. We show that transport inequalities, similar to the one derived by Talagrand [30] for the Gaussian measure, are implied by logarithmic Sobolev inequalities. Conversely, Talagrand's inequality implies a logarithmic Sobolev inequality if the density of the measure is approximately log-concave, in a precise sense. All constants are independent of the dimension, and optimal in certain cases. The proofs are based on partial dierential equations, and an interpolation inequality involving the Wasserstein distance, the entropy functional and the Fisher information. Contents 1. Introduction 1 2. Main results 5 3. Heuristics 11 4. Proof of Theorem 1 18 5. Proof of Theorem 3 24 6. An application of Theorem 1 30 7. Linearizations 31 Appendix A. A nonlinear approximation argument 35 References 36 1. Introduction Let M be a smooth complete Riemannian manifold of dimension n, with the geodesic distance d(x; y) = inf 8 < : s Z 1 0 j _ w(t)j 2 dt; w 2 C 1 ((0; 1); M); w(0) = x; w(1) = y 9 ...

As part of our study of convergence to equilibrium for spatially inhomogeneous kinetic equations, begun in [13], we derive estimates on the rate of convergence to equilibrium for solutions of the Boltzmann equation, like O(t-&infin;). Our results hold...

. -- 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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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

. We prove convergence to equilibrium with explicit rates for various kinetic equations with relatively bad control of the distribution tails : in particular, Boltzmann-type equations with (smoothed) soft potentials. We compensate the lack of uniform in time estimates by the use of precise logarithmic Sobolevtype inequalities, and the assumption that the initial datum decays rapidly at large velocities. Our method not only gives explicit results on the times of convergence, but is also able to cover situations in which compactness arguments apparently do not apply (even mere convergence to equilibrium was an open problem for soft potentials). Contents 1. Introduction 1 2. The Fokker-Planck equation with weak drift 7 3. The Landau equation for mollified soft potentials 11 4. The Boltzmann equation for mollified soft potentials 21 References 26 1. Introduction We consider in this work the problem of trend to equilibrium for collisional kinetic equations of the form #f #t = Q(f) (1) w...

A new concept of Fisher-information is introduced through a cost function. That concept is used to obtain extensions and variants of transport and logarithmic Sobolev inequalities for general entropy functionals and transport costs. Our proofs rely on optimal mass transport from the Monge-Kantorovich theory. They express the convexity of entropy functionals with respect to suitably chosen paths on the set of probability measures.

It is shown that if X1,X2,... are independent and identically distributed square-integrable random variables then the entropy of the normalized sum