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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The Wasserstein geometry of non-linear sigma models and the Hamilton-Perelman Ricci flow
TL;DR: In this paper, the relation between non-linear sigma models and Ricci flow theory is discussed, and a rigorous model for the embedding of Ricci flows into the renormalization group flow for non linear sigmoid models is provided.
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Modified log-Sobolev inequalities and isoperimetry
TL;DR: In this article, the authors show that a probability measure can be defined to satisfy an inequality of the type (i.e., if the cost function is convex and the cost functions are concave) under broad assumptions on the costs and costs.
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Molecules as metric measure spaces with Kato-bounded Ricci curvature
Batu Güneysu,Max von Renesse +1 more
TL;DR: In this paper, it was shown that the metric measure space given by the Coulomb potential has a Bakry-Emery-Ricci tensor, which is an element of the Kato class induced by the metric.
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Dissipation of information in channels with input constraints
Yury Polyanskiy,Yihong Wu +1 more
TL;DR: This paper investigates channels with an average input cost constraint and finds that, while the contraction coefficient typically equals one (no contraction), the information nevertheless dissipates.
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Transport inequalities for random point measures
TL;DR: In this article, the authors derive transport-entropy inequalities for mixed binomial point processes and for Poisson point processes, and explore the consequences of these inequalities in terms of concentration of measure and modified logarithmic Sobolev inequalities.
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.