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
Citations
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A note on Talagrand’s transportation inequality and logarithmic Sobolev inequality
TL;DR: In this article, sufficient conditions for Talagrand's transportation information inequality and for the logarithmic Sobolev inequality were given for the case where the Bakry-Emery curvature is not lower bounded.
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Optimal Transport for Gaussian Mixture Models
TL;DR: An optimal mass transport framework on the space of Gaussian mixture models as a submanifold of probability densities equipped with the Wasserstein metric is introduced, which provides natural ways to compare, interpolate, and average Gaussia mixture models.
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Sharp geometric and functional inequalities in metric measure spaces with lower Ricci curvature bounds
Fabio Cavalletti,Andrea Mondino +1 more
TL;DR: For metric measure spaces satisfying the reduced curvature-dimension condition CD∗(K,N) as discussed by the authors, a series of sharp functional inequalities under the additional "essentially nonbranching" assumption were proved.
Posted Content
Certified dimension reduction in nonlinear Bayesian inverse problems
TL;DR: A dimension reduction technique for Bayesian inverse problems with nonlinear forward operators, non-Gaussian priors, and non- Gaussian observation noise is proposed and an analysis that enables control of the posterior approximation error due to this sampling is provided.
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Equivalent semigroup properties for the curvature-dimension condition
Feng-Yu Wang,Feng-Yu Wang +1 more
TL;DR: In this article, the curvature-dimension condition of the associated generator was used to derive the first eigenvalue, the log-Harnack inequality, the heat kernel estimates, and the HWI inequality for diffusion semigroups.
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.