Generalization of an Inequality by Talagrand and Links with the Logarithmic Sobolev Inequality
Felix Otto,Cédric Villani +1 more
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.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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Entropy dissipation semi-discretization schemes for Fokker-Planck equations
TL;DR: In this article, a semi-discretization scheme was proposed to approximate nonlinear Fokker-Planck equations by exploiting the gradient flow structures with respect to the 2-Wasserstein metric.
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A family of functional inequalities: Lojasiewicz inequalities and displacement convex functions
Adrien Blanchet,Jérôme Bolte +1 more
TL;DR: In this article, the equivalence between the gradient and functional type Łoja\-sie\-wicz inequalities was shown for displacement convex functionals in the probability space equipped with the Monge-Kantorovich metric.
Dissertation
Analysis and Geometry of RCD spaces via the Schrödinger problem
TL;DR: In this article, a schema d'approximation robuste jusqu'au deuxieme ordre and different de Brenier-McCann is presented, which permet d'etablir the formule de derivation du deux-ieme Ordre le long des geodesiques Wasserstein din le cadre de espaces RCD* de dimension finie.
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From Super Poincar\'e to Weighted Log-Sobolev and Entropy-Cost Inequalities
TL;DR: In this article, the authors derived weighted log-Sobolev inequalities from a class of super Poincar\'e inequalities for a complete connected Riemannian manifold.
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Trajectorial Otto calculus
TL;DR: In this paper, the authors revisited the variational characterization of diffusion as entropic gradient flux and provided for it a probabilistic interpretation based on stochastic calculus.
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.