Generalization of an Inequality by Talagrand and Links with the Logarithmic Sobolev Inequality
Felix Otto,Cédric Villani +1 more
Reads0
Chats0
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
More filters
Journal ArticleDOI
Prekopa-Leindler type inequalities on Riemannian manifolds, Jacobi fields, and optimal transport
TL;DR: In this paper, the authors investigated Prekopa-Leindler type inequalities on a Riemannian manifold M equipped with a measure with density e V where the potential V and the Ricci curvature satisfy Hessx V + Ricx I for all x 2 M, with some 2 R.
Journal ArticleDOI
Concentration inequalities for dependent Random variables via the martingale method
TL;DR: In this article, the martingale method is used to establish concentration inequalities for a class of dependent random sequences on a countable state space, with the constants in the inequalities expressed in terms of certain mixing coefficients.
Posted Content
Trend to equilibrium and particle approximation for a weakly selfconsistent Vlasov-Fokker-Planck equation
TL;DR: In this article, a Vlasov-Fokker-planck equation governing the evolution of the density of interacting and diffusive matter in the space of positions and velocities is considered and a probabilistic interpretation is used to obtain convergence towards equilibrium in Wasserstein distance with an explicit exponential rate.
Journal ArticleDOI
Perelman’s entropy formula for the Witten Laplacian on Riemannian manifolds via Bakry–Emery Ricci curvature
TL;DR: In this paper, an analogue of Perelman's and Ni's entropy formula for the heat equation associated with the Witten Laplacian on complete Riemannian manifolds via the Bakry-Emery Ricci curvature is studied.
Journal ArticleDOI
On quadratic transportation cost inequalities
Patrick Cattiaux,Arnaud Guillin +1 more
TL;DR: In this paper, the Poincare inequality was shown to be equivalent to the quadratic transportation cost inequalities, and new families of inequalities were introduced for relative entropy, which are equivalent to poincare inequalities for T 2 but not the logarithmic Sobolev inequality.
References
More filters
Book
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.
Journal ArticleDOI
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.
Journal ArticleDOI
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.
Journal ArticleDOI
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.