Generalization of an Inequality by Talagrand and Links with the Logarithmic Sobolev Inequality
Felix Otto,Cédric Villani +1 more
TLDR
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 of quantum channels on non-commutative transportation metric spaces
Li Gao,Cambyse Rouzé +1 more
TL;DR: In this paper, the coarse Ricci curvature of a quantum channel is defined as a non-commutative transportation cost in the spirit of [N. Gozlan and C. Leonard 2006], which gives a unified approach to different quantum Wasserstein distances.
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Private (Stochastic) Non-Convex Optimization Revisited: Second-Order Stationary Points and Excess Risks
TL;DR: In this paper , the authors proposed a new framework that utilizes two different kinds of gradient oracles to estimate the gradient of one point and the gradient difference between two points, and showed that the regularized exponential mechanism can closely match previous empirical and population risk bounds.
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Talagrand’s quadratic transportation cost inequalities for reflected SPDEs driven by space–time white noise
Ruinan Li,Yumeng Li +1 more
TL;DR: In this paper, the authors used Girsanov's transformation to prove Talagrand's quadratic transportation cost inequalities for the solutions of reflected SPDEs under the uniform norm on the continuous path space.
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On Prekopa-Leindler inequalities on metric-measure spaces
TL;DR: In this article, the Ricci curvature lower bound for metric spaces satisfying Borell-Brascamp-Lieb inequalities was shown to be equivalent to the lower bound on a Riemannian manifold.
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Stationary solutions and local equations for interacting diffusions on regular trees
Daniel Lacker,Jiacheng Zhang +1 more
TL;DR: In this article, the authors study the invariant measures of infinite systems of stochastic differential equations (SDEs) indexed by the vertices of a regular tree and derive existence and uniqueness results for the local equation and infinite SDE system.
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.