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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Global fluctuations for 1D log-gas dynamics
TL;DR: In this paper, the authors studied the hydrodynamic limit in the macroscopic regime of the coupled system of stochastic differential equations, and showed that fluctuations around the limit are Gaussian and satisfy an explicit PDE.
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Geometrical aspects of entropy production in stochastic thermodynamics based on Wasserstein distance
TL;DR: In this paper, the authors studied the relationship between optimal transport theory and stochastic thermodynamics for the Fokker-Planck equation and derived a lower bound on the partial entropy production by the Wasserstein distance as a generalization of the second law of information thermodynamics.
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Transportation Cost-Information Inequality for Stochastic Wave Equation
Yumeng Li,Xinyu Wang +1 more
TL;DR: In this article, the authors proved a Talagrand's transportation cost information inequality for the law of stochastic wave equation in spatial dimension $d=3$ driven by the Gaussian random field, white in time and correlated in space, on the continuous path space with respect to the weighted $L^{2}$ -norm on the weighted path space.
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Kurdyka-Lojasiewicz-Simon inequality for gradient flows in metric spaces
Daniel Hauer,José M. Mazón +1 more
TL;DR: In this article, the authors studied the trend to equilibrium of gradient flows in metric spaces in the entropy and metric sense, to establish decay rates, finite time of extinction, and to characterize Lyapunov stable equilibrium points.
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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.