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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 110 (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.
Ricci curvature for metricmeasure spaces via optimal transport”, to appear
 Ann. of Math
"... Abstract. We define a notion of a measured length space X having nonnegative NRicci 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) ..."
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Cited by 82 (9 self)
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Abstract. We define a notion of a measured length space X having nonnegative NRicci 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 GromovHausdorff 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
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 56 (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...
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 41 (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 ...
Sobolev inequalities in disguise
 Indiana Univ. Math. J
, 1995
"... We present a simple and direct proof of the equivalence of various functional inequalities such as Sobolev or Nash inequalities. This proof applies in the context of Riemannian or subelliptic geometry, as well as on graphs and to certain nonlocal Sobolev norms. It only uses elementary cutoff argu ..."
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Cited by 38 (4 self)
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We present a simple and direct proof of the equivalence of various functional inequalities such as Sobolev or Nash inequalities. This proof applies in the context of Riemannian or subelliptic geometry, as well as on graphs and to certain nonlocal Sobolev norms. It only uses elementary cutoff arguments. This method has interesting consequences concerning Trudinger type inequalities. 1. Introduction. On R n, the classical Sobolev inequality [27] indicates that, for every smooth enough function f with compact support,
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
Interpolated inequalities between exponential and Gaussian, Orlicz hypercontractivity and isoperimetry
, 2004
"... ..."
Rate of convergence for ergodic continuous Markov processes : Lyapunov versus Poincaré
 J. Func. Anal
, 1996
"... Abstract. We study the relationship between two classical approaches for quantitative ergodic properties: the first one based on Lyapunov type controls and popularized by Meyn and Tweedie, the second one based on functional inequalities (of Poincaré type). We show that they can be linked through new ..."
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Cited by 25 (14 self)
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Abstract. We study the relationship between two classical approaches for quantitative ergodic properties: the first one based on Lyapunov type controls and popularized by Meyn and Tweedie, the second one based on functional inequalities (of Poincaré type). We show that they can be linked through new inequalities (LyapunovPoincaré inequalities). Explicit examples for diffusion processes are studied, improving some results in the literature. The example of the kinetic FokkerPlanck equation recently studied by HérauNier, HelfferNier and Villani is in particular discussed in the final section.
On Logarithmic Sobolev Inequalities For Continuous Time Random Walks On Graphs
 Probab. Theory Related Fields
, 2000
"... . 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 ..."
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Cited by 24 (2 self)
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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...