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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Ricci curvature, entropy and optimal transport
TL;DR: This lecture notes on the interplay between optimal transport and Riemannian geometry discusses geometric properties of general metric measure spaces satisfying this convexity condition.
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Improved Langevin Monte Carlo for stochastic optimization via landscape modification
Michael C.H. Choi,Youjia Wang +1 more
TL;DR: In this article , a Langevin Monte Carlo (LMC) algorithm was proposed for gradient-based sampling with a modified Gibbs distribution, where the landscape of the transformed landscape is a transformed version of that of the original landscape.
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Rigidity of some functional inequalities on RCD spaces.
TL;DR: In this paper, the authors studied the case of equality and proved a rigidity theorem concerning the 1-Bakry-Emery inequality in the non-smooth setting, which unifies and extends the results of Carlen-Kerce, Morgan, Bouyrie, and Ohta-Takatsu.
Wasserstein-p Bounds in the Central Limit Theorem Under Local Dependence
Tianle Liu,Morgane Austern +1 more
TL;DR: In this paper , the authors derived optimal rates in the central limit theorem for the Wasserstein-p distance for m-dependent random fields and U-statistics under conditions on the moments and the dependency neighborhoods.
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New results on relative entropy production, time reversal, and optimal control of time-inhomogeneous nonequilibrium diffusion processes
TL;DR: In this paper, the relative entropy of the time-inhomogeneous diffusion process with respect to the transient invariant probability measures is derived for both Brownian dynamics and Langevin dynamics.
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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.