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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Concentration of quantum states from quantum functional and Talagrand inequalities
Cambyse Rouzé,Nilanjana Datta +1 more
TL;DR: In this paper, the authors introduce quantum generalizations of T1 and T2, making use of appropriate quantum versions of the Wasserstein distances, one recently defined by Carlen and Maas and the other defined by us.
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Coarse-graining of non-reversible stochastic differential equations: quantitative results and connections to averaging
TL;DR: This work extends recent results on effective reversible dynamics by conditional expectations to the setting of general non-reversible processes with non-constant diffusion coefficient and proves relative entropy and Wasserstein error estimates for the difference between the time marginals of the effective and original dynamics.
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A quantitative log-Sobolev inequality for a two parameter family of functions
Emanuel Indrei,Diego Marcon +1 more
TL;DR: In this paper, a sharp, dimension-free stability result for the classical logarithmic Sobolev inequality for a two-parameter family of functions was proved for a certain class of log $C^{1,1}$ functions.
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A sharp uniform-in-time error estimate for Stochastic Gradient Langevin Dynamics
Lei Li,Yuliang Wang +1 more
TL;DR: A sharp uniform-in-time error estimate is established for the Stochastic Gradient Langevin Dynamics (SGLD) and an O ( η ) bound is obtained for the distance between the SGLD iteration and the invariant distribution of the Langevin diffusion, in terms of Wasserstein or total variation distances.
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A new approach to posterior contraction rates via Wasserstein dynamics
TL;DR: This paper presents a new approach to the classical problem of quantifying posterior contraction rates (PCRs) in Bayesian statistics based on Wasserstein distance, and it leads to two main contributions which improve on the existing literature of PCRs.
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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.