Entropies, convexity, and functional inequalities

TL;DR

This paper unifies and extends Phi-Sobolev inequalities, demonstrating their broad applicability in various probabilistic and geometric contexts through convexity-based proofs.

Contribution

It provides a comprehensive synthesis of Phi-Sobolev inequalities, including new extensions and stability results under convexity assumptions.

Findings

Inequalities hold for hyper-contractive diffusions and log-concave measures.

Stability under tensor products, convolution, and perturbations established.

Simple convexity-based proofs are provided for broad classes of measures.

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.