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We obtain estimates on the exponential rate of decay of the relative entropy from equilibrium for Markov processes with a non-local infinitesimal generator. We adapt some of the ideas coming from the Bakry-Emery approach to this setting. In particular, we obtain volume- independent lower bounds for the Glauber dynamics of interacting point particles and for various classes of hardcore models.

Ce document présente une synthèse des travaux menés sur les inégalités fonctionnelles et leurs liens avec le phénomène de concentration de la mesure.

International audience; In Avantaggiati and Loreti [Ric. Mat. 57(2):171-202, 2008] we studied the Cauchy problem for a class of Hamilton-Jacobi equations with initial data verifying the Lipschitz condition. In this paper we extend those results to the case in which the initial data are lower semicontinuous [in the following lsc], and are lower bounded and semiconvex. Here we prove hypercontractivity results and new Logarithm Sobolev Inequalities (shortly, LSI).

International audience; We describe conditions on non-gradient drift diffusion Fokker-Planck equations for its solutions to converge to equilibrium with a uniform exponential rate in Wasserstein distance. This asymptotic behaviour is related to a functional inequality, which links the distance with its dissipation and ensures a spectral gap in Wasserstein distance. We give practical criteria for this inequality and compare it to classical ones. The key point is to quantify the contribution of the diffusion term to the rate of convergence, which to our knowledge is a novelty.

We consider the Fast Diffusion Equation u t = Δ u m , m 1 , posed in a bounded smooth domain Ω ⊂ R d with homogeneous Dirichlet conditions. It is known that in the exponent range m s = ( d − 2 ) + / ( d + 2 ) m 1 all bounded positive solutions u ( t , x ) of such problem extinguish in a finite time T = T ( u ) , and also that such solutions approach a separate variable solution u ( t , x ) ∼ ( T − t ) 1 / ( 1 − m ) S ( x ) , as t → T − . Here, we are interested in describing the behaviour of the solutions near the extinction time in that range of exponents. We first show that the convergence v ( x , t ) = u ( t , x ) ( T − t ) − 1 / ( 1 − m ) to S ( x ) takes place uniformly in the relative error norm. Then, we study the question of rates of convergence of the rescaled flow, i.e., v → S . For m close to 1 we get such rates by means of entropy methods and weighted Poincare inequalities. The analysis of the latter point makes an essential use of fine properties of the associated stationary elliptic problem − Δ S m = c S in the limit m → 1 , and such a study has an independent interest.

We prove a stochastic formula for the Gaussian relative entropy in the spirit of Borell's formula for the Laplace transform. As an application, we give unified and short proofs of a number of functional inequalities., Comment: 18 pages

Given a birth-death process on $\mathbb {N}$ with semigroup $(P_t)_{t\geq0}$ and a discrete gradient ${\partial}_u$ depending on a positive weight $u$, we establish intertwining relations of the form ${\partial}_uP_t=Q_t\,{\partial}_u$, where $(Q_t)_{t\geq0}$ is the Feynman-Kac semigroup with potential $V_u$ of another birth-death process. We provide applications when $V_u$ is nonnegative and uniformly bounded from below, including Lipschitz contraction and Wasserstein curvature, various functional inequalities, and stochastic orderings. Our analysis is naturally connected to the previous works of Caputo-Dai Pra-Posta and of Chen on birth-death processes. The proofs are remarkably simple and rely on interpolation, commutation, and convexity., Comment: Published in at http://dx.doi.org/10.3150/12-BEJ433 the Bernoulli (http://isi.cbs.nl/bernoulli/) by the International Statistical Institute/Bernoulli Society (http://isi.cbs.nl/BS/bshome.htm)

We give a direct proof of a functional Santalo inequality due to Fradelizi and Meyer. This provides a new proof of the Blaschke-Santalo inequality. The argument combines a logarithmic form of the Prekopa-Leindler inequality and a partition theorem of Yao and Yao., Comment: 6 pages, file might be slightly different from the published version

