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11
Exponential integrability and transportation cost related to logarithmic Sobolev inequalities
 J. Funct. Anal
, 1999
"... We study some problems on exponential integrability, concentration of measure, and transportation cost related to logarithmic Sobolev inequalities. On the real line, we then give a characterization of those probability measures which satisfy these inequalities 1999 Academic Press Key Words: logarith ..."
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Cited by 80 (4 self)
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We study some problems on exponential integrability, concentration of measure, and transportation cost related to logarithmic Sobolev inequalities. On the real line, we then give a characterization of those probability measures which satisfy these inequalities 1999 Academic Press Key Words: logarithmic Sobolev inequalities; exponential integrability; concentration of measure; transportation inequalities.
Interpolated inequalities between exponential and Gaussian, Orlicz hypercontractivity and isoperimetry
, 2004
"... ..."
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 14 (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.
Weighted Poincarétype inequalities for Cauchy and other convex measures
 Annals of Probability
, 2007
"... Brascamp–Liebtype, weighted Poincarétype and related analytic inequalities are studied for multidimensional Cauchy distributions and more general κconcave probability measures (in the hierarchy of convex measures). In analogy with the limiting (infinitedimensional logconcave) Gaussian model, the ..."
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Cited by 9 (1 self)
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Brascamp–Liebtype, weighted Poincarétype and related analytic inequalities are studied for multidimensional Cauchy distributions and more general κconcave probability measures (in the hierarchy of convex measures). In analogy with the limiting (infinitedimensional logconcave) Gaussian model, the weighted inequalities fully describe the measure concentration and large deviation properties of this family of measures. Cheegertype isoperimetric inequalities are investigated similarly, giving rise to a common weight in the class of concave probability measures under consideration. 1. Introduction. The
CugliandoloKurchan equations for dynamics of spinglasses
, 2006
"... Abstract. We study the Langevin dynamics for the family of spherical pspin disordered meanfield models of statistical physics. We prove that in the limit of system size N approaching infinity, the empirical state correlation and integrated response functions for these Ndimensional coupled diffusio ..."
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Cited by 4 (1 self)
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Abstract. We study the Langevin dynamics for the family of spherical pspin disordered meanfield models of statistical physics. We prove that in the limit of system size N approaching infinity, the empirical state correlation and integrated response functions for these Ndimensional coupled diffusions converge almost surely and uniformly in time, to the nonrandom unique strong solution of a pair of explicit nonlinear integrodifferential equations, first introduced by Cugliandolo and Kurchan.
Poincaré inequalities for non euclidean metrics and . . .
, 2007
"... In this paper, we consider Poincaré inequalities for non euclidean metrics on R d. These inequalities enable us to derive precise dimension free concentration inequalities for product measures. This technique is appropriate for a large scope of concentration rate: between exponential and gaussian ..."
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Cited by 2 (0 self)
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In this paper, we consider Poincaré inequalities for non euclidean metrics on R d. These inequalities enable us to derive precise dimension free concentration inequalities for product measures. This technique is appropriate for a large scope of concentration rate: between exponential and gaussian and beyond. We give different equivalent functional forms of these Poincaré type inequalities in terms of transportationcost inequalities and infimum convolution inequalities. Workable sufficient conditions are given and a comparison is made with generalized BecknerLatalaOleszkiewicz inequalities.
Ultracontractivity And Supercontractivity Of Markov Semigroups
"... By using perturbation arguments, a sufficient condition is presented for the ultracontractivity of symmetric diffusion semigroups. As a consequence, a result suggested by D. Stroock is proved: let P t be generated by L = + rV with V = r ( > 0; > 2) on a complete connected Riemannian manifold M , whe ..."
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By using perturbation arguments, a sufficient condition is presented for the ultracontractivity of symmetric diffusion semigroups. As a consequence, a result suggested by D. Stroock is proved: let P t be generated by L = + rV with V = r ( > 0; > 2) on a complete connected Riemannian manifold M , where is the Riemannian distance function from a fixed point, then P t is ultracontractive provided the Ricci curvature is bounded below. Furthermore, kP t k1!1 exp[ 1 + 2 t =( 2) ] for some 1 ; 2 > 0: Next, it is shown that, for a di usion semigroup (not necessarily symmetric) with an invariant probability measure, if the curvature of its generator is bounded from below, then the ultracontractivity is equivalent to kP t exp[ 2 ]k 1 0: Especially, the above estimate of kP t k1!1 holds if L 2 c1 c2 for some c1 ; c2 > 0 and > 2: This estimate is sharp as is shown by examples given at the end of the paper. Corresponding results are proven for supe...
LIMITING DYNAMICS FOR SPHERICAL MODELS OF SPIN GLASSES WITH MAGNETIC FIELD
, 806
"... Abstract. We study the Langevin dynamics for the family of spherical spin glass models of statistical physics, in the presence of a magnetic field. We prove that in the limit of system size N approaching infinity, the empirical state correlation, the response function, the overlap and the magnetizat ..."
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Abstract. We study the Langevin dynamics for the family of spherical spin glass models of statistical physics, in the presence of a magnetic field. We prove that in the limit of system size N approaching infinity, the empirical state correlation, the response function, the overlap and the magnetization for these Ndimensional coupled diffusions converge to the nonrandom unique strong solution of four explicit nonlinear integrodifferential equations, that generalize the system proposed by Cugliandolo and Kurchan in the presence of a magnetic field. We then analyze the system and provide a rigorous derivation of the FDT regime in a large area of the temperaturemagnetization plane. 1.
Measure concentration through nonLipschitz observables and functional inequalities
"... E l e c t r o n ..."
Author manuscript, published in "Probability measures on groups, Mumbai: Inde (2004)" Functional inequalities for Markov semigroups
, 2009
"... 1 Abstract: In these notes, we describe some of the most interesting inequalities related to Markov semigroups, namely spectral gap inequalities, Logarithmic Sobolev inequalities and Sobolev inequalities. We show different aspects of their meanings and ..."
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1 Abstract: In these notes, we describe some of the most interesting inequalities related to Markov semigroups, namely spectral gap inequalities, Logarithmic Sobolev inequalities and Sobolev inequalities. We show different aspects of their meanings and