On Φ-entropies and Φ-Sobolev inequalities

Our aim is to provide a short and self contained synthesis which unify various related and unrelated works involving what we call Φ-Sobolev functional inequalities. Such inequalities can be seen as an inclusive interpolation between Poincaré and logarithmic Sobolev inequalities. In addition to the known material, some extensions are provided and improvements are given for some aspects. We show that under simple convexity assumptions on Φ, such inequalities hold in a lot of situations, including hyper-contractive diffusions, uniformly strictly log-concave measures, paths space of Brownian Motion on Riemannian Manifolds, paths space of Lévy processes and infinitely divisible laws. The proofs are simple and relies essentially on convexity. We end up by a short parallel inspired by the analogy with Boltzmann-Shannon’s entropy in Information Theory.

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