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Generalization Of An Inequality By Talagrand, And Links With The Logarithmic Sobolev Inequality
 J. Funct. Anal
, 2000
"... . 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 logconcave, in a ..."
Abstract

Cited by 121 (10 self)
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. 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 logconcave, 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 ...
Logarithmic Sobolev inequality and finite markov chains
, 1996
"... This is an expository paper on the use of logarithmic Sobolev inequalities for bounding rates of convergence of Markov chains on finite state spaces to their stationary distributions. Logarithmic Sobolev inequalities complement eigenvalue techniques and work for nonreversible chains in continuous ti ..."
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Cited by 113 (11 self)
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This is an expository paper on the use of logarithmic Sobolev inequalities for bounding rates of convergence of Markov chains on finite state spaces to their stationary distributions. Logarithmic Sobolev inequalities complement eigenvalue techniques and work for nonreversible chains in continuous time. Some aspects of the theory simplify considerably with finite state spaces and we are able to give a selfcontained development. Examples of applications include the study of a Metropolis chain for the binomial distribution, sharp results for natural chains on the box of side n in d dimensions and improved rates for exclusion processes. We also show that for most rregular graphs the logSobolev constant is of smaller order than the spectral gap. The logSobolev constant of the asymmetric twopoint space is computed exactly as well as the logSobolev constant of the complete graph on n points.
On the Trend to Global Equilibrium for Spatially Inhomogeneous Kinetic Systems: The Boltzmann Equation
 Comm. Pure Appl. Math
, 2003
"... 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∞). Our results hold... ..."
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Cited by 82 (6 self)
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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∞). Our results hold...
Hypercontractivity Of HamiltonJacobi Equations
 J. Math. Pures Appl
, 2000
"... .  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 HamiltonJacobi equations. By the infimumconvolution description of the Hamilt ..."
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Cited by 57 (11 self)
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.  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 HamiltonJacobi equations. By the infimumconvolution description of the HamiltonJacobi 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 BrunnMinkowki 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...
Contractions in the 2Wasserstein Length Space and Thermalization of Granular Media
, 2004
"... An algebraic decay rate is derived which bounds the time required for velocities to equilibrate in a spatially homogeneous flowthrough model representing the continuum limit of a gas of particles interacting through slightly inelastic collisions. This rate is obtained by reformulating the dynamical ..."
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Cited by 56 (19 self)
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An algebraic decay rate is derived which bounds the time required for velocities to equilibrate in a spatially homogeneous flowthrough model representing the continuum limit of a gas of particles interacting through slightly inelastic collisions. This rate is obtained by reformulating the dynamical problem as the gradient flow of a convex energy on an infinitedimensional manifold. An abstract theory is developed for gradient flows in length spaces, which shows how degenerate convexity (or even nonconvexity) — if uniformly controlled — will quantify contractivity (limit expansivity) of the flow.
From BrunnMinkowski To BrascampLieb And To Logarithmic Sobolev Inequalities
 Geom. Funct. Anal
"... .  We develop several applications of the BrunnMinkowki inequality in the Pr'ekopaLeindler form. In particular, we show that an argument of B. Maurey may be adapted to deduce from the Pr'ekopaLeindler inequality the BrascampLieb inequality for stricly convex potentials. We deduce similarly the ..."
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Cited by 42 (2 self)
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.  We develop several applications of the BrunnMinkowki inequality in the Pr'ekopaLeindler form. In particular, we show that an argument of B. Maurey may be adapted to deduce from the Pr'ekopaLeindler inequality the BrascampLieb 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'ekopaLeinder 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 nonnegative 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 ...
Interpolated inequalities between exponential and Gaussian, Orlicz hypercontractivity and isoperimetry
, 2004
"... ..."
The Geometry of Markov Diffusion Generators
, 1998
"... 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'evyGromov isoperimetric inequ ..."
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Cited by 33 (4 self)
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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'evyGromov 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 minicourse 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
Solution of Shannon’s problem on the monotonicity of entropy
 J. Amer. Math. Soc
, 2004
"... It is shown that if X1,X2,... are independent and identically distributed squareintegrable random variables then the entropy of the normalized sum ..."
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Cited by 27 (1 self)
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It is shown that if X1,X2,... are independent and identically distributed squareintegrable random variables then the entropy of the normalized sum
On The Trend To Equilibrium For Some Dissipative Systems With Slowly Increasing A Priori Bounds
 J. Statist. Phys
"... . We prove convergence to equilibrium with explicit rates for various kinetic equations with relatively bad control of the distribution tails : in particular, Boltzmanntype equations with (smoothed) soft potentials. We compensate the lack of uniform in time estimates by the use of precise logarithm ..."
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Cited by 23 (9 self)
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. We prove convergence to equilibrium with explicit rates for various kinetic equations with relatively bad control of the distribution tails : in particular, Boltzmanntype 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 FokkerPlanck 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...