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447
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 184 (9 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.
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 179 (15 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.
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. ..."
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Cited by 175 (9 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.
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 165 (13 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&infin;). Our results hold...
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 134 (32 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...
Twodimensional KellerSegel model: Optimal critical mass and qualitative properties of the solutions
 J. DIFF. EQNS
, 2006
"... 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 130 (15 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.
Best constants for GagliardoNirenberg inequalities and applications to nonlinear diffusions
, 2001
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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 122 (35 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.
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 113 (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.
Concentration of the Spectral Measure for Large Matrices
 ELECTRONIC COMMUNICATIONS IN PROBABILITY
, 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 fo ..."
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Cited by 103 (14 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 noncommutative functionals of random matrices.