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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Fokker-Planck Dynamics and Entropies for the Normalized Ricci Flow
TL;DR: In this article, the authors consider the geometry of the space of probability measures endowed with Wasserstein distance and characterize two monotonic functionals for the volume-normalized Ricci flow.
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Equivalent Semigroup Properties for Curvature-Dimension Condition
TL;DR: In this article, the curvature-dimension condition of the associated generator was used to derive the first eigenvalue, the log-Harnack inequality, the heat kernel estimates, and the HWI inequality for diffusion semigroups.
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Functional inequalities for forward and backward diffusions
Daniel Bartl,Ludovic Tangpi +1 more
TL;DR: In this article, the authors derive Talagrand's $T_2$ inequality on the path space w.r.t. the maximum norm for various stochastic processes.
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Stability of Gibbs Posteriors from the Wasserstein Loss for Bayesian Full Waveform Inversion
Matthew M. Dunlop,Yunan Yang +1 more
TL;DR: Recently, the Wasserstein loss function has been proven to be effective when applied to deterministic full-waveform inversion (FWI) problems as discussed by the authors, and the authors consider the application of this loss function in Ba...
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Kinetic Theory for Residual Neural Networks
TL;DR: A microscopic simplified residual neural network model is studied as the limit of infinitely many inputs, which leads to kinetic formulations of the SimResNet and those with respect to sensitivities and steady states are analyzed.
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.