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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Quadratic transportation inequalities for sdes with measurable drift
TL;DR: In this article, it was shown that the law of X satisfies a uniform quadratic transportation inequality when b is measurable and sigma is in an appropriate Sobolev space.
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The Poincar\'e inequality and quadratic transportation-variance inequalities
TL;DR: In this article, the Poincare inequality is shown to be equivalent to the quadratic transportation-variance inequality, which is known as the Lyapunov condition.
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A stochastic model of grain boundary dynamics: A Fokker-Planck perspective.
TL;DR: In this article, a Fokker-Planck model for the evolution of the planar grain boundary network is proposed, which considers anisotropic grain boundary energy which depends on lattice misorientation and takes into account mobility of the triple junctions.
Dissertation
Functional and transport inequalities and their applications to concentration of measure
TL;DR: In this article, the authors studied functional and transportation inequalities connected to the concentration of measure phenomenon and obtained improved (dimension-free) two-level concentration for products of such measures.
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Dimension-free log-Sobolev inequalities for mixture distributions
TL;DR: In this paper, it was shown that Gaussian convolutions of measures with bounded support enjoy dimension-free log-Sobolev inequalities in various settings of interest, in particular for Gaussian distributions.
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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.
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
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.