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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Transport inequalities for log-concave measures, quantitative forms and applications
TL;DR: In this article, simple techniques based on monotone mass transport that allow to obtain transport-type inequalities for any log-concave probability measure, and for more general measures as well, are discussed.
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The W-entropy formula for the Witten Laplacian on manifolds with time dependent metrics and potentials
Songzi Li,Xiang-Dong Li +1 more
TL;DR: In this paper, the W-entropy formula for the Witten Laplacian via warped product on Riemannian manifolds has been shown to be equivalent to the optimal logarithmic Sobolev constant on compact manifolds equipped with the K-super m-dimensional Bakry Emery Ricci flow.
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Optimal transport and Ricci curvature for metric-measure spaces
TL;DR: In this paper, the authors survey the work of Lott-Villani and Sturm on lower Ricci curvature bounds for metric-measure spaces, and present a survey of the results.
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Asymptotic behavior for doubly degenerate parabolic equations
TL;DR: In this paper, Agueh et al. used mass transportation inequalities to study the asymptotic behavior for a class of doubly degenerate parabolic equations of the form (1) ∂ρ ∂t = div ρ∇c ∗ ∇ F′(ρ)+V in (0,∞)×Ω, and ρ(t=0)=ρ 0 in {0}×ε, where Ω is R n, or a bounded domain of R n in which case ρ ∇c∗
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Mean-Field Langevin Dynamics : Exponential Convergence and Annealing
TL;DR: This work studies the annealed dynamics, and shows that for a noise decaying at a logarithmic rate, the dynamics converges in value to the global minimizer of the unregularized objective function.
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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.