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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Fokker–Planck Equations for a Free Energy Functional or Markov Process on a Graph
TL;DR: In this paper, the authors show that the corresponding Fokker-planck equation is a system of N nonlinear ordinary differential equations defined on a Riemannian manifold of probability distributions whose inner product is generated by a 2-Wasserstein distance.
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The Wasserstein Gradient Flow of the Fisher Information and the Quantum Drift-diffusion Equation
TL;DR: In this article, the global existence of non-negative variational solutions to the drift diffusion evolution equation was proved under variational boundary condition, and the long-time behavior of the solutions was studied.
On the trend to equilibrium for the fokker-planck equation : an interplay between physics and functional analysis
Peter A. Markowich,C. Villani +1 more
TL;DR: In this paper, the authors make connections between the problem of trend to equilibrium for the Fokker-Planck equation of statistical physics and several inequalities from functional analysis, like logarithmic Sobolev or Poincare inequalities, together with some inequalities arising in the context of concentration of measures, introduced by Talagrand, or in the study of Gaussian isoperimetry.
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Functional inequalities, thick tails and asymptotics for the critical mass Patlak–Keller–Segel model
TL;DR: In this paper, the authors investigate the long time behavior of the critical mass Patlak-Keller-Segel equation and find basins of attraction for them using an entropy functional.
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A new class of transport distances between measures
TL;DR: In this paper, the authors introduce a new class of distances between nonnegative Radon measures, which are modeled on the dynamical characterization of the Kantorovich-Rubinstein-Wasserstein distances and provide a wide family interpolating between the Wasserstein and the homogeneous Sobolev distances.
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.