Entropies, convexity, and functional inequalities, On $\Phi $-entropies and $\Phi $-Sobolev 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 Poincaré 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 Lévy 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.