Weak curvature conditions on metric graphs

TL;DR

This paper explores weak lower curvature bounds on metric graphs through heat semigroup estimates, establishing equivalences among various weak curvature conditions and discussing potential applications to the Schrödinger bridge problem.

Contribution

It introduces a novel framework connecting weak curvature conditions via heat semigroup estimates, variational inequalities, and convexity on metric graphs.

Findings

Proves equivalence between weak Bakry-Émery curvature, EVIs, and geodesic convexity.

Provides a regularization method for absolutely continuous curves on metric graphs.

Discusses potential applications to Schrödinger bridge problems.

Abstract

Starting from pointwise gradient estimates for the heat semigroup, we study three characterizations of weak lower curvature bounds on metric graphs. More precisely, we prove the equivalence between a weak notion of the Bakry-\'Emery curvature condition, a weak Evolutionary Variational Inequality and a weak form of geodesic convexity. The proof is based on a careful regularization of absolutely continuous curves together with an explicit representation of the Cheeger energy. We conclude with a brief discussion on possible applications to the Schr\"odinger bridge problem on metric graphs.