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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A Riemannian interpolation inequality a la Borell, Brascamp and Lieb
TL;DR: In this article, a concavity estimate is derived for interpolations between L 1 (M) mass densities on a Riemannian manifold, which sheds new light on the theorems of Prekopa, Leindler, Borell, Brascamp and Lieb.
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A survey of the Schrödinger problem and some of its connections with optimal transport
TL;DR: In this paper, the authors present the Schrodinger problem and some of its connections with optimal transport, and give a user's guide to the problem and a survey of the related literature.
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Gradient flows of the entropy for finite Markov chains
TL;DR: In this paper, the authors construct a metric W on the set of probability measures on a finite set X and show that with respect to this metric, the law of the continuous time Markov chain evolves as the gradient flow of the entropy.
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From Brunn-Minkowski to Brascamp-Lieb and to logarithmic Sobolev inequalities
Sergey G. Bobkov,Michel Ledoux +1 more
TL;DR: In this article, the authors developed several applications of the Brunn-Minkowski inequality in the Prekopa-Leindler form for convex potentials and obtained new results in this context.
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Transportation cost-information inequalities and applications to random dynamical systems and diffusions
TL;DR: In this article, a characterization of the L1-transportation cost-information inequality on a metric space and some appropriate sufficient condition to transportation cost information inequalities for dependent sequences are given. And applications to random dynamical systems and diffusions are studied.
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.