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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Rédacteurs en chef / Chief Editors
Thierry Bodineau,Route de Saclay,Lorenzo Zambotti,Comité de Rédaction,V. B Aladi,M. Barlow,G. Blanchard,B. Collins,I. Corwin,F. Delarue,F. Flandoli,Giambattista Giacomin,Alice Guionnet,Martin Hairer,M. Hoffmann,A. Holroyd,G. Miermont,Leonid Mytnik,E. Perkins,G. Pete,Z. Shi,Bálint Tóth +21 more
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A Feynman-Kac approach for Logarithmic Sobolev Inequalities
TL;DR: In this paper, a method based on Feynman-kac semigroups for logarithmic Sobolev inequalities is presented, which goes beyond the Bakry-Emery criterion and allows to investigate high-dimensional effects on the optimal log-rithmic SOP.
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Instability results for the logarithmic Sobolev inequality and its application to related inequalities
TL;DR: In this article, the authors construct a sequence of centered probability measures such that the deficit of the logarithmic Sobolev inequality converges to zero but the relative entropy and the moments do not.
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Subgaussian 1-cocycles on discrete groups
Marius Junge,Qiang Zeng +1 more
TL;DR: In this article, it was shown that the spectral gap inequality implies the $L_p$ Poincar\'e inequalities with constant $C\sqrt{p}$ for 1-cocycles on countable discrete groups under Bakry-Emery's $Gamma_2$-criterion.
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Irregularity of Distribution in Wasserstein Distance
TL;DR: In this paper, the Wasserstein-p distance was derived for the irregularity of distribution of sequences on the interval and circle, and the Erdős-Turan inequality was established.
References
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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.