In this thesis, I have studied the heat kernel, the heat semigroup and the associated functional inequalities on three model spaces in subelliptic geometry. The main interest of this study is in fact to develop new methods and new technics that we hope we can extend in subelliptic geometry. The real goal is to understand a notion of Ricci curvature bounded below by a constant in subelliptic geometry. Here, the three model spaces are 3-dimensional Lie groups: the Heisenberg group, the SU(2) group and the SL(2,R) group endowed with a sublaplacian: a left-invariant second order differential opertor, self-adjoint with respect to the Haar mesure of the group, which is not elliptic but hypoelliptic from Hörmander's results. My results first concern the setting of explicit formulas for the above heat kernels. Then I focus on the generalisation in subelliptic geometry of the notion of Ricci curvature bounded below by a constant. In this way, I give a generalized Bakry-Emery curvature-dimension criterion which under some antisymetrical conditions satisfied by our model spaces, implies some Li-Yau type estimates. I also study the setting and the consequences of subcommutation inequalities between the gradient and the semigroup. In particular, I give two new proofs of the H.Q.Li inequality on the Heisenberg group.; Dans cette thèse, j'ai étudié le noyau et le semi-groupe de la chaleur ainsi que les inégalités fonctionnelles associées sur trois espaces modèles de la géométrie sous-elliptique. Cette étude a en fait pour principal objectif de développer et tester de nouvelles techniques et méthodes que l'on espère ensuite pouvoir étendre en géométrie sous-elliptique. Le but avoué est de comprendre en géométrie sous-elliptique une notion de courbure de Ricci minorée par une constante. Ici, les trois espaces modèles sont des groupes de Lie de dimension 3: le groupe de Heisenberg, le groupe SU(2) et le groupe SL(2,R), que l'on munit d'un sous-laplacien: un opérateur différentiel du second ordre invariant à gauche essentiellement auto-adjoint pour la mesure de Haar du groupe qui n'est pas elliptique mais hypoelliptique d'après des résultats de Hörmander. Mes résultats portent tout d'abord sur l'obtention de formules explicites pour les noyaux de la chaleur associés. J'ai ensuite introduit un critère de courbure-dimension de Bakry-Emery généralisé qui, sous certaines conditions d'antisymétrie vérifiées sur nos espaces modèles, permet l'obtention d'estimées du type de Li-Yau. Je me suis enfin intéressé à l'établissement et l'étude d'inégalités de sous-commutation entre le gradient et le semi-groupe de la chaleur. J'ai notamment donné deux nouvelles démonstrations de l'inégalité de H.Q.Li sur le groupe de Heisenberg.

In this thesis, I have studied the heat kernel, the heat semigroup and the associated functional inequalities on three model spaces in subelliptic geometry. The main interest of this study is in fact to develop new methods and new technics that we hope we can extend in subelliptic geometry. The real goal is to understand a notion of Ricci curvature bounded below by a constant in subelliptic geometry. Here, the three model spaces are 3-dimensional Lie groups: the Heisenberg group, the SU(2) group and the SL(2,R) group endowed with a sublaplacian: a left-invariant second order differential opertor, self-adjoint with respect to the Haar mesure of the group, which is not elliptic but hypoelliptic from Hörmander's results. My results first concern the setting of explicit formulas for the above heat kernels. Then I focus on the generalisation in subelliptic geometry of the notion of Ricci curvature bounded below by a constant. In this way, I give a generalized Bakry-Emery curvature-dimension criterion which under some antisymetrical conditions satisfied by our model spaces, implies some Li-Yau type estimates. I also study the setting and the consequences of subcommutation inequalities between the gradient and the semigroup. In particular, I give two new proofs of the H.Q.Li inequality on the Heisenberg group.; Dans cette thèse, j'ai étudié le noyau et le semi-groupe de la chaleur ainsi que les inégalités fonctionnelles associées sur trois espaces modèles de la géométrie sous-elliptique. Cette étude a en fait pour principal objectif de développer et tester de nouvelles techniques et méthodes que l'on espère ensuite pouvoir étendre en géométrie sous-elliptique. Le but avoué est de comprendre en géométrie sous-elliptique une notion de courbure de Ricci minorée par une constante. Ici, les trois espaces modèles sont des groupes de Lie de dimension 3: le groupe de Heisenberg, le groupe SU(2) et le groupe SL(2,R), que l'on munit d'un sous-laplacien: un opérateur différentiel du second ordre invariant à gauche essentiellement auto-adjoint pour la mesure de Haar du groupe qui n'est pas elliptique mais hypoelliptique d'après des résultats de Hörmander. Mes résultats portent tout d'abord sur l'obtention de formules explicites pour les noyaux de la chaleur associés. J'ai ensuite introduit un critère de courbure-dimension de Bakry-Emery généralisé qui, sous certaines conditions d'antisymétrie vérifiées sur nos espaces modèles, permet l'obtention d'estimées du type de Li-Yau. Je me suis enfin intéressé à l'établissement et l'étude d'inégalités de sous-commutation entre le gradient et le semi-groupe de la chaleur. J'ai notamment donné deux nouvelles démonstrations de l'inégalité de H.Q.Li sur le groupe de Heisenberg.

