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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Wasserstein Distance, Fourier Series and Applications
TL;DR: In this paper, the Wasserstein metric has been studied from the perspective of Fourier analysis and applications, and it has been shown that the distance between the distribution of quadratic residues in a finite field can be bounded by a factor at most a factor.
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Certified dimension reduction in nonlinear Bayesian inverse problems
TL;DR: In this paper , the posterior distribution is approximated by a ridge function, i.e., a map which depends nontrivially only on a few linear combinations of the parameters.
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Exponential Contraction in Wasserstein Distances for Diffusion Semigroups with Negative Curvature
Feng-Yu Wang,Feng-Yu Wang +1 more
TL;DR: In this paper, the exponential contraction of the Bakry-Emery curvature of diffusion semigroups with negative curvature was shown to be bounded by a positive constant if and only if the curvature is larger than 1.
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The Kantorovich and variation distances between invariant measures of diffusions and nonlinear stationary Fokker-Planck-Kolmogorov equations
TL;DR: In this article, the authors obtained upper bounds for the total variation distance and the quadratic Kantorovich distance between stationary distributions of two diffusion processes with different drifts, and applied them to nonlinear stationary Fokker-Planck-Kolmogorov equations.
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On the time-dependent Fisher information of a density function
Omid Kharazmi,Majid Asadi +1 more
TL;DR: In this paper, a time-dependent version of Fisher information distance between the densities of two nonnegative random variables is proposed and several properties of the proposed measures and their relations to other statistical measures are investigated.
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.