Generalization of an Inequality by Talagrand and Links with the Logarithmic Sobolev Inequality
Felix Otto,Cédric Villani +1 more
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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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Degenerate Fokker-Planck Equations : Bismut Formula, Gradient Estimate and Harnack Inequality
TL;DR: In this article, successful couplings for degenerate diffusion processes, explicit derivative formula and Harnack type inequalities are presented for solutions to a class of degenerate Fokker-Planck equations on
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Modified logarithmic Sobolev inequalities and transportation inequalities
TL;DR: In this paper, a class of modified logarithmic Sobolev inequalities interpolating between Poincar\'e and LSS inequalities was presented, which are suitable for measures of the type $exp(exp(-|x|^\al)$ or more complex $exp((exp(|x||^ \al\log^\beta(2+|x |))$ ($\al\in]1,2[$ and be\in\dR$) which lead to new concentration inequalities.
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The one dimensional free Poincaré inequality
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TL;DR: In this article, the authors discuss the natural candidate for the one-dimensional free Poincare inequality in the planar limit of a matrix model and discuss the relations with the other free functional inequalities, namely, the free transportation and Log-Sobolev inequalities.
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Talagrand Inequality on Free Path Space and Application to Stochastic Reaction Diffusion Equations
TL;DR: In this article, the transportation cost inequality on the free path space of Markov processes was established by using a split argument due to [1], and the general result was applied to stochastic reaction diffusion equations with random initial values.
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Gradient flow structure and exponential decay of the sandwiched Rényi divergence for primitive Lindblad equations with GNS-detailed balance
Yu Cao,Jianfeng Lu,Yulong Lu +2 more
TL;DR: It is shown that the sandwiched R\'enyi divergence of any order ${\alpha} \in (0, \infty)$ decays exponentially fast under the time-evolution of such a Lindblad equation.
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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.