The $\Gamma$-calculus of Bakry Emery and $\Phi$-entropy inequalities

Abstract

Based on a proof of the Gross Logarithmic Sobolev inequality we analyze the ingredients which are important for the so called $\Gamma$-calculus of Bakry Emery. We then establish the framework of the $\Gamma$-calculus and discuss curvature dimension condtitions for Markov Diffusion operators. Afterwards, under curvature dimension condtion conditions, we deduce $\Phi$-Entropy inequalities, among which are for instance Poincaré inequalities and Logarithmic Sobolev inequalities. We finally discuss how this can be used to get the hypercontractivity of the corresponding semigroup or for the study of long time behaviour of solutions to Fokker-Planck type equations.