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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Wasserstein geometry and Ricci curvature bounds for Poisson spaces
TL;DR: In this article , the authors studied the geometry of the configuration space over a complete and separable metric base space, endowed with the Poisson measure π and showed that the relative entropy is a complete geodesic space.
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Equivalence of the Poincar\'e inequality with a transport-chi-square inequality in dimension one
TL;DR: In this article, the Poincare inequality is equivalent to a transport-chi-square inequality linking the square of the quadratic Wasserstein distance with the chi-square pseudo-distance.
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Optimal Transportation and Functional Inequalities
TL;DR: In this article, the authors investigated the relation between optimal transportation and functional inequalities in the Markov operator framework and showed that the Ricci curvature lower bound of optimal transport can be derived from the corresponding functional inequalities.
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Equivalent Harnack and Gradient Inequalities for Pointwise Curvature Lower Bound
TL;DR: In this paper, an explicit log-Harnack inequality with local geometry quantities is established for (sub-Markovian) diffusion semigroups on a Riemannian manifold (possibly with boundary).
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Integrability of optimal mappings
TL;DR: In this paper, the integrability of optimal mappings T taking a probability measure μ to another measure g · μ was studied, where T minimizes the cost function c and μ satisfies some special inequalities related to c (the infimum convolution inequality or the logarithmic c-Sobolev inequality).
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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.