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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On the Law of Large Numbers for the Empirical Measure Process of Generalized Dyson Brownian Motion
TL;DR: In this article, the authors studied the generalized Dyson Brownian motion (GDBM) of an interacting N-particle system with logarithmic Coulomb interaction and general potential V. Under reasonable condition on V, they proved the existence and uniqueness of strong solution to SDE for GDBM.
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Ricci curvature, isoperimetry and a non-additive entropy
TL;DR: This work presents an isoperimetric interpretation of the non-extensive parameter and comment on further features of the system that can be probed through this tensor.
Journal ArticleDOI
A Stein deficit for the logarithmic Sobolev inequality
TL;DR: In this article, lower bounds on the Gaussian logarithmic Sobolev inequality in terms of the Stein characterization of Gaussian distribution were derived based on the representation of the relative Fisher information along the Ornstein-Uhlenbeck semigroup.
Book ChapterDOI
Some Remarks on Free Energy and Coarse-Graining
TL;DR: Results are presented on the computation of the stress-strain relation for one-dimensional chains of atoms, and the construction of an effective dynamics for a scalar coarse-grained variable when the complete system evolves according to the overdamped Langevin equation.
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Equality in the logarithmic Sobolev inequality
Shin-ichi Ohta,Asuka Takatsu +1 more
TL;DR: In this article, it was shown that the 1-dimensional Gaussian space is necessarily split off, similarly to the rigidity results of Cheng-Zhou on the spectral gap as well as Morgan on the isoperimetric inequality.
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.