In this paper we study Hopf-Lax formulas, hypercontractivity, ultracontractivity, logarithmic Sobolev inequalities for a class of first order Hamilton-Jacobi equations.

International audience; A periodic perturbation of a Gaussian measure modifies the sharp constants in Poincaré and logarithmic Sobolev inequalities in the homogenization limit, that is, when the period of a periodic perturbation converges to zero. We use variational techniques to determine the homogenized constants and get optimal convergence rates towards equilibrium of the solutions of the perturbed diffusion equations. The study of these sharp constants is motivated by the study of the stochastic Stokes' drift. It also applies to Brownian ratchets and molecular motors in biology. We first establish a transport phenomenon. Asymptotically, the center of mass of the solution moves with a constant velocity, which is determined by a doubly periodic problem. In the reference frame attached to the center of mass, the behaviour of the solution is governed at large scale by a diffusion with a modified diffusion coefficient. Using the homogenized logarithmic Sobolev inequality, we prove that the solution converges in self-similar variables attached to the center of mass to a stationary solution of a Fokker-Planck equation modulated by a periodic perturbation with fast oscillations, with an explicit rate. We also give an asymptotic expansion of the traveling diffusion front corresponding to the stochastic Stokes' drift with given potential flow.

The presented works concern three autonomous topics :(1) Biological models and Statistics : comparmental models, population pharmacokinetics and pharmacodynamics, estimators for stochastic inverse problems, nonlinear mixed effects models, mixture models, EM and ICF type algorithms, graphical covariance models, modelling in Cancerology, point processes and particles, queueing systems, Feynman-Kac formulas(2) Functional inequalities: Sobolev type inequalities, concentration of measure, isoperimetry, role of convexity in entropic inequalities, tensorization, heat kernels, Heisenberg group and hypoelliptic dynamics, queueing systems, mixtures of distributions (3) Random matrices: spectrum of random Markov matrices, graphs with random weights, Wigner, Marchenko-Pastur, and Girko-Bai type theorems, convergence of extremal eigenvalues, rank one deformations.The most frequent concept here is the notion of Markov dynamics. In the first part, the compartmental models give rise to such dynamics. The second part is related to the geometry and trend to equilibrium of Markov dynamics. The third part is devoted to random Markov dynamics. However, these three parts cannot be reduced to the study of certain aspects of Markov dynamics. The content ranges from concrete applications to abstract theoretical problems, and makes use of various concepts and techniques from Mathematical Analysis, Probability Theory and Statistics.; Les travaux présentés concernent trois thématiques autonomes :(1) Modèles biologiques et statistique : modèles compartimentaux, pharmacocinétique et pharmacodynamie de population, estimateurs pour problèmes inverses stochastiques, modèles non-linéaires à effets mixtes, modèles de mélanges, algorithmes de type EM et ICF, modèles graphiques de covariance, modélisation en cancérologie, processus ponctuels, particules, files d'attentes, renormalisation de processus markoviens inhomogènes et formules de Feynman-Kac(2) Inégalités fonctionnelles : inégalités de type Sobolev, concentration de la mesure, isopérimétrie rôle de la convexité dans les inégalités entropiques, tensorisation, noyau de la chaleur, groupe d'Heisenberg et dynamiques hypoelliptiques, files d'attentes, mélanges de lois (3) Matrices aléatoires : spectre des matrices markoviennes aléatoires, graphes à poids aléatoires, théorèmes de type Wigner, Marchenko-Pastur, et Girko-Bai, convergence des valeurs propres extrémales, déformations de rang un.Le concept le plus récurrent ici est celui de dynamique markovienne. Dans la première partie, ce sont les modèles à compartiments de la pharmacologie qui sont liés à de telles dynamiques. La seconde partie traite d'inégalités fonctionnelles associées à la vitesse et à la géométrie de dynamiques markoviennes. Enfin, la troisième partie traite de dynamiques markoviennes aléatoires. Ces trois parties ne se réduisent pas à l'étude de facettes de problèmes markoviens. Leur contenu balaye un spectre à la fois théorique et appliqué, et met en oeuvre des techniques et des concepts variés issus de l'analyse, des probabilités, et de la statistique.

