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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Modified logarithmic Sobolev inequalities for canonical ensembles
TL;DR: In this article, the modified logarithmic Sobolev inequalities for canonical ensembles with superquadratic single-site potential were shown to converge in Wasserstein distance W p for Kawasaki dynamics on the Ginzburg−Landau's model.
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Transportation Cost Inequality on Path Spaces with Uniform Distance
TL;DR: In this paper, a class of reversible infinite dimensional diffusion processes on Riemnnian manifolds was constructed from a sequence of independent Wright-Fisher diffusion processes, and the transportation-cost inequality with respect to the uniform distance was established.
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A Dual Formula for the Noncommutative Transport Distance
TL;DR: In this article , a duality formula that can be understood as a quantum version of the dual Benamou-Brenier formulation of the Wasserstein distance in terms of subsolutions of a Hamilton-Jacobi-Bellmann equation is presented.
Sharp High-dimensional Central Limit Theorems for Log-concave Distributions
Xiao Fang,Yuta Koike +1 more
TL;DR: A new Gaussian coupling inequality is developed that gives almost dimension-free bounds for projected versions of p -Wasserstein distance for every p > 2 and a Cram´er type moderate deviation result for this normal approximation error is given.
Nesterov smoothing for sampling without smoothness
TL;DR: A novel sampling algorithm is proposed for a class of non-smooth potentials by approximating them by smooth potentials using a technique that is akin to Nesterov smoothing, and the accuracy of the algorithm is guaranteed.
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.