La courbure de Ricci grossière d’un processus markovien sur un espace polonais est définie comme un taux de contraction local de la distance de Wasserstein W1 entre les lois du processus partant de deux points distincts. La première partie de cette thèse traite de résultats valables dans le cas d’espaces polonais quelconques. On montre que l’infimum de la courbure de Ricci grossière est un taux de contraction global du semigroupe du processus pour la distance W1. Quoiqu’intuitif, ce résultat est difficile à démontrer en temps continu. La preuve de ce résultat, ses conséquences sur le trou spectral du générateur font l’objet du chapitre 1. Un autre résultat intéressant, faisant intervenir les valeurs de la courbure de Ricci grossière en différents points, et pas seulement son infimum, est un résultat de concentration des mesures d’équilibre, valable uniquement en temps discret. Il sera traité dans le chapitre 2. La seconde partie de cette thèse traite du cas particulier des diffusions sur les variétés riemanniennes. Une formule est donnée permettant d’obtenir la courbure de Ricci grossière à partir du générateur. Dans le cas où la métrique est adaptée à la diffusion, nous montrons l’existence d’un couplage entre les trajectoires tel que la courbure de Ricci grossière est exactement le taux de décroissance de la distance entre ces trajectoires. Le trou spectral du générateur de la diffusion est alors plus grand que la moyenne harmonique de la courbure de Ricci. Ce résultat peut être généralisé lorsque la métrique n’est pas celle induite par le générateur, mais il nécessite une hypothèse contraignante, et la courbure que l'on doit considérer est plus faible. / The coarse Ricci curvature of a Markov process on a Polish space is defined as a local contraction rate of the W1 Wasserstein distance between the laws of the process starting at two different points. The first part of this thesis deals with results holding in the case of general Polish spaces. The simplest of them is that the infimum of the coarse Ricci curvature is a global contraction rate of the semigroup of the process for the W1 distance between probability measures. Though intuitive, this result is diffucult to prove in continuous time. The proof of this result, and the following consequences for the spectral gap of the generator are the subject of Chapter 1. Another interesting result, using the values of the coarse Ricci curvature at different points, and not only its infimum, is a concentration result for the equilibrium measures, only holding in a discrete time framework. That will be the topic of Chapter 2. The second part of this thesis deals with the particular case of diffusions on Riemannian manifolds. A formula is given, allowing to get the coarse Ricci curvature from the generator of the diffusion. In the case when the metric is adapted to the diffusion, we show the existence of a coupling between the paths starting at two different points, such that the coarse Ricci curvature is exactly the decreasing rate of the distance between these paths. We can then show that the spectral gap of the generator is at least the harmonic mean of the Ricci curvature. This result can be generalized when the metric is not the one induced by the generator, but it needs a very restricting hypothesis, and the curvature we have to choose is smaller.

During the last half century there has been a resurgence of interest in Monge's 18th century mass transportation problem, with most of the activity limited to continuous spaces. This thesis, consequently, develops techniques based on mass transportation for the purpose of obtaining tight concentration inequalities in a discrete setting. Such inequalities on n-fold products of graphs, equipped with product measures, have been well investigated using combinatorial and probabilistic techniques, the most notable being martingale techniques. The emphasis here, is instead on the analytic viewpoint, with the precise contribution being as follows. We prove that the modified log-Sobolev inequality implies the transportation inequality in the first systematic comparison of the modified log-Sobolev inequality, the Poincar inequality, the transportation inequality, and a new variance transportation inequality. The duality shown by Bobkov and Gtze of the transportation inequality and a generating function inequality is then utilized in finding the asymptotically correct value of the subgaussian constant of a cycle, regardless of the parity of the length of the cycle. This result tensorizes to give a tight concentration inequality on the discrete torus. It is interesting in light of the fact that the corresponding vertex isoperimetric problem has remained open in the case of the odd torus for a number of years. We also show that the class of bounded degree expander graphs provides an answer, in the affirmative, to the question of whether there exists an infinite family of graphs for which the spread constant and the subgaussian constant differ by an order of magnitude. Finally, a candidate notion of a discrete Ricci curvature for finite Markov chains is given in terms of the time decay of the Wasserstein distance of the chain to its stationarity. It can be interpreted as a notion arising naturally from a standard coupling of Markov chains. Because of its natural definition, ease of calculation, and tensoring property, we conclude that it deserves further investigation and development. Overall, the thesis demonstrates the utility of using the mass transportation problem in discrete isoperimetric and functional inequalities.

[pt] Seja V um conjunto de n pontos no espaço euclidiano X de dimensão d. Pelo teorema de Johnson-Lindenstrauss, existe uma projeção entre X e Y, outro espaço de dimensão k bastante menor, com a propriedade que as distâncias entre imagens de pontos de V sejam mantidas dentro de um fator c arbitrariamente próximo de 1. O teorema apresenta uma relação entre d, k e c, indicando a possibilidade de dramáticas reduções de dimensão para representações fidedignas de V. A demonstração emprega as Grassmannianas, as variedades de subespaços de dimensão k em X. São construídas cartas e uma medida homogênea em relação à ação natural do grupo ortogonal na Grassmanniana. O resultado segue estimando através de gaussianas certas integrais de caráter fortemente geométrico. / [en] Let V be a set of n points in the Euclidean space X of dimension d. The Johnson-Lindenstrauss theorem states that there is a projection between X a and Y, another Euclidean space of a smaller dimension k, with the property that images of points of X under projection do not differ by more that a multiplicative factor c arbitrarily close to 1. The theorem presents a relation among d, k and c, indicating the possibility of dramatic dimensional reduction of very faithful representations of V. The proof makes use of Grassmanians, the manifolds consisting of subspaces of dimension k in X. In the text, charts are presented, together with a measure which is homogeneous with respect to the natural action of the orthogonal group on the Grassmanian. The result follows by taking estimates using gaussians of certain integrals with a strong geometric flavor.