These notes survey some aspects of discrete-time chaotic calculus and its applications, based on the chaos representation property for i.i.d. sequences of random variables. The topics covered include the Clark formula and predictable representation, anticipating calculus, covariance identities and functional inequalities (such as deviation and logarithmic Sobolev inequalities), and an application to option hedging in discrete time., Originally published in at http://dx.doi.org/10.1214/08-PS139 the Probability Surveys (http://www.i-journals.org/ps/) by the Institute of Mathematical Statistics (http://www.imstat.org) This version includes a revision of Section 9

Mixtures are convex combinations of laws. Despite this simple definition, a mixture can be far more subtle than its mixed components. For instance, mixing Gaussian laws may produce a potential with multiple deep wells. We study in the present work fine properties of mixtures with respect to concentration of measure and Sobolev type functional inequalities. We provide sharp Laplace bounds for Lipschitz functions in the case of generic mixtures, involving a transportation cost diameter of the mixed family. Additionally, our analysis of Sobolev type inequalities for two-component mixtures reveals natural relations with some kind of band isoperimetry and support constrained interpolation via mass transportation. We show that the Poincar\'e constant of a two-component mixture may remain bounded as the mixture proportion goes to 0 or 1 while the logarithmic Sobolev constant may surprisingly blow up. This counter-intuitive result is not reducible to support disconnections, and appears as a reminiscence of the variance-entropy comparison on the two-point space. As far as mixtures are concerned, the logarithmic Sobolev inequality is less stable than the Poincar\'e inequality and the sub-Gaussian concentration for Lipschitz functions. We illustrate our results on a gallery of concrete two-component mixtures. This work leads to many open questions., Comment: Corrections. To appear in Annales de l'Institut Henri Poincare (AIHP)

International audience; In a statistical mechanics model with unbounded spins, we prove uniqueness of the Gibbs measure under various assumptions on finite volume functional inequalities. We follow the approach of G. Royer (1999) and obtain uniqueness by showing convergence properties of a Glauber-Langevin dynamics. The result was known when the measures on the box $[-n,n]^d$ (with free boundary conditions) satisfied the same logarithmic Sobolev inequality. We generalize this in two directions: either the constants may be allowed to grow sub-linearly in the diameter, or we may suppose a weaker inequality than log-Sobolev, but stronger than Poincaré. We conclude by giving a heuristic argument showing that this could be the right inequalities to look at.

Via Phi-Sobolev inequalities, we give some sharp integrability conditions on $F$ for the large deviation principle of the empirical mean $\frac{1}{T}{\int_{0}^{T}{F(X_{s})}ds}$ for large time $T$, where $F$ is unbounded with values in some separable Banach space. Several examples are provided.

It is known that the couple formed by the two dimensional Brownian motion and its L\'evy area leads to the heat kernel on the Heisenberg group, which is one of the simplest sub-Riemannian space. The associated diffusion operator is hypoelliptic but not elliptic, which makes difficult the derivation of functional inequalities for the heat kernel. However, Driver and Melcher and more recently H.-Q. Li have obtained useful gradient bounds for the heat kernel on the Heisenberg group. We provide in this paper simple proofs of these bounds, and explore their consequences in terms of functional inequalities, including Cheeger and Bobkov type isoperimetric inequalities for the heat kernel., Comment: Minor corrections

International audience; This note is concerned with intermediate inequalities which interpolate between the logarithmic Sobolev and the Poincaré inequalities. For such generalized Poincaré inequalities we improve upon the known constants from the literature. Cette note est consacrée à des inégalités intermédiaires qui interpolent entre les inégalités de Sobolev logarithmiques et les inégalités de Poincaré. Pour de telles inégalités de Poincaré généralisées, nous améliorons les constantes données dans la littérature.

