Concentration for the Hamming distance may also be deduced from this approach.

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.

International audience; We provide yet another proof of the Otto-Villani theorem from the log Sobolev inequality to the Talagrand transportation cost inequality valid in arbitrary metric measure spaces. The argument relies on the recent development [2] identifying gradient flows in Hilbert space and in Wassertein space, emphasizing one key step as precisely the root of the Otto-Villani theorem. The approach does not require the doubling property or the validity of the local Poincare' inequality.

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.

In this paper we discuss the natural candidate for the one dimensional free Poincar\'e inequality. Two main strong points sustain this candidacy. One is the random matrix heuristic and the other the relations with the other free functional inequalities, namely, the free transportation and Log-Sobolev inequalities. As in the classical case the Poincar\'e is implied by the others. This investigation is driven by a nice lemma of Haagerup which relates logarithmic potentials and Chebyshev polynomials. The Poincar\'e inequality revolves around the counting number operator for the Chebyshev polynomials of first kind with respect to the arcsine law on $[-2,2]$. This counting number operator appears naturally in a representation of the minimum of the logarithmic potential with external fields as well as in the perturbation of logarithmic energy with external fields, which is the essential connection between all these inequalities., Comment: Some things were corrected and a new proposition added. This will appear in Transactions of AM

We define a natural Lie algebra structure associated to a Markov generator which extends the "carre du champ" operator and its iterated gradients. While the iterated gradients may be used in expansions ofthe variance of a function of the demain, this more general structure is needed in formai expansions of the entropy of a function.

We analyze from the viewpoint of an abstract Markov operator recent results by Nualart and Peccati, and Nourdin and Peccati, on the fourth moment as a condition on a Wiener chaos to have a distribution close to Gaussian. In particular, we are led to introduce a notion of chaos associated to a Markov operator through its iterated gradients and present conditions on the (pure) point spectrum for a sequence of chaos eigenfunctions to converge to a Gaussian distribution. Convergence to gamma distributions may be examined similarly., Comment: Published in at http://dx.doi.org/10.1214/11-AOP685 the Annals of Probability (http://www.imstat.org/aop/) by the Institute of Mathematical Statistics (http://www.imstat.org)

We survey several Talagrand type inequalities and their application to influences with the tool of hypercontractivity for both discrete and continuous, and product and non-product models. The approach covers similarly by a simple interpolation the framework of geometric influences recently developed by N. Keller, E. Mossel and A. Sen. Geometric Brascamp-Lieb decompositions are also considered in this context.

Si M est une vari6t6 riemannienne complete non compacte, il est maintenant bien connu que l'in~galit6 isop~rim6trique I AI (N1)/N 0 , off Pt est le noyau de la chaleur sur M ([17], voir aussi [6]). Une r6ciproque partielle ~ ce r6sultat est annonc~e dans [20]: si M est ~ g~om6trie born6e (par exemple courbure de Ricci minor6e et ~ rayon d'injectivit~ positif), et si sup~,ycMPt(X , y )= O(t--N/2), t--+~-O0 (dans cette situation on a de toute fa~on sup~,yeMPt(X,y)= O(t--(dimM)/2), t-*O), alors IAI(N'-I)/N'

We present a simple analytic proof of the inequality of P. Buser showing the equivalence of the first eigenvalue of a compact Riemannian manifold without boundary and Cheeger's isoperimetric constant under a lower bound on the Ricci curvature. Our tools are the Li-Yau inequality and ideas of Varopoulos in his functional approach to isoperimetric inequalities and heat kernel estimates on groups and manifolds. The method is easily modified to yield a logarithmic isoperimetric inequality involving the hypercontractivity constant of the manifold. 1. BUSER'S INEQUALITY Throughout this paper, M will denote a compact Riemannian manifold without boundary of dimension n. We denote by ,u the normalised Riemannian measure on M, by A the Laplace operator, and by Vf the gradient of a smooth function f on M with Riemannian length lVfl. The first nontrivial eigenvalue Al of the Laplacian is characterised via the min-max theorem by the Poincare type inequality AlJ 2dl? 0, for all f with f f da = 0, where 11 . I[P is the LP-norm (1 0, is the heat semigroup on M. In 1970, Cheeger [C] introduced an isoperimetric constant to bound below the first eigenvalue Al . Set h = inf a((A) u (A) where the infimum runs over all open subsets A with ,u(A) < I and smooth boundary AA, and where a(.) denotes the (n 1)-dimensional measure. Received by the editors October 14, 1992. 1991 Mathematics Subject Classification. Primary 58G1 1, 58G25, 53C99, 49Q15. ( 1994 American Mathematical Society 0002-9939/94 $1.00 + $.25 per page

In a remarkable series of works, E. Milman recently showed how to reverse the usual hierarchy, and deduce from measure concentration inequali- ties, dimension free isoperimetric type inequalities in spaces with non-negative (Ricci) curvature. The results cover two basic instances, linear isoperimetry under arbitrarily slow concentration, and logarithmic strengthenings above the linear case under exponential concentration. The proofs are developed in a Rie- mannian (with density) context making use of isoperimetric minimizers and refined tools from geometric measure theory. In this note, we present simple semigroup arguments to cover the super-linear case, of potential usefulness in more general settings. A particular emphasis is put on functional inequalities for heat kernel measures.

