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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On Optimal Matching of Gaussian Samples
Michel Ledoux,Michel Ledoux +1 more
TL;DR: In this article, Ambrosio, Stra, and Trevisan showed that the left-hand side provides the correct order for some numerical constant C > 0, where W2 is the quadratic Kantorovich metric.
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Human behavior and lognormal distribution. A kinetic description
TL;DR: In recent years, it has been increasing evidence that lognormal distributions are widespread in physical and biological sciences, as well as in various phenomena of economics and social sciences.
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A characterization of dimension free concentration in terms of transportation inequalities
TL;DR: In this paper, it was shown that a probability measure concentrates independently of the dimension like a gaussian measure if and only if it verifies Talagrand's transportation-cost inequality.
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A curved brunn-minkowski inequality on the discrete hypercube, or: what is the ricci curvature of the discrete hypercube? ∗
Yann Ollivier,Cédric Villani +1 more
TL;DR: This work compares two approaches to Ricci curvature on nonsmooth spaces in the case of the discrete hypercube and gets new results of a combinatorial and probabilistic nature, including a curved Brunn--Minkowski inequality on the discretehypercube.
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ℒ-optimal transportation for Ricci flow
TL;DR: In this paper, the notion of L-optimal transportation was introduced and used to construct a natural monotonic quantity for Ricci flow which includes a selection of other monotonicity results, including some key discoveries of Perelman [13] (both related to entropy and to L-length).
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.