Characterization of Talagrand's transport-entropy inequalities in metric spaces

We give a characterization of transport-entropy inequalities in metric spaces. As an application we deduce that such inequalities are stable under bounded perturbation (Holley-Stroock perturbation Lemma)

Bounds on the deficit in the logarithmic Sobolev inequality

The de cit in the logarithmic Sobolev inequality for the Gaussian measure is considered and estimated by means of transport and information-theoretic distances

Criteria for entropic curvature on graph spaces

This paper presents local criteria for lower bounds on entropic curvature of graph spaces along Schr\"odinger bridges at zero temperature, according to the definition given by the second named author in [31], in the continuity of the work by C. L\'eonard [19] inspired by the Lott-Sturm-Villani theory. A graph space is defined as a quadruple $(\mathcal{X},d,L,m)$ where $\mathcal{X}$ is the set of vertices, $d$ is the combinatorial distance and $m$ is a reversible reference measure with respect to a generator $L$ of a Markov semi-group on $\mathcal{X}$. The criteria are given by local optimization problems on balls of radius two, depending only on the generator $L$ and on the discrete structure of these balls. General tensorization properties of the criteria are presented for the study of the Cartesian product of graphs. This approach is robust since it applies to a wide range of graph spaces and also for any measure $m$, including measures with interaction potential like Ising models. We introduce a very large class of structured graphs, for which the local criteria give non-negative entropic curvature for the uniform measure. A Bonnet-Myers type of theorem also ensures that any such graph with positive entropic curvature is finite. On any graph space, positive entropic curvature provides transport-entropy inequalities with so-called weak optimal transport costs, as well as Poincar\'e or modified log-Sobolev inequalities for the renormalized probability measure $\mu = m/m(\mathcal{X})$. These inequalities are related to refined concentration properties of the measure $\mu$, speed of convergence of semi-groups to the measure $\mu$. We also present examples of graphs with negative curvature. Some comparisons of our results with other notions of curvature are established, such as Bakry-Emery curvature conditions [7,8], Ollivier or Lin-Lu-Yau's curvature [28,20].Comment: 62 pages, 2 figure

Kantorovich duality for general transport costs and applications

We introduce a general notion of transport cost that encompasses many costs used in the literature (including the classical one and weak transport costs introduced by Talagrand and Marton in the 90's), and prove a Kantorovich type duality theorem. As a by-product we obtain various applications in different directions: we give a short proof of a result by Strassen on the existence of a martingale with given marginals, we characterize the associated transport-entropy inequalities together with the log-Sobolev inequality restricted to convex/concave functions. Some explicit examples of discrete measures satisfying weak transport-entropy inequalities are also given

A new characterization of Talagrand's transport-entropy inequalities and applications

We show that Talagrand's transport inequality is equivalent to a restricted logarithmic Sobolev inequality. This result clarifies the links between these two important functional inequalities. As an application, we give the first proof of the fact that Talagrand's inequality is stable under bounded perturbations.Comment: Published in at http://dx.doi.org/10.1214/10-AOP570 the Annals of Probability (http://www.imstat.org/aop/) by the Institute of Mathematical Statistics (http://www.imstat.org

Phénomènes de concentration en grande dimension, transport de mesure et inégalités fonctionnelles.

My research is focused on functional inequalities related to the concentration of measure phenomenon, in particular the transport-entropy inequalities, also called transport inequalities. The chapters of this document are articulated around this inequality and its relation with other functional inequalities. We present an abstract general version of the measure concentration principle in order to unify the different results of my research. Moreover some elementary arguments are given for a better understanding of the mathematical tools. The first chapter introduces the concentration of measure principle. It's general setting allows to compare many results depending on the involved cost function. The second chapter is devoted to the so-called weak transport inequality related to the general concentration of measure principle. We also present the links between logarithmic Sobolev inequalities and transport inequalities in metric spaces. The third chapter develops different results around the so-called barycentric costs, the notion of convex order on probability measures and weak-transport inequalities associated to this barycentric cost.Chapter IV is based on examples of ``universal'' transport inequalities as the so-called Csizár-Kullback-Pinsker inequality and, by tensorization, their applications to Bernstein type of deviation inequalities for suprema of empirical processes. We also present recent transport inequalities obtained for the uniform law on the symmetric group obtained by using different tensorization arguments. Chapter V is about the Poincaré inequality and its characterization in terms of a weak dimension-free concentration principles.Chapter VI concerns the notion of curvature on metric spaces and functional inequalities as the so-called displacement convexity property.We explain how a concentration property may imply a logarithmic Sobolev inequality or a Poincaré inequality when this convexity property holds on a geodesic space. In another direction, we try to extend the notion of curvature to discrete spaces by considering a similar convexity property of entropy on discrete spaces. The last chapter is devoted to improved versions of the logarithmic Sobolev inequality and the HWI inequality for the standard Gaussian measure on $\R^n$

