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
Citations
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Trajectorial dissipation and gradient flow for the relative entropy in Markov chains
TL;DR: The steepest descent and gradient flow properties for the variance, the relative entropy, and appropriate generalizations are studied along with their respective geometries under conditions of detailed balance, leading to a very direct proof for the HWI inequality of Otto and Villani in the present context.
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Fokker-Planck equation with respect to heat measures on loop groups
Shizan Fang,Jinghai Shao +1 more
TL;DR: In this article, the authors used Wasserstein distance variational method to solve the associated heat equation for a given data of finite entropy, where the Dirichlet form on the loop group L e (G ) with respect to the heat measure defines a Laplacian Δ DM on L e(G ).
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A variational approach to nonlinear and interacting diffusions
Marc Arnaudon,Pierre Del Moral +1 more
TL;DR: In this article, a variational calculus is presented to analyze the stability and the propagation of chaos properties of nonlinear and interacting diffusions, combining gradient flow estimates with backward stochastic interpolations, Lyapunov linearization techniques as well as spectral theory.
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A trajectorial approach to relative entropy dissipation of McKean–Vlasov diffusions: Gradient flows and HWBI inequalities
TL;DR: In this paper , a trajectorial version of the relative entropy dissipation identity for McKean-Vlasov diffusions is proposed, which is based on time-reversal of diffusions and Lions' differential calculus over Wasserstein space.
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A Decentralized Approach to Bayesian Learning.
TL;DR: This work proposes a collaborative Bayesian learning algorithm taking the form of decentralized Langevin dynamics in a non-convex setting and shows that the initial KL-divergence between the Markov Chain and the target posterior distribution is exponentially decreasing while the error contributions from the additive noise is decreasing in polynomial time.
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.