This thesis consists of four papers and focuses on function spaces related to first-order analysis in abstract metric measure spaces. The classical (i.e., Sobolev) theory in Euclidean spaces makes use of summability of distributional gradients, whose definition depends on the linear structure of Rn. In metric spaces, we can replace the distributional gradients by (weak) upper gradients that control the functions’ behavior along (almost) all rectifiable curves, which gives rise to the so-called Newtonian spaces. The summability condition, considered in the thesis, is expressed using a general Banach function lattice quasi-norm and so an extensive framework is built. Sobolev-type spaces (mainly based on the Lp norm) on metric spaces, and Newtonian spaces in particular, have been under intensive study since the mid-1990s. In Paper I, the elementary theory of Newtonian spaces based on quasi-Banach function lattices is built up. Standard tools such as moduli of curve families and the Sobolev capacity are developed and applied to study the basic properties of Newtonian functions. Summability of a (weak) upper gradient of a function is shown to guarantee the function’s absolute continuity on almost all curves. Moreover, Newtonian spaces are proven complete in this general setting. Paper II investigates the set of all weak upper gradients of a Newtonian function. In particular, existence of minimal weak upper gradients is established. Validity of Lebesgue’s differentiation theorem for the underlying metric measure space ensures that a family of representation formulae for minimal weak upper gradients can be found. Furthermore, the connection between pointwise and norm convergence of a sequence of Newtonian functions is studied. Smooth functions are frequently used as an approximation of Sobolev functions in analysis of partial differential equations. In fact, Lipschitz continuity, which is (unlike <img src="http://www.diva-portal.org/cgi-bin/mimetex.cgi?%5Cmathcal%7BC%7D%5E1" />-smoothness) well-defined even for functions on metric spaces, often suffices as a regularity condition. Thus, Paper III concentrates on the question when Lipschitz functions provide good approximations of Newtonian functions. As shown in the paper, it suffices that the function lattice quasi-norm is absolutely continuous and a fractional sharp maximal operator satisfies a weak norm estimate, which it does, e.g., in doubling Poincaré spaces if a non-centered maximal operator of Hardy–Littlewood type is locally weakly bounded. Therefore, such a local weak boundedness on rearrangement-invariant spaces is explored as well. Finer qualitative properties of Newtonian functions and the Sobolev capacity get into focus in Paper IV. Under certain hypotheses, Newtonian functions are proven to be quasi-continuous, which yields that the capacity is an outer capacity. Various sufficient conditions for local boundedness and continuity of Newtonian functions are established. Finally, quasi-continuity is applied to discuss density of locally Lipschitz functions in Newtonian spaces on open subsets of doubling Poincaré spaces.

Dans cette thèse, nous considérons les chemins possibles pour donner une mesure quantitative du fait que deux espaces ne sont pas quasi-isométriques. De ce point de vue quantitatif, on reprend la définition de quasi-isométrie et on propose une notion de “croissance de distorsion quasi-isométrique” entre deux espaces métriques. Nous révisons notre article [32] où une borne supérieure optimale pour le lemme de Morse est donnée, avec la variante duale que nous appelons Anti-Morse Lemma, et leurs applications.Ensuite, nous nous concentrons sur des bornes inférieures sur la croissance de distorsion quasi-isométrique pour des espaces métriques hyperboliques. Dans cette classe, les espaces de $L^p$-cohomologie fournissent des invariants de quasi-isométrie utiles et les constantes de Poincaré des boules sont leur incarnation quantitative. Nous étudions comment les constantes de Poincaré sont transportées par quasi-isométries. Dans ce but, nous introduisons la notion de transnoyau. Nous calculons les constantes de Poincaré pour les métriques localement homogènes de la forme $dt^2+\sum_ie^{2\mu_it}dx_i^2$, et donnons une borne inférieure sur la croissance de distorsion quasi-isométrique entre ces espaces.Cela nous permet de donner des exemples présentant différents type de croissance de distorsion quasi-isométrique, y compris un exemple sous-linéaire (logarithmique). / In this thesis we discuss possible ways to give quantitative measurement for two spaces not being quasi-isometric. From this quantitative point of view, we reconsider the definition of quasi-isometries and propose a notion of ``quasi-isometric distortion growth'' between two metric spaces. We revise our article [32] where an optimal upper-bound for Morse Lemma is given, together with the dual variant which we call Anti-Morse Lemma, and their applications.Next, we focus on lower bounds on quasi-isometric distortion growth for hyperbolic metric spaces. In this class, $L^p$-cohomology spaces provides useful quasi-isometry invariants and Poincar\'e constants of balls are their quantitative incarnation. We study how Poincar\'e constants are transported by quasi-isometries. For this, we introduce the notion of a cross-kernel. We calculate Poincar\'e constants for locally homogeneous metrics of the form $dt^2+\sum_ie^dx_i^2$, and give a lower bound on quasi-isometric distortion growth among such spaces.This allows us to give examples of different quasi-isometric distortion growths, including a sublinear one (logarithmic).

