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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Multilevel Optimal Transport: a Fast Approximation of Wasserstein-1 distances
TL;DR: A fast algorithm for the calculation of the Wasserstein-1 distance, which is a particular type of optimal transport distance with homogeneous of degree one ground metric, is proposed, built on multilevel primal-dual algorithms.
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Nonlinear geometric analysis on Finsler manifolds
Shin-ichi Ohta,Shin-ichi Ohta +1 more
TL;DR: In this article, a survey article on recent progress of comparison geometry and geometric analysis on Finsler manifolds of weighted Ricci curvature bounded below is given, along with some gradient estimates, functional inequalities, and isoperimetric inequalities.
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Equivalent Harnack and gradient inequalities for pointwise curvature lower bound
TL;DR: In this article, an explicit log-Harnack inequality with local geometry quantities is established for (sub-Markovian) di usion semigroups on a Riemannian manifold (possibly with boundary).
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Private Convex Optimization via Exponential Mechanism
TL;DR: The Gaussian Differential Privacy (GDP) of the exponential mechanism if the loss function is strongly convex and the perturbation is Lipschitz is proved using the isoperimetric inequality for strongly log-concave measures.
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Lipschitzian norm estimate of one-dimensional Poisson equations and applications
Hacène Djellout,Liming Wu +1 more
TL;DR: In this article, the authors identify explicit the norme Lipschitzienne of the solution of the Poisson Equation of Poisson $-\mathcal {L}G=g$ en terme de different normes de g, ou $\mathcal{L}$ est l’operateur de Sturm-Liouville ou le generateur d'une diffusion non singuliere sur un intervalle.
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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.