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52
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.
Fastest Mixing Markov Chain on A Graph
 SIAM REVIEW
, 2003
"... We consider a symmetric random walk on a connected graph, where each edge is labeled with the probability of transition between the two adjacent vertices. The associated Markov chain has a uniform equilibrium distribution; the rate of convergence to this distribution, i.e. the mixing rate of the Mar ..."
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Cited by 90 (15 self)
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We consider a symmetric random walk on a connected graph, where each edge is labeled with the probability of transition between the two adjacent vertices. The associated Markov chain has a uniform equilibrium distribution; the rate of convergence to this distribution, i.e. the mixing rate of the Markov chain, is determined by the second largest (in magnitude) eigenvalue of the transition matrix. In this paper we address the problem of assigning probabilities to the edges of the graph in such a way as to minimize the second largest magnitude eigenvalue, i.e., the problem of finding the fastest mixing Markov chain on the graph. We show that
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...
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
"... ..."
Isoperimetric Constants For Product Probability Measures
 Ann. Probab
, 1995
"... A dimension free lower bound is found for isoperimetric constants of product probability measures. From this, some analytic inequalities are derived. 1 Introduction Let (X; d) be a metric space equipped with a nonatomic, separable Borel probability measure ¯. In the present paper we study the qua ..."
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Cited by 28 (5 self)
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A dimension free lower bound is found for isoperimetric constants of product probability measures. From this, some analytic inequalities are derived. 1 Introduction Let (X; d) be a metric space equipped with a nonatomic, separable Borel probability measure ¯. In the present paper we study the quantity Is(¯) = inf ¯ + (A) min(¯(A); 1 \Gamma ¯(A)) (1:1) Key words: Isoperimetry, Poincar'e Inequalities, Cheeger's inequality, Khinchine Kahane inequality, Holder's Inequality y Research supported in part by the ISF grant NZX000 and NZX300. This author enjoyed the hospitality of the Faculty of Wiskunde and Informatica, Free University of Amsterdam, while part of this research was carried out. z Research supported in part by an NSF Postdoctoral Fellowship. This author enjoyed the hospitality of Le Cermics, ENPC, France, of the Steklov Mathematical Institute (Sankt Petersburg branch) and of the Department of Mathematics, University of Syktyvkar, Russia, while part of this research...
Quasifactorization of the entropy and logarithmic Sobolev inequalities for Gibbs random fields
, 2001
"... . We show that the entropy functional exhibits a quasi{factorization property with respect to a pair of weakly dependent {algebras. As an application we give a simple proof that the Dobrushin and Shlosman's complete analyticity condition, for a Gibbs specication with nite range summable interaction, ..."
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Cited by 21 (0 self)
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. We show that the entropy functional exhibits a quasi{factorization property with respect to a pair of weakly dependent {algebras. As an application we give a simple proof that the Dobrushin and Shlosman's complete analyticity condition, for a Gibbs specication with nite range summable interaction, implies uniform logarithmic Sobolev inequalities. This result has been previously proven using several dierent techniques. The advantage of our approach is that it relies almost entirely on a general property of the entropy, while very little is assumed on the Dirichlet form. No topology is introduced on the single spin space, thus discrete and continuous spins can be treated in the same way. Key Words: Entropy, Logarithmic Sobolev inequalities, Gibbs measures Mathematics Subject Classication 2000: 82B20, 82C20, 39B62 v1.0 1. Introduction Logarithmic Sobolev inequalities have been introduced in [Gr1] where it has been shown that Z R d f 2 (x) log jf(x)j d (dx) Z R d jr...
INTRINSIC ULTRACONTRACTIVITY OF NONSYMMETRIC DIFFUSION SEMIGROUPS IN BOUNDED DOMAINS
 TOHOKU MATH. J.
, 2008
"... We extend the concept of intrinsic ultracontractivity to nonsymmetric semigroups and prove the intrinsic ultracontractivity of the Dirichlet semigroups of nonsymmetric second order elliptic operators in bounded Lipschitz domains. ..."
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Cited by 21 (18 self)
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We extend the concept of intrinsic ultracontractivity to nonsymmetric semigroups and prove the intrinsic ultracontractivity of the Dirichlet semigroups of nonsymmetric second order elliptic operators in bounded Lipschitz domains.
Hypercontractivity for perturbed diffusion semigroups
 ANN. FAC. DES SC. DE TOULOUSE
, 2005
"... µ being a nonnegative measure satisfying some LogSobolev inequality, we give conditions on F for the Boltzmann measure ν = e −2F µ to also satisfy some LogSobolev inequality. This paper improves and completes the final section in [6]. A general sufficient condition and a general necessary conditio ..."
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Cited by 20 (14 self)
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µ being a nonnegative measure satisfying some LogSobolev inequality, we give conditions on F for the Boltzmann measure ν = e −2F µ to also satisfy some LogSobolev inequality. This paper improves and completes the final section in [6]. A general sufficient condition and a general necessary condition are given and examples are explicitly studied.