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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Talagrand’s T 2 -transportation Inequality w.r.t. a Uniform Metric for Diffusions
TL;DR: In this article, the authors used Girsanov transformation to establish the Talagrand's T 2 -inequality for diffusion on the path space C([0,N],ℝ d ) with respect to a uniform metric, with the constant independent of N.
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Global Well-posedness of the Spatially Homogeneous Kolmogorov–Vicsek Model as a Gradient Flow
TL;DR: In this paper, the authors consider the spatially homogeneous Kolmogorov-Vicsek model and prove the global existence and uniqueness of weak solutions to the Fokker-Planck equation.
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From large deviations to wasserstein gradient flows in multiple dimensions
TL;DR: In this paper, a new proof of the upper bound for the large deviation rate functional for the empirical distribution of independent Brownian particles with drift in arbitrary dimensions was presented, thereby generalising the result of Adams et al. to arbitrary dimensions.
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From concentration to logarithmic Sobolev and Poincaré inequalities
TL;DR: In this paper, it was shown that Gaussian concentration implies the logarithmic Sobolev inequality when the curvature is bounded from below, and also that exponential concentration implies Poincare inequality under null curvature condition.
Posted Content
Benamou-Brenier and duality formulas for the entropic cost on $RCD^*(K,N)$ spaces
Nicola Gigli,Luca Tamanini +1 more
TL;DR: In this paper, it was shown that the entropic cost of the Schrodinger problem admits a three-fold variational representation, in the spirit of the Benamou-Brenier formula for the Wasserstein distance, in line with Bobkov-Gentil-Ledoux and Otto-Villani results.
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.