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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Bounding relative entropy by the relative entropy of local specifications in product spaces
TL;DR: In this paper, an inequality between relative entropy and the sum of average conditional relative entropies of the following form was proved: for any density function $p^n(x^n)$ on $\Bbb R^n$, $D(pn||q^n)\leq Const.
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Hamilton-Jacobi equations on graph and applications
TL;DR: In this paper, the authors introduce a notion of gradient and an infimal convolution operator that extend properties of solutions of Hamilton Jacobi equations to more general spaces, in particular to graphs.
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Transportation inequalities for Markov kernels and their applications
TL;DR: In this paper, the authors studied the relationship between functional inequalities for a Markov kernel on a metric space X and inequalities of transportation distances on the space of probability measures P(X), and showed that contraction inequalities for these divergences are equivalent to reverse logarithmic Sobolev and Wang Harnack inequalities.
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Simulated annealing from continuum to discretization: a convergence analysis via the Eyring--Kramers law
Wenpin Tang,Xun Yu Zhou +1 more
TL;DR: In this paper, the convergence rate of continuous-time simulated annealing and its discretization for approximating the global optimum of a given function $f$ was studied and it was shown that the tail probability of the model is polynomial in time (resp in cumulative step size).
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One-Dimensional Fokker–Planck Equations and Functional Inequalities for Heavy Tailed Densities
TL;DR: In this paper , the problem of trend to equilibrium for one-dimensional Fokker-Planck equations modeling socio-economic problems, and functional inequalities of the type of Poincaré, Wirtinger and logarithmic Sobolev, with weight, for probability densities with polynomial tails are discussed.
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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.
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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.