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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Displacement convexity of generalized relative entropies. II
Shin-ichi Ohta,Asuka Takatsu +1 more
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Integral criteria for transportation cost inequalities
TL;DR: In this paper, the authors provide a characterization of a large class of transportation-cost inequalities in terms of exponential integrability of the cost function under the reference probability measure, which completely extend the previous works by Djellout, Guillin and Wu (DGW03) and Bolley and Villani (BV03).
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Displacement convexity of entropy and related inequalities on graphs
TL;DR: In this article, the authors introduce the notion of an interpolating path on the set of probability measures on finite graphs and derive Prekopa-Leindler type inequalities, a Talagrand transport-entropy inequality, certain HWI type as well as log-Sobolev type inequalities in discrete settings.
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Bernstein type's concentration inequalities for symmetric Markov processes
TL;DR: In this article, the authors established Bernstein-type concentration inequalities for empirical means of functions of the Markov process, where the objective function is an unbounded observable of the symmetric Markov processes.
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An Optimal Transport View of Schrödinger's Equation
TL;DR: The Schrodinger equation is a lift of Newton's third law of motion on the space of probability measures, where derivatives are taken with respect to the Wasserstein Riemannian metric as discussed by the authors.
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.