Generalization of an Inequality by Talagrand and Links with the Logarithmic Sobolev Inequality
Felix Otto,Cédric Villani +1 more
TLDR
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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Modified logarithmic Sobolev inequalities on R
Franck Barthe,Cyril Roberto +1 more
TL;DR: In this paper, Gentil, Guillin and Miclo provided a sufficient condition for a measure on the real line to satisfy a modified logarithmic Sobolev inequality, thus extending the criterion of Bobkov and Gotze.
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From super Poincaré to weighted log-Sobolev and entropy-cost inequalities
Feng-Yu Wang,Feng-Yu Wang +1 more
TL;DR: In this article, the authors derived weighted log-Sobolev inequalities from a class of super Poincare inequalities and proved that they are equivalent if the curvature of the corresponding generator is bounded below.
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Wasserstein information matrix
Wuchen Li,Wuchen Li,Jiaxi Zhao +2 more
TL;DR: Wasserstein score functions and covariance operators in statistical models are introduced and Wasserstein-Cramer-Rao bounds for estimations are established, and the asymptotic behaviors and efficiency of estimators are considered.
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The mean field Schr\"odinger problem: ergodic behavior, entropy estimates and functional inequalities
TL;DR: In this paper, the mean field Schrodinger problem (MFSP) is studied in terms of an optimization problem with marginal constraints whose objective function is the large deviation rate function associated with a system of weakly dependent Brownian particles.
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Some Geometric PDEs Related to Hydrodynamics and Electrodynamics
TL;DR: In this paper, several geometric PDEs and their relationship with Hydrodydenics and classical Electrodynamics are discussed. But the main focus of this paper is on the Vlasov-Maxwell system.
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.