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 (one-dimensional) free Brunn–Minkowski inequality
TL;DR: Ledoux et al. as discussed by the authors presented a one-dimensional version of the functional form of the geometric Brunn-Minkowski inequality in free (non-commutative) probability theory, which was used to establish free analogues of the logarithmic Sobolev and transportation cost inequalities for strictly convex potentials.
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Ultracontractive bounds on Hamilton–Jacobi solutions
TL;DR: In this paper, the equivalence between Euclidean-type Sobolev inequality and an ultracontractive control of the Hamilton-Jacobi equations was shown to be equivalent to the Varopoulos theorem.
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Uniform long-time and propagation of chaos estimates for mean field kinetic particles in non-convex landscapes
Arnaud Guillin,Pierre Monmarché +1 more
TL;DR: In this article, the trend to equilibrium in large time is studied for a large particle system associated to a Vlasov-Fokker-Planck equation, and the convergence rate is proven to be independent from the number of particles.
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Displacement interpolations from a Hamiltonian point of view
TL;DR: In this paper, the authors study displacement interpolations from the point of view of Hamiltonian systems and give a unifying approach to the above mentioned results, and generalize this result to the Finsler manifolds and manifolds with a Ricci flow background.
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Dimensional contraction via Markov transportation distance
TL;DR: It is proved that a new distance between probability measures is introduced that satisfies the same properties for a general Markov semigroup as the Wasserstein distance does in the specific case of the Euclidean heat semigroup, namely dimensional contraction properties and Evolutional variational inequalities.
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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.