Generalized Bakry-Émery curvature condition and equivalent entropic inequalities in groups

Abstract

We study a generalization of the Bakry-Émery pointwise gradient estimate for the heat semigroup and its equivalence with some entropic inequalities along the heat flow and Wasserstein geodesics for metric-measure spaces with a suitable group structure. Our main result applies to Carnot groups of any step and to the SU(2) group.