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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The Quasi Curvature‐Dimension Condition with Applications to Sub‐Riemannian Manifolds
TL;DR: In this article, a quasi-convex relaxation of the Lott-Sturm-Villani condition was introduced, called the Quasi Curvature-Dimension condition (Q,K,N) for ideal sub-Riemannian manifolds.
Book ChapterDOI
Statistical Description of Human Addiction Phenomena
TL;DR: In this paper, the authors studied the evolution in time of the statistical distribution of some addiction phenomena in a system of individuals and built up a novel class of Fokker-Planck equations describing relaxation of the probability density solution towards a generalized Gamma density.
Book ChapterDOI
Entropic Ricci Curvature for Discrete Spaces
TL;DR: In this paper, the Ricci curvature for discrete spaces has been studied for a metric which is similar to (but different from) the 2-Wasserstein metric, based on geodesic convexity properties of the relative entropy along geodesics.
Posted Content
A semigroup approach to Finsler geometry: Bakry--Ledoux's isoperimetric inequality
TL;DR: In this article, the authors developed the celebrated semigroup approach a la Bakry et al on Finsler manifolds, where natural Laplacian and heat semigroup are nonlinear, based on the Bochner-Weitzenbock formula established by Sturm and the author.
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Dimensional improvements of the logarithmic Sobolev, Talagrand and Brascamp-Lieb inequalities
TL;DR: In this article, the authors consider dimensional improvements of the logarithmic Sobolev, Talagrand and Brascamp-Lieb inequalities, which lead to sharpened concentration properties as well as refined contraction bounds, convergence to equilibrium and short time behaviour for the laws of solutions to stochastic differential equations.
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.
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.
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.