The gap between Gromov-vague and Gromov-Hausdorff-vague topology

TL;DR

This paper explores the relationship between Gromov-vague and Gromov-Hausdorff-vague topologies on metric measure spaces, establishing conditions for their convergence and analyzing their topological properties with applications to Galton-Watson trees.

Contribution

It systematically introduces and compares Gromov-vague and Gromov-Hausdorff-vague topologies, clarifies their relationship, and applies these concepts to Galton-Watson trees.

Findings

Convergence in Gromov-vague implies Gromov-Hausdorff-vague convergence under the lower mass-bound property.

The paper shows that certain spaces are not Polish in these topologies.

Constructs the Kallenberg-Kesten tree as a limit in Gromov-Hausdorff-vague topology.

Abstract

In Athreya, L\"ohr, Winter (2016), an invariance principle is stated for a class of strong Markov processes on tree-like metric measure spaces. It is shown that if the underlying spaces converge Gromov vaguely, then the processes converge in the sense of finite dimensional distributions. Further, if the underlying spaces converge Gromov-Hausdorff vaguely, then the processes converge weakly in path space. In this paper we systematically introduce and study the Gromov-vague and the Gromov-Hausdorff-vague topology on the space of equivalence classes of metric boundedly finite measure spaces. The latter topology is closely related to the Gromov-Hausdorff-Prohorov metric which is defined on different equivalence classes of metric measure spaces. We explain the necessity of these two topologies via several examples, and close the gap between them. That is, we show that convergence in Gromov-vague topology implies convergence in Gromov-Hausdorff-vague topology if and only if the so-called lower mass-bound property is satisfied. Furthermore, we prove and disprove Polishness of several spaces of metric measure spaces in the topologies mentioned above (summarized in Figure~1). As an application, we consider the Galton-Watson tree with critical offspring distribution of finite variance conditioned to not get extinct, and construct the so-called Kallenberg-Kesten tree as the weak limit in Gromov-Hausdorff-vague topology when the edge length are scaled down to go to zero.