Let P-n(+) denote the set of all n x n nonnegative matrices. For a function f : R-+(m) -> R+ and matrices [GRAPHICS] define [GRAPHICS] For each A epsilon P-n(+) we denote its spectral radius by rho(A) and its max eigenvalue by mu(A). In a previous paper, all functions f which satisfy rho(f(A(1),...,A(m))) equal to or less than f(rho(A(1)),...,rho(A(m))), ____n epsilon N, ___(sic)A(1),...,A(m) epsilon P-n(+) and some functions which satisfy f(rho(A(1)),...,rho(A(m))) equal to or less than rho(f(A(1),...,A(m))), ____n epsilon N, ___(sic)A(1),...,A(m) epsilon P-n(+) were characterized. Here, for an interval I in R+, we characterize those functions f satisfying mu(f(A(1),...,A(m))) equal to or less than f(mu(A(1)),...,mu(A(m))), ____n epsilon N, ___(sic)A(1),...,A(m) epsilon I-nn as well as the functions satisfying f(0) = 0 and f(mu(A(1)),...,mu(A(m))) equal to or less than mu(f(A(1),...,A(m))), ____n epsilon N, ___(sic)A(1),...,A(m) epsilon I-nn (c) 2007 Elsevier Inc. All rights reserved.

In this work, we apply several functional inequalities (Poincaré,logarithmic Sobolev etc.) to solve two problems. First, we study aninhomogeneous diffusion, akin to the discrete simulated annealingalgorithm, in the spirit of a paper by L. Miclo. We show that thisdiffusion converges under weaker hypotheses than what was assumed inprevious work. In particular, the potential governing the drift isallowed to grow very slowly at infinity. In a second part, we turn toa model of statistical mechanics with unbounded spins, recentlystudied by T. Bodineau and B. Helffer, N. Yoshida and G. Royer (amongothers). We clarify links between mixing properties, uniqueness of theGibbs measure, and functional inequalities. We show in particular thatthe infinite volume tempered Gibbs measure is unique, provided thatthe finite volume measures for one boundary condition satisfy aBeckner inequality.; Dans cette thèse, nous utilisons différentes inégalités fonctionnelles(Poincaré, Sobolev logarithmique, etc.) pour étudier deux questions.Nous appliquons d'abord des inégalités affaiblies à l'étude d'unediffusion inhomogène, analogue continu de l'algorithme de recuitsimulé, dans la lignée d'un travail de L. Miclo. Nous montrons unrésultat de convergence de la diffusion, sous des hypothèses plusfaibles que celles posées précédemment : le potentiel dans lequel ladiffusion évolue peut croître très lentement à l'infini.Dans le cadre d'un modèle de mécanique statistique à spins non-bornés,en nous basant sur des résultats de T. Bodineau et B. Helffer, N. Yoshidaet G. Royer, nous éclaircissons ensuite les liens entre différentesinégalités fonctionnelles, des propriétés de mélange et l'unicité dela mesure de Gibbs en volume infini. Nous montrons en particulierl'unicité si les mesures en volume fini et pour une seule conditionaux bords vérifient uniformément une inégalité de Beckner.

Accepted for publication in ESAIM/PS.; International audience; This article provides entropic inequalities for binomial-Poisson distributions, derived from the two point space. They appear as local inequalities of the M/M/$\infty$ queue. They describe in particular the exponential dissipation of $\Phi$-entropies along this process. This simple queueing process appears as a model of ``constant curvature'', and plays for the simple Poisson process the role played by the Ornstein-Uhlenbeck process for Brownian Motion. Some of the inequalities are recovered by semi-group interpolation. Additionally, we explore the behaviour of these entropic inequalities under a particular scaling, which sees the Ornstein-Uhlenbeck process as a fluid limit of M/M/$\infty$ queues. Proofs are elementary and rely essentially on the development of a ``$\Phi$-calculus''.

In this paper, we investigate Poincare type inequalities for the functions having zero mean value on the whole boundary of a Lipschitz domain or on a measurable part of the boundary. We find exact and easily computable constants in these inequalities for some basic domains (rectangles, cubes, and right triangles) and discuss applications of the inequalities to quantitative analysis of partial differential equations. Copyright © 2014 John Wiley & Sons, Ltd.

We show that a strong mixing condition implies a Bakry-Bobkov-Ledoux inequality for a probability measure on infinite-dimensional space.

