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

By Siva Athreya, Wolfgang Lohr, A. Winter. 2014

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In Athreya et al. (2015) 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.

Subfields: Computational Theory and Mathematics, Geometry and Topology, Management Science and Operations Research

View Paper: https://www.semanticscholar.org/paper/1299731fa4aee3328a4e2ae2ce59c27c2a9bf6bb