Generalized Bakry-\'Emery curvature condition and equivalent entropic inequalities in groups

TL;DR

This paper extends Bakry-Émery curvature conditions to groups like Carnot groups and SU(2), establishing their equivalence with entropic inequalities along heat flow and Wasserstein geodesics.

Contribution

It introduces a generalized Bakry-Émery curvature condition applicable to groups with a suitable structure, linking it to entropic inequalities in these spaces.

Findings

Applicable to Carnot groups of any step

Valid for the SU(2) group

Establishes equivalence between curvature conditions and entropic inequalities

Abstract

We study a generalization of the Bakry-\'Emery 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 $\mathbb{SU}(2)$ group.