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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Relating relative entropy, optimal transport and Fisher information: a quantum HWI inequality
Cambyse Rouzé,Nilanjana Datta +1 more
TL;DR: In this article, a quantum version of the Ricci lower bound is considered and it is shown that it implies a quantum HWI inequality from which the quantum functional and transportation cost inequalities follow.
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Transport-entropy inequalities on the line
TL;DR: In this article, a necessary and sufficient condition for transport-entropy inequalities in dimension one is given and a new example of a probability distribution verifying Talagrand's T2 inequality and not the logarithmic Sobolev inequality is constructed.
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Mean-field Langevin System, Optimal Control and Deep Neural Networks
TL;DR: A system of mean-field Langevin equations, the invariant measure of which is shown to be the optimal control of the initial problem under mild conditions, is introduced and endorses the solvability of the stochastic gradient descent algorithm for a wide class of deep neural networks.
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Some functional inequalities on non-reversible Finsler manifolds
TL;DR: In this article, the Bochner inequality on non-reversible Finsler manifolds was studied and the dimensional versions of the Poincare-Lichnerowicz inequality, the logarithmic Sobolev inequality, and the Sobolevev inequality were obtained by Cavalletti and Mondino.
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Transport proofs of weighted Poincar\'e inequalities for log-concave distributions
TL;DR: In this article, it was shown that the variance conjecture is true for increments of log-concave martingales, and the weighted Poincar'e inequalities for random vectors satisfying some centering conditions.
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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.
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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.