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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Robin Heat Semigroup and HWI Inequality on Manifolds with Boundary
TL;DR: In this article, a probabilistic formula for the semigroup generated by the reflecting diffusion process on a complete connected Riemannian manifold with boundary and its local time on the boundary is presented, where Hsu's gradient estimate and Bismut's derivative formula are established on a class of noncompact manifolds with boundary.
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The two-scale approach to hydrodynamic limits for non-reversible dynamics
Manh Hong Duong,Max Fathi +1 more
TL;DR: In this article, the authors generalize Grunewald's method to a family of non-reversible dynamics and obtain quantitative rates of convergence to the hydrodynamic limit for a weakly asymmetric version of the Ginzburg-Landau model endowed with Kawasaki dynamics.
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Transport proofs of some discrete variants of the Prékopa-Leindler inequality
TL;DR: In this paper, a transport proof of a discrete version of the displacement convexity of entropy on integers (Z) was given, and two discrete forms of the Prekopa-Leindler Inequality were obtained.
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Remark on the stability of Log-Sobolev inequality for Gaussian measure
TL;DR: In this paper, the authors bound the deficit in the logarithmic Sobolev Inequality and in the Talagrand transport-entropy Inequality for the Gaussian measure, in any dimension, by mean of a distance introduced by Bucur and Fragal.
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Problemi di ottimizzazione in teoria del trasporto di massa
TL;DR: In this paper, the authors discuss the relation between the teoria of trasporto di massa and the problem of ottimizzazione di forma, in which a data distribuzione of massa puo essere trasported in una configurazione diversa, ugualmente assegnata, minimizzando un funzionale that rappresenta the cost of the trasports.
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.