Generalization of an Inequality by Talagrand and Links with the Logarithmic Sobolev Inequality

By Felix Otto, Cédric Villani. 2000

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In Applied Mathematics

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A deterministic and computable Bernstein-von Mises theorem (2019, methodology): “Another possible path towards avoiding the normalization constants in the KL divergence is given by the Log-Sobolev Inequality (LSI; Bakry and Émery [1985], Otto and Villani [2000]).”
Kac’s program in kinetic theory (2011, result): “This notion is related but different from the notions of differentiability developed in the theory of gradient flow by Ambrosio, Otto, Villani and co-authors in [2, 41, 62], and from the one introduced by Lions in [46].”
Transportation cost-information inequalities and applications to random dynamical systems and diffusions (2004, methodology): “As the result of Otto and Villani [15] suggests that the log-Sobolev inequality implies the T2(C) inequality (that is proved on the smooth Riemannian manifold), we should have Px ∈ T2(C) on C([0, T ]) w.r.t. the following pseudo-metric, dH(γ1, γ2) := {‖γ1 − γ2‖H , if γ1 − γ2 ∈H, +∞, otherwise.”
The Wasserstein Gradient Flow of the Fisher Information and the Quantum Drift-diffusion Equation (2009, methodology): “The relationship between displacement convexity, Talagrand and logarithmic Sobolev inequalities has been deeply investigated by [41].”
Kinetic equilibration rates for granular media and related equations: entropy dissipation and mass transportation estimates (2003, methodology): “Nearly all of the forthcoming discussion follows Otto and Villani [28].”
Equivalent semigroup properties for the curvature-dimension condition (2010, neutral): “Since according to [3, 13] one has K 2 W ρ 2 (f (2)μ, μ)(2) ≤ μ(f 2 log f (2)) ≤ 2 K− μ(|∇f |(2)), it is clear that r ≤ 2 K .”
Prekopa-Leindler type inequalities on Riemannian manifolds, Jacobi fields, and optimal transport (2006, methodology): “We now rigorously prove the displacement semiconvexity of the Boltzmann type entropy in a Riemannian setting — another result derived formally by Otto and Villani [35].”
Logarithmic Sobolev inequalities for some nonlinear PDE's (2001, positive): “Key words: Interacting particle system, logarithmic Sobolev inequality, propagation of chaos, relative entropy, concentration of measure”
A large deviation approach to some transportation cost inequalities (2005, methodology): “It is shown in the articles by F. Otto and C. Villani [16] and by S. Bobkov, I. Gentil and M. Ledoux [1], that if µ satisfies the logarithmic Sobolev inequality, then it also satisfies T2.”
A characterization of dimension free concentration in terms of transportation inequalities (2008, background): “For example, it is the case when μ is a Log-concave probability measure (see [33], Corollary 3.”
On the geometry of metric measure spaces. II (2006, methodology): “See e.g. [Ta95], [Le01], [BG99], [ OV00 ], [DD02], [CMS01], [AT04], [F¨”
Modified logarithmic Sobolev inequalities and transportation inequalities (2004, methodology): “Otto and Villani proved that a logarithmic Sobolev inequality implies a transportation inequality with a quadratic cost (this is the case α = β = 2), see [ OV00 , BGL01].”
Quantitative Concentration Inequalities for Empirical Measures on Non-compact Spaces (2005, positive): “Sobolev inequality [23, 6, 30], and that they imply a Poincaré, or spectral gap, inequality [23, 6].”
Ricci curvature of Markov chains on metric spaces (2007, background): “Another notion of lower bound on Ricci curvature, valid for length spaces equipped with a measure, has been simultaneously introduced by Sturm [46], Lott and Villani [31], and Ohta [37] (see also [43] and [41]).”
Rapid Convergence of the Unadjusted Langevin Algorithm: Isoperimetry Suffices (2019, methodology): “It was first studied by Talagrand [58] for Gaussian ⌫, and extended by Otto and Villani [54] to all ⌫ satisfying LSI; namely, if ⌫ satisfies LSI with constant ↵ > 0, then ⌫ also satisfies Talagrand’s inequality with the same constant [54, Theorem 1].”