Weak curvature conditions on metric graphs

TL;DR

The paper develops a framework for weak lower curvature bounds on metric graphs by proving the equivalence of three notions—$BE_w(c,\infty)$, $EVI_w(c)$, and $RCD_w(c,\infty)$—through a careful regularization of Wasserstein geodesics and a detailed identification of Cheeger energy with Dirichlet energy on graphs. It proves that the heat flow on metric graphs can be viewed equivalently as the gradient flow of the Cheeger energy in $L^2$ and as the gradient flow of the entropy in Wasserstein space, establishing a weak Bakry-Émery gradient bound with a contraction property in $W_2$. Central to the argument are the notions of strongly regular curves and a two-tier regularization of absolutely continuous curves, which enable passage to the limit in the weak curvature inequalities. The results lay groundwork for extending RCD-type analysis to weak curvature-dimension settings on graphs and hint at applications to Schrödinger-type problems on networked spaces.

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-Émery 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 Schrödinger bridge problem on metric graphs.

Figures (1)

• Figure 3.1: Example of an extension of a metric graph for $\varepsilon>0$

Theorems & Definitions (39)

• Lemma 2.1
• proof
• Theorem 3.1
• proof
• Definition 3.2
• Proposition 3.3
• proof
• Theorem 3.4
• proof
• Theorem 3.5