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

Abstract

In Athreya, Löhr, 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.

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• Metrization of the Gromov-Hausdorff (-Prokhorov) Topology for Boundedly-Compact Metric Spaces
• Spaces of algebraic measure trees and triangulations of the circle
• The multiplicative coalescent, inhomogeneous continuum random trees, and new universality classes for critical random graphs
• Continuum Space Limit of the Genealogies of Interacting Fleming-Viot Processes on $\mathbb{Z}$
• Global lower mass-bound for critical configuration models in the heavy-tailed regime
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• Invariance principles for pruning processes of Galton-Watson trees
• Curvature-Dimension Condition Meets Gromov's $n$-Volumic Scalar Curvature
• A Unified Framework for Generalizing the Gromov-Hausdorff Metric
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• Limits of chordal graphs with bounded tree-width
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