El objetivo fundamental de este trabajo ha sido el estudio del sistema fundamental de paralelos de un cuerpo convexo (conjunto compacto y convexo) en el espacio euclídeo n-dimensional. Se ha llevado a cabo siguiendo tres líneas diferentes: el estudio del polinomio de Steiner y el polinomio alternado Steiner desde el punto de vista algebraico de sus raíces y la conjetura de Matheron; el estudio de la diferenciabilidad de las quermassintegrales asociadas al cuerpo con respecto al parámetro de definición del sistema completo de paralelos y, por último, el estudio de las quermassintegrales del cuerpo desde el punto de vista analítico proporcionado por la identificación del cuerpo convexo con su función soporte, las propiedades de ésta cuando el cuerpo es suficientemente suave y la desigualdad de Brunn-Minkowski. / The aim of this work consists on studying the full system of parallel bodies of a convex body (compact and convex set) in the n-dimensional Euclidean space. It has been carried out following three different lines of work: the study of the Steiner polynomial and the alternating Steiner polynomial from the algebraic point of view of its roots; the study of the differentiability of the quermassintegrals associated to the convex body with respect to the parameter that defines the full system of parallel bodies and finally, the study of the quermassintegrals from the analytical point of view provided by the identification of a convex body with its support function, the properties of this function when the body is smooth enough and the Brunn-Minkowski inequality

The topic of this thesis is a diffusion process on a potential landscape which is given by a smooth Hamiltonian function in the regime of small noise. The work provides a new proof of the Eyring-Kramers formula for the Poincaré inequality of the associated generator of the diffusion. The Poincaré inequality characterizes the spectral gap of the generator and establishes the exponential rate of convergence towards equilibrium in the L²-distance. This result was first obtained by Bovier et. al. in 2004 relying on potential theory. The presented approach in the thesis generalizes to obtain also asymptotic sharp estimates of the constant in the logarithmic Sobolev inequality. The optimal constant in the logarithmic Sobolev inequality characterizes the convergence rate to equilibrium with respect to the relative entropy, which is a stronger distance as the L²-distance and slightly weaker than the L¹-distance. The optimal constant has here no direct spectral representation. The proof makes use of the scale separation present in the dynamics. The Eyring-Kramers formula follows as a simple corollary from the two main results of the work: The first one shows that the associated Gibbs measure restricted to a basin of attraction has a good Poincaré and logarithmic Sobolev constants providing the fast convergence of the diffusion to metastable states. The second main ingredient is a mean-difference estimate. Here a weighted transportation distance is used. It contains the main contribution to the Poincaré and logarithmic Sobolev constant, resulting from exponential long waiting times of jumps between metastable states of the diffusion.

