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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A new family of transportation costs with applications to reaction–diffusion and parabolic equations with boundary conditions
TL;DR: In this paper, a family of transportation costs between non-negative measures is introduced to obtain parabolic and reaction-diffusion equations with drift, subject to Dirichlet boundary condition.
EDPs de difusión y transporte óptimo de masa
TL;DR: In this paper, aplicaciones que la teoria de transporte optimo de masa tiene en el comportamiento asintotico de EDPs are described.
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Lagrangian calculus for nonsymmetric diffusion operators
TL;DR: In this article, lower bounds for the Bakry-Emery Ricci tensor of nonsymmetric diffusion operators were characterized by convexity of entropy on the Wasserstein space, and a curvature-dimension condition for general metric measure spaces together with a square integrable $1$-form was defined.
From large deviations to Wasserstein gradient flows in multiple dimensions
TL;DR: In this article, a new proof of the upper bound for the large deviation rate functional for the empirical distribution of independent Brownian particles with drift in arbitrary dimensions was presented, thereby generalising the result of Adams et al. to arbitrary dimensions.
Mean field approximations via log-concavity
TL;DR: A new approach to deriving quantitative mean field approximations for any probability measure P on R with density proportional to e, for f strongly concave is proposed, based primarily on functional inequalities and the notion of displacement convexity from optimal transport.
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.