Entropies, convexity, and functional inequalities

Abstract

Our aim is to provide a short and self contained synthesis which generalise and unify various related and unrelated works involving what we call Phi-Sobolev functional inequalities. Such inequalities related to Phi-entropies can be seen in particular as an inclusive interpolation between Poincare and Gross logarithmic Sobolev inequalities. In addition to the known material, extensions are provided and improvements are given for some aspects. Stability by tensor products, convolution, and bounded perturbations are addressed. We show that under simple convexity assumptions on Phi, such inequalities hold in a lot of situations, including hyper-contractive diffusions, uniformly strictly log-concave measures, Wiener measure (paths space of Brownian Motion on Riemannian Manifolds) and generic Poisson space (includes paths space of some pure jumps Levy processes and related infinitely divisible laws). Proofs are simple and relies essentially on convexity. We end up by a short parallel inspired by the analogy with Boltzmann-Shannon entropy appearing in Kinetic Gases and Information Theories.

Formerly "On Phi-entropies and Phi-Sobolev inequalities". Author's www homepage: http://www.lsp.ups-tlse.fr/Chafai/

References in corpus (2)

• A General Divergence Measure for Monotonic Functions and Applications in Information Theory
• Interpolated inequalities between exponential and Gaussian, Orlicz hypercontractivity and isoperimetry

Cited by in corpus (24)

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• Intertwining and commutation relations for birth-death processes
• McKean-Vlasov SDEs in nonlinear filtering
• Subadditivity of Matrix phi-Entropy and Concentration of Random Matrices
• Monotonic Convergence in an Information-Theoretic Law of Small Numbers
• Higher Order Coercive Inequalities
• Dimensional contraction via Markov transportation distance
• Entropy dissipation estimates for the linear Boltzmann operator
• Exponential Decay of Matrix $\Phi$-Entropies on Markov Semigroups with Applications to Dynamical Evolutions of Quantum Ensembles
• Characterisations of Matrix and Operator-Valued $\Phi$-Entropies, and Operator Efron-Stein Inequalities
• Concentration inequalities for bounded functionals via generalized log-Sobolev inequalities
• On Poincare and logarithmic Sobolev inequalities for a class of singular Gibbs measures
• Functional inequalities: nonlinear flows and entropy methods as a tool for obtaining sharp and constructive results
• Poincaré type inequalities for compact degenerate pure jump Markov processes
• On the strong consistency of asymptotic M-estimators
• From Boltzmann to random matrices and beyond
• Improving Beckner's bound via Hermite functions
• Inequalities involving Aharonov-Bohm magnetic potentials in dimensions 2 and 3
• Modified log-Sobolev inequality for a compact PJMP with degenerate jumps
• Trajectorial dissipation of $\Phi$-entropies for interacting particle systems
• Wasserstein geometry and Ricci curvature bounds for Poisson spaces