Effects of reflux laryngitis on laryngeal chemoreflexes in newborn lambs

It has been suggested that reflux laryngitis (RL) is involved in apneas-bradycardias of the newborn. The aim of the present study was to develop a unique RL model in newborn lambs to test the hypothesis that RL enhances the cardiorespiratory components of the laryngeal chemoreflexes (LCR) in the neonatal period. Gastric juice surrogate (2 ml of normal saline solution with HCl pH 2 + pepsin 300 U/ml) (RL group, n = 6) or normal saline (control group, n = 6) was repeatedly injected onto the posterior aspect of the larynx, 3 times a day for 6 consecutive days, via a retrograde catheter introduced into the cervical esophagus. Lambs instilled with gastric juice surrogate presented clinical signs of RL, as well as moderate laryngitis on histological observation. Laryngeal chemoreflexes were thereafter induced during sleep by injection of 0.5 ml of HCl (pH 2), ewe's milk, distilled water or saline into the laryngeal vestibule via a chronic, transcutaneous supraglottal catheter. Overall, RL led to a significantly greater respiratory inhibition compared with the control group during LCR, including longer apnea duration (P = 0.01), lower minimal respiratory rate (P = 0.002), and a more prominent decrease in arterial hemoglobin saturation (SpO(2)) (P = 0.03). No effects were observed on cardiac variables. In conclusion, 1) our unique neonatal ovine model presents clinical and histological characteristics of RL; and 2) the presence of RL in newborn lambs increases the respiratory inhibition observed with LCR, at times leading to severe apneas and desaturations

Effects of moderate hyperbilirubinemia on nutritive swallowing and swallowing-breathing coordination in preterm lambs

Background: Hyperbilirubinemia (HB) occurs in 90% of preterm newborns. HB induces acute neurological disorders (somnolence, abnormal tone, feeding difficulties, auditory dysfunction) and alterations in respiratory control. These findings suggest brainstem neurotoxicity that could also affect swallowing centers. Objective: To test the hypothesis that HB impairs nutritive swallowing (NS) and swallowing-breathing coordination. Methods: Two groups of preterm lambs (born 14 days prior to term), namely control (n = 6) and HB (n = 5), were studied. On day 5 of life (D0), moderate HB (150-250 µmol/l) was induced during 17 h in the HB group. Swallowing was assessed via recording of pharyngeal pressure and respiration by respiratory inductance plethysmography and pulse oximetry. The effect of HB on NS was assessed during standardized bottle-feeding. A second recording was performed 48 h after recovery from HB (D3). Results: Swallows were less frequent (p = 0.003) and of smaller volume (p = 0.01) in HB lambs while swallowing frequency was decreased (p = 0.004). These differences disappeared after HB normalization. Swallowing-breathing coordination was impaired in HB lambs, with a decrease in percent time with NS burst-related apneas/hypopneas at D0 and D3. Simultaneously, HB lambs tended to experience more severe desaturations (<80%) during bottle-feeding. Finally, following bottle-feeding, the respiratory rate was significantly lower, along with an increased apnea duration in HB lambs. Conclusions: Swallowing and swallowing-breathing coordination are altered by acute moderate HB in preterm lambs. Decreased efficiency at bottle-feeding is accompanied by continuation of breathing during swallow bursts, which may promote lung aspiration