Abstract. We define a notion of a measured length space X having nonnegative N-Ricci curvature, for N ∈ [1, ∞), or having ∞-Ricci curvature bounded below by K, for K ∈ R. The definitions are in terms of the displacement convexity of certain functions on the associated Wasserstein metric space P2(X) of probability measures. We show that these properties are preserved under measured Gromov-Hausdorff limits. We give geometric and analytic consequences. This paper has dual goals. One goal is to extend results about optimal transport from the setting of smooth Riemannian manifolds to the setting of length spaces. A second goal is to use optimal transport to give a notion for a measured length space to have Ricci curvature bounded below. We refer to [10] and [41] for background material on length spaces and optimal transport, respectively. Further bibliographic notes on optimal transport are in Appendix G. In the present introduction we motivate the questions that we address and we state the main results. To start on the geometric side, there are various reasons to try to extend notions of curvature from smooth Riemannian manifolds to more general spaces. A fairly general

. -- 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 infimumconvolution 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-Minkowki inequalities and for the exponential measure. 1. Introduction The fundamental work by L. Gross [Gr] put forward the equivalence between logarithmic Sobolev inequalities and hypercontractivity of the associated heat semigroup. Let us consider for example a probability measure on the Borel sets of R n satisfying the logarithmic Sobolev inequality ae Ent (f 2 ) 2 Z jrf j 2 d (1:1) for some ae ? 0 and all smooth eno...

. --- We develop several applications of the Brunn-Minkowki inequality in the Pr'ekopa-Leindler form. In particular, we show that an argument of B. Maurey may be adapted to deduce from the Pr'ekopaLeindler inequality the Brascamp-Lieb inequality for stricly 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. 1. Introduction The Pr'ekopa-Leinder inequality is a functional form of the geometric BrunnMinkowski inequality which indicates that whenever t; s ? 0, t + s = 1, and u, v, w are non-negative measurable functions on R n such that for all x; y 2 R n , w \Gamma tx + sy) u(x) t v(y) s ; then Z wdx `Z udx ' t `Z vdx ' s : (1:1) Applied to the characteristic functions of bounded measurable sets A and B in R ...

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. ---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. 1. Introduction The classical logarithmic Sobolev inequality for Brownian motion B = (B t ) t0 in IR d [Gr] indicates that for all functionals F in the domain of the Malliavin gradient operator D : L 2(\Omega ; IP) ! L 2(\Omega \Theta [0; T ]; IP\Omega dt), IE(F 2 log F 2 ) \Gamma IE(F 2 ) log IE(F 2 ) 2 IE `Z T 0 jD t F j 2 dt ' : (1:1) In particular, if F = f(B t 1 ; : : : ; B t n ), 0 = t 0 t 1 \Delta \Delta \Delta t n for some smooth function f : (IR d ) n ! IR, D t F = n X i=1 r i F I ft...

CONTENTS INTRODUCTION p. 4 1. GEOMETRIC ASPECTS OF DIFFUSION GENERATORS p. 8 1.1 Semigroups and generators p. 8 1.2 Curvature and dimension p. 13 1.3 Functional inequalities p. 18 2. INFINITE DIMENSIONAL GENERATORS p. 22 2.1 Logarithmic Sobolev inequalities p. 22 2.1 L'evy-Gromov isoperimetric inequality p. 24 3. SHARP SOBOLEV INEQUALITIES AND COMPARISON THEOREMS p. 29 3.1 Sobolev inequalities p. 29 3.2 Myers's diameter theorem p. 33 3.3 Eigenvalues comparison theorems p. 36 4. SOBOLEV INEQUALITIES AND HEAT KERNEL BOUNDS p. 41 4.1 Equivalent Sobolev inequalities p. 41 4.2 Logarithmic Sobolev inequalities and hypercontractivity p. 45 4.3 Optimal heat kernel bounds p. 47 4.4 Rigidity properties p. 52 REFERENCES p. 56 3 These notes form a summary of a mini-course given at the Eidgenossische Technische Hochschule in Zurich in November 1998. They aim to present some of the basic ideas in the geometric investigation of Markov diffusion generators, as developed in the last decade by

Abstract. — We analyze recent proofs of decay of correlations and logarithmic Sobolev inequalities for unbounded spin systems in the perturbative regime developed by B. Zegarlinski, N. Yoshida, B. Helffer, Th. Bodineau. We investigate to this task a simple analytic model. Proofs are short and self-contained. Let µ be a probability measure on R satisfying, for some constant C> 0 and for every smooth enough function f on R, either the Poincaré (or spectral gap) inequality Varµ(f) ≤ C f ′2 dµ where Varµ(f) is the variance of f with respect to µ (see below), or the logarithmic

Abstract. µ being a nonnegative measure satisfying some Log-Sobolev inequality, we give conditions on F for the Boltzmann measure ν = e −2F µ to also satisfy some Log-Sobolev inequality. This paper improves and completes the final section in [6]. A general sufficient condition and a general necessary condition are given and examples are explicitly studied. Résumé. µ étant une mesure positive satisfaisant une inégalité de Sobolev logarithmique, nous donnons des conditions sur F pour que la mesure de Boltzmann ν = e −2F µ satisfasse également une telle inégalité (améliorant et complétant ainsi la dernière partie de [6]). Les conditions obtenues sont illustrées par des exemples.

We extend the concept of intrinsic ultracontractivity to non-symmetric semigroups and prove the intrinsic ultracontractivity of the Dirichlet semigroups of non-symmetric second order elliptic operators in bounded Lipschitz domains.

This article provides entropic inequalities for binomial-Poisson distributions, derived from the two point space. They appear as local inequalities of the M/M/ ∞ queue. They describe in particular the exponential dissipation of Φ-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/ ∞ queues. Proofs are elementary and rely essentially on the development of a “Φ-calculus”.