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70
Interpolated inequalities between exponential and Gaussian, Orlicz hypercontractivity and isoperimetry
, 2004
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A logarithmic Sobolev form of the LiYau parabolic inequality
 Revista Mat. Iberoamericana
"... Abstract. – We present a finite dimensional version of the logarithmic Sobolev inequality for heat kernel measures of nonnegatively curved diffusion operators that contains and improves upon the LiYau parabolic inequality. This new inequality is of interest already in Euclidean space for the stand ..."
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Cited by 57 (4 self)
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Abstract. – We present a finite dimensional version of the logarithmic Sobolev inequality for heat kernel measures of nonnegatively curved diffusion operators that contains and improves upon the LiYau parabolic inequality. This new inequality is of interest already in Euclidean space for the standard Gaussian measure. The result may also be seen as an extended version of the semigroup commutation properties under curvature conditions. It may be applied to reach optimal Euclidean logarithmic Sobolev inequalities in this setting. Exponential Laplace differential inequalities through the Herbst argument furthermore yield diameter bounds and dimensional estimates on the heat kernel volume of balls. 1. Introduction and
Some Applications of Mass Transport to GaussianType Inequalities
, 2002
"... As discovered by Brenier, mapping through a convex gradient gives the optimal transport in Rn. In the present article, this map is used in the setting of Gaussianlike measures to derive an inequality linking entropy with mass displacement by a straightforward argument. As a consequence, logarithmic ..."
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Cited by 50 (6 self)
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As discovered by Brenier, mapping through a convex gradient gives the optimal transport in Rn. In the present article, this map is used in the setting of Gaussianlike measures to derive an inequality linking entropy with mass displacement by a straightforward argument. As a consequence, logarithmic Sobolev and transport inequalities are recovered. Finally, a result of Caffarelli on the Brenier map is used to obtain Gaussian correlation inequalities.
Weak Poincaré inequalities and L2convergence rates of Markov semigroups
 J. Funct. Anal
"... In order to describe L2convergence rates slower than exponential, the weak Poincare ́ inequality is introduced. It is shown that the convergence rate of a Markov semigroup and the corresponding weak Poincare ́ inequality can be determined by each other. Conditions for the weak Poincare ́ inequality ..."
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Cited by 47 (8 self)
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In order to describe L2convergence rates slower than exponential, the weak Poincare ́ inequality is introduced. It is shown that the convergence rate of a Markov semigroup and the corresponding weak Poincare ́ inequality can be determined by each other. Conditions for the weak Poincare ́ inequality to hold are presented, which are easy to check and which hold in many applications. The weak Poincaré inequality is also studied by using isoperimetric inequalities for diffusion and jump processes. Some typical examples are given to illustrate the general results. In particular, our results are applied to the stochastic quantization of field theory in finite volume. Moreover, a sharp criterion of weak Poincare ́ inequalities is presented for Poisson measures on configuration spaces.
On the role of convexity in isoperimetry, spectralgap and concentration
 Invent. Math
"... We show that for convex domains in Euclidean space, Cheeger’s isoperimetric inequality, spectral gap of the Neumann Laplacian, exponential concentration of Lipschitz functions, and the apriori weakest requirement that Lipschitz functions have arbitrarily slow uniform taildecay, are all quantitativ ..."
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Cited by 47 (12 self)
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We show that for convex domains in Euclidean space, Cheeger’s isoperimetric inequality, spectral gap of the Neumann Laplacian, exponential concentration of Lipschitz functions, and the apriori weakest requirement that Lipschitz functions have arbitrarily slow uniform taildecay, are all quantitatively equivalent (to within universal constants, independent of the dimension). This substantially extends previous results of Maz’ya, Cheeger, Gromov– Milman, Buser and Ledoux. As an application, we conclude a sharp quantitative stability result for the spectral gap of convex domains under convex perturbations which preserve volume (up to constants) and under maps which are “onaverage ” Lipschitz. We also provide a new characterization (up to constants) of the spectral gap of a convex domain, as one over the square of the average distance from the “worst ” subset having half the measure of the domain. In addition, we easily recover and extend many previously known lower bounds on the spectral gap of convex domains, due to Payne–Weinberger, Li–Yau, Kannan– Lovász–Simonovits, Bobkov and Sodin. The proof involves estimates on the diffusion semigroup following Bakry–Ledoux and a result from Riemannian Geometry on the concavity of the isoperimetric profile. Our results extend to the more general setting of Riemannian manifolds with density which satisfy the CD(0, ∞) curvaturedimension condition of BakryÉmery. 1
Martingale Representation And A Simple Proof Of Logarithmic Sobolev Inequalities On Path Spaces
, 1997
"... We show how the ClarkOconeHaussmann formula for Brownian motion on a compact Riemannian manifold put forward by S. Fang in his proof of the spectral gap inequality for the OrnsteinUhlenbeck operator on the path space can yield in a very simple way the logarithmic Sobolev inequality on the same sp ..."
