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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Analyzing the Generalization Capability of SGLD Using Properties of Gaussian Channels.
TL;DR: In this paper, a generalization bound for stochastic gradient Langevin dynamics (SGLD) was derived by connecting SGLD with Gaussian channels found in information and communication theory.
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A Perspective on Lindblad’s Non-Equilibrium Entropy
Erik Aurell,Ryoichi Kawai +1 more
TL;DR: In this article , the authors summarize the contents of this book, and provide a perspective on its relation to later developments in statistical physics and quantum physics, concluding that Lindblad's proposal for thermodynamic entropy foreshadows much more recent investigations into optimal quantum transport which is a current research focus in several fields.
Differential Geometric Heuristics for Riemannian Optimal Mass Transportation
TL;DR: In this article, the authors give an account on Otto's geometrical heuristics for realizing, on a compact Riemannian manifold M,t heL 2 Wasserstein distance restricted to smooth positive probability measures, as a Riemmannian distance.
Talagrand's transportation inequality for SPDEs with locally monotone drifts
Ruinan Li,Xinyu Wang +1 more
TL;DR: In this article , the authors proved transportation inequalities for a class of non-linear monotone stochastic partial differential equations (SPDEs) driven by Wiener noise, including the Burgers type equation and the Navier-Stokes equation.
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Entropy-type inequalities for generalized Gamma densities
TL;DR: In this paper, the authors investigated the relaxation to equilibrium of the solution of a class of one-dimensional linear Fokker-Planck type equations that have been recently considered in connection with the study of addiction phenomena in a system of individuals.
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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.
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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.