Various miscellaneous functional inequalities are deduced for the so-called generalized inverse trigonometric and hyperbolic functions. For instance, functional inequalities for sums, difference and quotient of generalized inverse trigonometric and hyperbolic functions are given, as well as some Gr\"unbaum inequalities with the aid of the classical Bernoulli inequality. Moreover, by means of certain already derived bounds, bilateral bounding inequalities are obtained for the generalized hypergeometric ${}_3F_2$ Clausen function., Comment: 12 pages

The paper is devoted to the maximum principles for functional equations in the space of measurable essentially bounded functions. The necessary and sufficient conditions for validity of corresponding maximum principles are obtained in a form of theorems about functional inequalities similar to the classical theorems about differential inequalities of the Vallee Poussin type. Assertions about the strong maximum principle are proposed. All results are also true for difference equations, which can be considered as a particular case of functional equations. The problems of validity of the maximum principles are reduced to nonoscillation properties and disconjugacy of functional equations. Note that zeros and nonoscillation of a solution in a space of discontinuous functions are defined in this paper. It is demonstrated that nonoscillation properties of functional equations are connected with the spectral radius of a corresponding operator acting in the space of essentially bounded functions. Simple sufficient conditions of nonoscillation, disconjugacy and validity of the maximum principles are proposed. The known nonoscillation results for equation in space of functions of one variable follow as a particular cases of these assertions. It should be noted that corresponding coefficient tests obtained on this basis cannot be improved. Various applications to nonoscillation, disconjugacy and the maximum principles for partial differential equations are proposed.

We describe conditions on non-gradient drift diffusion Fokker–Planck equations for its solutions to converge to equilibrium with a uniform exponential rate in Wasserstein distance. This asymptotic behaviour is related to a functional inequality, which links the distance with its dissipation and ensures a spectral gap in Wasserstein distance. We give practical criteria for this inequality and compare it to classical ones. The key point is to quantify the contribution of the diffusion term to the rate of convergence, in any dimension, which to our knowledge is a novelty.

In this paper, by defining a fuzzy inner product on the dual of real fuzzy Hilbert space, we show that the dual of fuzzy Hilbert space is also fuzzy Hilbert space. Then the concepts of fuzzy linear operator and fuzzy isomentric are defined. It is shown that each real fuzzy Hilbert space is isometricaly isomorphic to the dual of the fuzzy Hilbert space. Publisher's Version

In this work, we apply several functional inequalities (Poincaré,logarithmic Sobolev etc.) to solve two problems. First, we study aninhomogeneous diffusion, akin to the discrete simulated annealingalgorithm, in the spirit of a paper by L. Miclo. We show that thisdiffusion converges under weaker hypotheses than what was assumed inprevious work. In particular, the potential governing the drift isallowed to grow very slowly at infinity. In a second part, we turn toa model of statistical mechanics with unbounded spins, recentlystudied by T. Bodineau and B. Helffer, N. Yoshida and G. Royer (amongothers). We clarify links between mixing properties, uniqueness of theGibbs measure, and functional inequalities. We show in particular thatthe infinite volume tempered Gibbs measure is unique, provided thatthe finite volume measures for one boundary condition satisfy aBeckner inequality.; Dans cette thèse, nous utilisons différentes inégalités fonctionnelles(Poincaré, Sobolev logarithmique, etc.) pour étudier deux questions.Nous appliquons d'abord des inégalités affaiblies à l'étude d'unediffusion inhomogène, analogue continu de l'algorithme de recuitsimulé, dans la lignée d'un travail de L. Miclo. Nous montrons unrésultat de convergence de la diffusion, sous des hypothèses plusfaibles que celles posées précédemment : le potentiel dans lequel ladiffusion évolue peut croître très lentement à l'infini.Dans le cadre d'un modèle de mécanique statistique à spins non-bornés,en nous basant sur des résultats de T. Bodineau et B. Helffer, N. Yoshidaet G. Royer, nous éclaircissons ensuite les liens entre différentesinégalités fonctionnelles, des propriétés de mélange et l'unicité dela mesure de Gibbs en volume infini. Nous montrons en particulierl'unicité si les mesures en volume fini et pour une seule conditionaux bords vérifient uniformément une inégalité de Beckner.

In this paper we intend to give a comprehensive approach of functional inequalities for diffusion processes under some ''curvature'' assumptions. Our notion of curvature coincides with the usual $\Gamma_2$ curvature of Bakry and Emery in the case of a (reversible) drifted Brownian motion, but differs for more general diffusion processes. Our approach using simple coupling arguments together with classical stochastic tools, allows us to obtain new results, to recover and to extend already known results, giving in many situations explicit (though non optimal) bounds. In particular, we show new results for gradient/semigroup commutation in the log concave case. Some new convergence to equilibrium in the granular media equation is also exhibited.