Generalization of an Inequality by Talagrand and Links with the Logarithmic Sobolev Inequality
Felix Otto,Cédric Villani +1 more
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In this paper, it was shown that transport inequalities, similar to the one derived by M. Talagrand (1996, Geom. Funct. Anal. 6, 587-600) for the Gaussian measure, are implied by logarithmic Sobolev inequalities.About:
This article is published in Journal of Functional Analysis.The article was published on 2000-06-01 and is currently open access. It has received 1080 citations till now. The article focuses on the topics: Sobolev inequality & Interpolation inequality.read more
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Uniform Long-Time and Propagation of Chaos Estimates for Mean Field Kinetic Particles in Non-convex Landscapes
Arnaud Guillin,Pierre Monmarché +1 more
TL;DR: In this article, the trend to equilibrium in large time is studied for a large particle system associated to a Vlasov-Fokker-Planck equation, and the convergence rate is proven to be independent from the number of particles.
Birth-death dynamics for sampling: Global convergence, approximations and their asymptotics
TL;DR: In this paper , the authors study a continuum birth-death dynamics and provide weaker hypotheses under which the probability density of the birthdeath governed by Kullback-Leibler divergence or by $\chi^2$ divergence converge exponentially fast to the Gibbs equilibrium measure, with a universal rate that is independent of the potential barrier.
Inégalités de Poincaré et de Gross pour les mesures de Bernoulli, de Poisson, et de Gauss
TL;DR: Les inegalites de Sobolev logarithmiques doivent leur nom a un article celebre de Gross paru en 1975 as discussed by the authors, which obtenues for les mesures de Poisson et de Gauss, par tensorisation infinie, a partir d'inegalites optimales for des mesures of Bernoulli sur l'espace a deux points.
Posted Content
Mass transport and variants of the logarithmic Sobolev inequality
TL;DR: In this article, the optimal transportation approach to modified log-Sobolev inequalities and isoperimetric inequalities is developed and sufficient conditions for such inequalities are given, some of them are new even in the classical logSobolerv case, and 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.
Dissertation
Phénomènes de concentration en grande dimension, transport de mesure et inégalités fonctionnelles.
TL;DR: In this paper, a general version of the measure concentration principle is presented and some elementary arguments are given for a better understanding of the mathematical tools, including functional inequalities related to the concentration of measure phenomenon, in particular the transport-entropy inequalities.
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Measure theory and fine properties of functions
TL;DR: In this article, the authors define and define elementary properties of BV functions, including the following: Sobolev Inequalities Compactness Capacity Quasicontinuity Precise Representations of Soboleve Functions Differentiability on Lines BV Function Differentiability and Structure Theorem Approximation and Compactness Traces Extensions Coarea Formula for BV Functions isoperimetric inequalities The Reduced Boundary The Measure Theoretic Boundary Gauss-Green Theorem Pointwise Properties this article.
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Polar Factorization and Monotone Rearrangement of Vector-Valued Functions
TL;DR: In this paper, it was shown that for every vector-valued function u Lp(X, p; Rd) there is a unique polar factorization u = V$.s, where $ is a convex function defined on R and s is a measure-preserving mapping from (x, p) into (Q, I. I), provided that u is nondegenerate.
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The geometry of dissipative evolution equations: the porous medium equation
TL;DR: In this paper, the authors show that the porous medium equation has a gradient flow structure which is both physically and mathematically natural, and they use the intuition and the calculus of Riemannian geometry to quantify this asymptotic behavior.
Book
Topological methods in hydrodynamics
Vladimir I. Arnold,Boris Khesin +1 more
TL;DR: A group theoretical approach to hydrodynamics is proposed in this article, where the authors consider the hydrodynamic geometry of diffeomorphism groups and the principle of least action implies that the motion of a fluid is described by geodesics on the group in the right-invariant Riemannian metric given by the kinetic energy.
Journal ArticleDOI
The variational formulation of the Fokker-Planck equation
TL;DR: The Fokker-Planck equation as mentioned in this paper describes the evolution of the probability density for a stochastic process associated with an Ito Stochastic Differential Equation.