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Open AccessJournal ArticleDOI

Generalization of an Inequality by Talagrand and Links with the Logarithmic Sobolev Inequality

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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.
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Journal ArticleDOI

A new characterization of Talagrand’s transport-entropy inequalities and applications

TL;DR: In this article, it was shown that Talagrand's transport inequality is equivalent to a restricted logarithmic Sobolev inequality and that it is stable under bounded perturbations.
Book ChapterDOI

Lecture Notes on Gradient Flows and Optimal Transport

TL;DR: In this article, the strongest variational formulation for gradient flows of geodesically λ-convex functionals in metric spaces is presented, with applications to diffusion equations in Wasserstein spaces of probability measures.
Journal ArticleDOI

From log Sobolev to Talagrand: A quick proof

TL;DR: In this paper, the authors provide yet another proof of the Otto-Villani theorem from the log Sobolev inequality to the Talagrand transportation cost inequality valid in arbitrary metric measure spaces.
Journal ArticleDOI

Concentration for multidimensional diffusions and their boundary local times

TL;DR: In this article, it was shown that probability laws of certain multidimensional semimartingales which includes time-inhomogenous diffusions, under suitable assumptions, satisfy quadratic transportation cost inequality under the uniform metric.
Posted Content

Exponential Contraction in Wasserstein Distances for Diffusion Semigroups with Negative Curvature

TL;DR: In this paper, it was shown that the Bakry-Emery curvature of diffusion semigroups with negative curvature is bounded by a positive constant if and only if W_p(mu_1P_t, \mu_2P-t) is bounded below by a constant.
References
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Book

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.
Journal ArticleDOI

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.
Journal ArticleDOI

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

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.
Journal ArticleDOI

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.