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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Bakry-\'Emery curvature-dimension condition and Riemannian Ricci curvature bounds
TL;DR: In this article, the authors provide synthetic and abstract notions of lower Ricci curvature bounds for Riemannian energy measure spaces, and show that the tensorization property of these spaces is equivalent to the stability property of the Sturm-Gromov-Hausdorff convergence.
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Localization and Tensorization Properties of the Curvature-Dimension Condition for Metric Measure Spaces
TL;DR: Bacher and Sturm as mentioned in this paper proved the tensorization property of the curvature-dimension condition for metric measure spaces, including explicit dependence of constants, and added some detailed calculations, and comment on assumptions and conjectures concerning the local-to-global statement.
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Concentration of Measure Inequalities in Information Theory, Communications and Coding
Maxim Raginsky,Igal Sason +1 more
TL;DR: This monograph focuses on some of the key modern mathematical tools that are used for the derivation of concentration inequalities, on their links to information theory, and on their various applications to communications and coding.
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Partial differential equations and stochastic methods in molecular dynamics
Tony Lelièvre,Gabriel Stoltz +1 more
TL;DR: This review describes how techniques from the analysis of partial differential equations can be used to devise good algorithms and to quantify their efficiency and accuracy.
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Weighted Csiszár-Kullback-Pinsker inequalities and applications to transportation inequalities
François Bolley,Cédric Villani +1 more
TL;DR: In this article, the authors deduisons des inegalites de transport faisant intervenir certaines distances de Wasserstein, retrouvant en particulier l'equivalence d'une inegalite T 1 and de l'existence d'un moment carre-exponentiel.
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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.
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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.