International audience; This paper builds upon several recent works, where semigroup proofs of Brascamp-Lieb inequalities are provided in various settings (Euclidean space, spheres, and symmetric groups). Our aim is two-fold. Firstly, we provide a general, unifying, framework based on Markov generators, in order to cover a variety of examples of interest going beyond previous investigations. Secondly, we put forward the combinatorial reasons for which unexpected exponents occur in these inequalities. Related superadditivity of information and entropy inequalities are also studied.

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.

International audience; We establish various small deviation inequalities for the extremal (soft edge) eigenvalues in the beta-Hermite and beta-Laguerre ensembles. In both settings, upper bounds on the variance of the largest eigenvalue of the anticipated order follow immediately.

Ce travail presente un analogue de la relation de recurrence de Harer et Zagier pour les moments de l’Ensemble Orthogonal Gaussien sous la forme d’une recurrence a cinq termes. La demonstration s’appuie sur des integrations par parties gaussiennes et des equations differentielles sur les transformees de Laplace. Une relation similaire est etablie pour l’Ensemble Symplectique Gaussien. Comme dans le cas complexe, cette relation s’interprete comme une formule de recurrence pour le nombre de cartes enracinees a nombre de faces et de cotes donne plongees dans des surfaces localement orientees. Cette relation de recurrence sur les moments fournit egalement une borne sur la loi de la plus grande valeur propre de l’Ensemble Orthogonal Gaussien et, par comparaison de moments, de familles de matrices de Wigner.

Brascamp--Lieb-type, weighted Poincar\'{e}-type and related analytic inequalities are studied for multidimensional Cauchy distributions and more general $\kappa$-concave probability measures (in the hierarchy of convex measures). In analogy with the limiting (infinite-dimensional log-concave) Gaussian model, the weighted inequalities fully describe the measure concentration and large deviation properties of this family of measures. Cheeger-type isoperimetric inequalities are investigated similarly, giving rise to a common weight in the class of concave probability measures under consideration., Comment: Published in at http://dx.doi.org/10.1214/08-AOP407 the Annals of Probability (http://www.imstat.org/aop/) by the Institute of Mathematical Statistics (http://www.imstat.org)

The goal of this note is to show that some convolution type inequalities from Harmonic Analysis and Information Theory, such as Young's convolution inequality (with sharp constant), Nelson's hypercontractivity of the Hermite semi-group or Shannon's inequality, can be reduced to a simple geometric study of frames of $\R^2$. We shall derive directly entropic inequalities, which were recently proved to be dual to the Brascamp-Lieb convolution type inequalities.

Let M be an arbitrary Hermitian matrix of order n, and k be a positive integer less than or equal to n. We show that if k is large, the distribution of eigenvalues on the real line is almost the same for almost all principal submatrices of M of order k. The proof uses results about random walks on symmetric groups and concentration of measure. In a similar way, we also show that almost all k x n submatrices of M have almost the same distribution of singular values., 6 pages

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.

Dispersion of monosized drops downstream a point injection in a grid-generated turbulence is studied. Influence of extra bodyforces is also investigated by use of ferrofluid drops and magnetic field. Datas are obtained through LDV and given for fluid and particles mean and fluctuant velocities. Presence probability repartition for particles downstream the injection is obtained by LDV counting.

A 12-month-old boy with 25% of total body surface area (TBSA) burned, i.e. with a burn sized 950 cm or 30 cm 31.5 cm involving the lower extremities and buttocks, was admitted to our burns unit (Figs. 1 and 2). The burn was the result of negligence. Past history and family history were unremarkable. The child showed both psychomotor and growth retardation, with low percentages for height (71 cm) and weight (7 kg), attributed to his poor environmental situation. Because of an initial delay in resuscitation, the boy presented with a picture of moderate shock for the first 48 h after burn. Treatment with oxacillin was started prophylactically on post-burn day (PBD) 2. On PBD 3, he vomited a small

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.

In this work, we study concentration properties for vector valued maps. In particular, we describe inequalities which capture the exact dimensional behavior of Lipschitz maps with values in Rk. To this task, we study in particular a domination principle for projections which might be of independent interest. We further compare our conclusions to earlier results by Pinelis in the Gaussian case, and discuss extensions to the infinite dimensional setting.

In these notes, we survey developments on the asymptotic behavior of the largest eigenvalues of random matrix and random growth models, and describe the corresponding known non-asymptotic exponential bounds. We then discuss some elementary and accessible tools from measure concentration and functional analysis to reach some of these quantitative inequalities at the correct small deviation rate of the fluctuation theorems. Results in this direction are rather fragmentary. For simplicity, we mostly restrict ourselves to Gaussian models.