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Cited by 42 (1 self)
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We show how the ClarkOconeHaussmann formula for Brownian motion on a compact Riemannian manifold put forward by S. Fang in his proof of the spectral gap inequality for the OrnsteinUhlenbeck operator on the path space can yield in a very simple way the logarithmic Sobolev inequality on the same space. By an appropriate integration by parts formula the proof also yields in the same way a logarithmic Sobolev inequality for the path space equipped with a general diffusion measure as long as the torsion of the corresponding Riemannian connection satisfies Driver's total antisymmetry condition. 1 Introduction Let ! = (! t ) t0 be a standard Brownian motion starting from the origin with values in IR n and denote by W 0 (IR n ) the path space of continuous functions from [0; 1] to IR n starting from the origin. We denote further by IE expectation with respect to the law ¯ (the Wiener measure) of ! on W 0 (IR n ). Gross [G] proved the following logarithmic Sobolev inequality holds: ...
inequalities on manifolds with boundary and applications
 J. Math. Pures Appl
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Spectral Gap, Logarithmic Sobolev Constant, and Geometric Bunds
 Surveys in Diff. Geom., Vol. IX, 219–240, Int
, 2004
"... We survey recent works on the connection between spectral gap and logarithmic Sobolev constants, and exponential integrability of Lipschitz functions. In particular, tools from measure concentration are used to describe bounds on the diameter of a (compact) Riemannian manifold and of Markov ch ..."
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Cited by 38 (0 self)
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We survey recent works on the connection between spectral gap and logarithmic Sobolev constants, and exponential integrability of Lipschitz functions. In particular, tools from measure concentration are used to describe bounds on the diameter of a (compact) Riemannian manifold and of Markov chains in terms of the first eigenvalue of the Laplacian and the logarithmic Sobolev constant. We examine similarly dimension free isoperimetric bounds using these parameters.
Mass transport and variants of the logarithmic Sobolev inequality
 J. Geometric Analysis
"... Abstract. We develop the optimal transportation approach to modified logSobolev inequalities and to isoperimetric inequalities. Various sufficient conditions for such inequalities are given. Some of them are new even in the classical logSobolev case. The idea behind many of these conditions is tha ..."
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Cited by 30 (0 self)
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Abstract. We develop the optimal transportation approach to modified logSobolev inequalities and to isoperimetric inequalities. Various sufficient conditions for such inequalities are given. Some of them are new even in the classical logSobolev case. The idea behind many of these conditions is that measures with a nonconvex potential may enjoy such functional inequalities provided they have a strong integrability property that balances the lack of convexity. In addition, several known criteria are recovered in a simple unified way by transportation methods and generalized to the Riemannian setting.
Discrete Isoperimetric And PoincaréType Inequalities
, 1996
"... We study some discrete isoperimetric and Poincar'etype inequalities for product probability measures ¯ n on the discrete cube f0; 1g n and on the lattice Z n . In particular we prove sharp lower estimates for the product measures of 'boundaries ' of arbitrary sets in the discre ..."
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Cited by 29 (2 self)
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We study some discrete isoperimetric and Poincar'etype inequalities for product probability measures ¯ n on the discrete cube f0; 1g n and on the lattice Z n . In particular we prove sharp lower estimates for the product measures of 'boundaries ' of arbitrary sets in the discrete cube. More generally, we characterize those probability distributions ¯ on Z which satisfy these inequalities on Z n . The class of these distributions can be described by a certain class of monotone transforms of the twosided exponential measure. A similar characterization of distributions on R which satisfy Poincar'e inequalities on the class of convex functions is proved in terms of variances of suprema of linear processes. 1 Introduction For a number p 2 (0; 1), let ¯ p denote the Bernoulli measure on f0; 1g with probabilities q = 1 \Gamma p and p, respectively, and let ¯ n p denote the corresponding product measure on f0; 1g n . Denote by s i (x) the 'neighbour' of x 2 f0; 1g n obtained b...