Ma thèse de doctorat se concentre principalement sur le comportement en temps long des processus de Markov, les inégalités fonctionnelles et les techniques relatives. Plus spécifiquement, Je vais présenter les taux de convergence sous-exponentielle explicites des processus de Markov dans deux approches : la méthode Meyn-Tweedie et l’hypocoercivité (faible). Le document se divise en trois parties. Dans la première partie, Je vais présenter quelques résultats importants et des connaissances connexes. D’abord, un aperçu de mon domaine de recherche sera donné. La convergence exponentielle (ou sous-exponentielle) des chaînes de Markov et des processus de Markov (à temps continu) est un sujet d’actualité dans la théorie des probabilité. La méthode traditionnelle développée et popularisée par Meyn-Tweedie est largement utilisée pour ce problème. Dans la plupart des résultats, le taux de convergence n’est pas explicite, et certains d’entre eux seront brièvement présentés. De plus, la fonction de Lyapunov est cruciale dans l’approche Meyn-Tweedie, et elle est aussi liée à certaines inégalités fonctionnelles (par exemple, inégalité de Poincaré). Cette relation entre fonction de Lyapounov et inégalités fonctionnelles sera donnée avec les résultats au sens L2. En outre, pour l’exemple de l’équation cinétique de Fokker-Planck, un résultat de convergence exponentielle explicite de la solution sera introduite à la manière de Villani : l’hypocoercivité. Ces contenus sont les fondements de mon travail, et mon but est d’étudier la décroissance sous-exponentielle. La deuxième partie, fait l’objet d’un article écrit en coopération avec d’autres sur les taux de convergence sous-exponentielle explicites des processus de Markov à temps continu. Comme nous le savons, les résultats sur les taux de convergence explicites ont été donnés pour le cas exponentiel. Nous les étendons au cas sous-exponentielle par l’approche Meyn-Tweedie. La clé de la preuve est l’estimation du temps de passage dans un ensemble ”petite”, obtenue par Douc, Fort et Guillin, mais pour laquelle nous donnons une preuve plus simple. Nous utilisons aussi la construction du couplage et donnons une ergodicité sous exponentielle explicite. Enfin, nous donnons quelques applications numériques. Dans la dernière partie, mon second article traite de l’équation cinétique de Fokker-Planck. Je prolonge l’hypocoercivité à l’hypocoercivité faible qui correspond à inégalité de Poincaré faible. Grâce à cette extension, on peut obtenir le taux de convergence explicite de la solution, dans des cas sous-exponentiels. La convergence est au sens H1 et au sens L2. A la fin de ce document, j’étudie le cas de l’entropie relative comme Villani, et j’obtiens la convergence au sens de l’entropie. Enfin, Je donne deux exemples pour les potentiels qui impliquent l’inégalité de Poincaré faible ou l’inégalité de Sobolev logarithmique faible pour la mesure invariante. / My Ph.D dissertation mainly focuses on long time behavior of Markov processes, functional inequalities and related techniques. More specifically, I will present the computable sub-exponential convergence rate of the Markov process in two approaches : Meyn-Tweedie’s method and (weak) hypocoercivity. The paper consists of three parts. In the first part, I will introduce some important results and related knowledge. Firstly, overviews of my research field are given. Exponential (or subexponential) convergence of Markov chains and (continuous time) Markov processes is a hot issue in probability. The traditional method - Meyn-Tweedie’s approach is widely applied for this problem. Most of the results about convergence rate is not explicit, and some of them will be introduced briefly. In addition,Lyapunov function is crucial in Meyn-Tweendie’s aproach, and it is also related to some functional inequalities (for example, Poincar´e inequality). The relationship of them will be given with results in L2 sense. Furthermore, as a example of kinetic Fokker-Planck equation, a computable result of exponential convergence of the solution of it will be introduced in Villani’ way - hypocoercivity. These contents are foundations of my work, and my destination is to study the sub-exponential decay. In the second part, it is my article cooperated with others about subexponential convergence rate of continuous time Markov processes. As we all know, the explicit results of convergence rate is about the exponential case. We extend them to sub-exponential case in Meyn-Tweedie’s approach. The key of the proof is the estimation of the hitting time to small set which was got by Douc, Fort and Guillin, for which we also propose an alternative simpler proof. We also use coupling construction as others and give a quantitative sub-exponential ergodicity. At last, we give some calculations for examples. In the last part, my second article deal with the kinetic Fokker-Planck equation. I extend the hypocoercivity to weak hypocoercivity which correspond to weak Poincar´e inequality. Through the extension, one can get the computable rate of convergence of the solution, which is also sub-exponential case. The convergence is in H1 sense and in L2 sense. In the end of this paper, I study the relative entropy case as C.Villani, and get convergence in entropy. Finally, I give two examples for potentials that implies weak Poincar´e inequality or weak logarithmic Sobolve inequality for invarient measure.

The topic of this thesis is a diffusion process on a potential landscape which is given by a smooth Hamiltonian function in the regime of small noise. The work provides a new proof of the Eyring-Kramers formula for the Poincaré inequality of the associated generator of the diffusion. The Poincaré inequality characterizes the spectral gap of the generator and establishes the exponential rate of convergence towards equilibrium in the L²-distance. This result was first obtained by Bovier et. al. in 2004 relying on potential theory. The presented approach in the thesis generalizes to obtain also asymptotic sharp estimates of the constant in the logarithmic Sobolev inequality. The optimal constant in the logarithmic Sobolev inequality characterizes the convergence rate to equilibrium with respect to the relative entropy, which is a stronger distance as the L²-distance and slightly weaker than the L¹-distance. The optimal constant has here no direct spectral representation. The proof makes use of the scale separation present in the dynamics. The Eyring-Kramers formula follows as a simple corollary from the two main results of the work: The first one shows that the associated Gibbs measure restricted to a basin of attraction has a good Poincaré and logarithmic Sobolev constants providing the fast convergence of the diffusion to metastable states. The second main ingredient is a mean-difference estimate. Here a weighted transportation distance is used. It contains the main contribution to the Poincaré and logarithmic Sobolev constant, resulting from exponential long waiting times of jumps between metastable states of the diffusion.