Results 1  10
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201
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 Talagrand's Deviation Inequalities For Product Measures
, 1996
"... We present a new and simple approach to some of the deviation inequalities for product measures deeply investigated by M. ..."
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Cited by 83 (0 self)
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We present a new and simple approach to some of the deviation inequalities for product measures deeply investigated by M.
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.
Entropy dissipation methods for degenerate parabolic problems and generalized Sobolev inequalities
 MONATSH. MATH
, 1999
"... We analyse the largetime asymptotics of quasilinear (possibly) degenerate parabolic systems in three cases: 1) scalar problems with confinement by a uniformly convex potential, 2) unconfined scalar equations and 3) unconfined systems. In particular we are interested in the rate of decay to equilibr ..."
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Cited by 75 (21 self)
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We analyse the largetime asymptotics of quasilinear (possibly) degenerate parabolic systems in three cases: 1) scalar problems with confinement by a uniformly convex potential, 2) unconfined scalar equations and 3) unconfined systems. In particular we are interested in the rate of decay to equilibrium or selfsimilar solutions. The main analytical tool is based on the analysis of the entropy dissipation. In the scalar case this is done by proving decay of the entropy dissipation rate and bootstrapping back to show convergence of the relative entropy to zero. As byproduct, this approach gives generalized Sobolevinequalities, which interpolate between the Gross logarithmic Sobolev inequality and the classical Sobolev inequality. The time decay of the solutions of the degenerate systems is analyzed by means of a generalisation of the Nash inequality. Porous media, fast diffusion, pLaplace and energy transport systems are included in the considered class of problems. A generalized Csiz...
The BrunnMinkowski inequality
 Bull. Amer. Math. Soc. (N.S
, 2002
"... Abstract. In 1978, Osserman [124] wrote an extensive survey on the isoperimetric inequality. The BrunnMinkowski inequality can be proved in a page, yet quickly yields the classical isoperimetric inequality for important classes of subsets of R n, and deserves to be better known. This guide explains ..."
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Cited by 74 (5 self)
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Abstract. In 1978, Osserman [124] wrote an extensive survey on the isoperimetric inequality. The BrunnMinkowski inequality can be proved in a page, yet quickly yields the classical isoperimetric inequality for important classes of subsets of R n, and deserves to be better known. This guide explains the relationship between the BrunnMinkowski inequality and other inequalities in geometry and analysis, and some applications. 1.
Concentration of the Spectral Measure for Large Matrices
, 2000
"... We derive concentration inequalities for functions of the empirical measure of eigenvalues for large, random, self adjoint matrices, with not necessarily Gaussian entries. The results presented apply in particular to nonGaussian Wigner and Wishart matrices. We also provide concentration bounds for ..."
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Cited by 65 (11 self)
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We derive concentration inequalities for functions of the empirical measure of eigenvalues for large, random, self adjoint matrices, with not necessarily Gaussian entries. The results presented apply in particular to nonGaussian Wigner and Wishart matrices. We also provide concentration bounds for non commutative functionals of random matrices. 1 Introduction and statement of results Consider a random N N Hermitian matrix X with i.i.d. complex entries (except for the symmetry constraint) satisfying a moment condition. It is well known since Wigner [28] that the spectral measure of N 1=2 X converges to the semicircle law. This observation has been generalized to a large class of matrices, e.g. sample covariance matrices of the form XRX where R is a deterministic diagonal matrix ([19]), band matrices (see [5, 16, 20]), etc. For the Wigner case, this convergence has been supplemented by Central Limit Theorems, see [15] for the case of Gaussian entries and [17], [22] for the gen...
Best constants for GagliardoNirenberg inequalities and applications to nonlinear diffusions
"... In this paper, we find optimal constants of a special class of GagliardoNirenberg type inequalities which turns out to interpolate between the classical Sobolev inequality and the Gross logarithmic Sobolev inequality. These inequalities provide an optimal decay rate (measured by entropy methods) of ..."
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Cited by 63 (16 self)
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In this paper, we find optimal constants of a special class of GagliardoNirenberg type inequalities which turns out to interpolate between the classical Sobolev inequality and the Gross logarithmic Sobolev inequality. These inequalities provide an optimal decay rate (measured by entropy methods) of the intermediate asymptotics of solutions to nonlinear diffusion equations.
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.
Twodimensional KellerSegel model: Optimal critical mass and qualitative properties of the solutions, in "Electron
 J. Diff. Eqns
"... The KellerSegel system describes the collective motion of cells which are attracted by a chemical substance and are able to emit it. In its simplest form it is a conservative driftdiffusion equation for the cell density coupled to an elliptic equation for the chemoattractant concentration. It is k ..."
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Cited by 52 (8 self)
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The KellerSegel system describes the collective motion of cells which are attracted by a chemical substance and are able to emit it. In its simplest form it is a conservative driftdiffusion equation for the cell density coupled to an elliptic equation for the chemoattractant concentration. It is known that, in two space dimensions, for small initial mass, there is global existence of solutions and for large initial mass blowup occurs. In this paper we complete this picture and give a detailed proof of the existence of weak solutions below the critical mass, above which any solution blowsup in finite time in the whole euclidean space. Using hypercontractivity methods, we establish regularity results which allow us to prove an inequality relating the free energy and its time derivative. For a solution with subcritical mass, this allows us to give for large times an “intermediate asymptotics ” description of the vanishing. In selfsimilar coordinates, we actually prove a convergence result to a limiting selfsimilar solution which is not a simple reflect of the diffusion. Résumé