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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Exponential Convergence Towards Stationary States for the 1D Porous Medium Equation with Fractional Pressure
TL;DR: In this paper, a functional inequality involving the entropy, entropy dissipation, and the Euclidean transport distance is derived for the Riesz potential in one dimension, which is used to deduce the exponential convergence of solutions in self-similar variables to the unique steady states.
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Pathwise Otto calculus
TL;DR: In this article, a probabilistic interpretation of diffusion as entropic gradient flux is provided for it based on stochastic calculus, and the Cordero-Erausquin version of the so-called HWI inequality relating relative entropy, Fisher information and Wasserstein distance.
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Transportation Cost Inequalities for Neutral Functional SDEs
Jianhai Bao,Chenggui Yuan +1 more
TL;DR: In this paper, the Girsanov-transformation argument is used to establish the quadratic transportation cost inequalities for a class of finite-dimensional neutral functional stochastic differential equations.
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Heat Kernels, Stochastic Processes and Functional Inequalities
TL;DR: The 2013 workshop on heat kernels, stochastic processes and functional inequalities as mentioned in this paper brought together leading researchers from analysis, probability and geometry and provided a unique opportunity for interaction of established and young scientists from these areas.
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Strong Kac's chaos in the mean-field Bose-Einstein Condensation
TL;DR: In this paper, a stochastic approach to the mean field limit in Bose-Einstein Condensation is described and the convergence of the ground state energy as well as of its components are established.
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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.