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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W-entropy formulas on super Ricci flows and Langevin deformation on Wasserstein space over Riemannian manifolds
Songzi Li,Xiang-Dong Li +1 more
TL;DR: In this article, the W-entropy for the heat equation associated with the Witten Laplacian on super-Ricci flows and Langevin deformation on the Wasserstein space over Riemannian manifolds was studied.
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Optimal scaling for the transient phase of Metropolis Hastings algorithms: The longtime behavior
TL;DR: In this paper, the authors consider the random walk Metropolis algorithm with Gaussian proposals and obtain a diffusive limit for each component of the Markov chain, where the target probability measure is the $n$-fold product of a one-dimensional law.
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Rates of decay to equilibria for p-Laplacian type equations
TL;DR: In this paper, the long-time asymptotics for p -Laplacian type equations ρ t = Δ p ρ m = div ( | ∇ ρm | p − 2 ∆ ∆ | p−1 ∆ m ) in R n are studied for p > 1 and m ≥ n − p + 1 n (p − 1 ).
Synthetic theory of ricci curvature bounds
TL;DR: Synthetic theory of Ricci curvature bounds is reviewed in this article, from the conditions which led to its birth, up to some of its latest developments, with a review of the most recent developments.
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.