Equivalence of Gromov-Prohorov- and Gromov's Box-Metric on the Space of Metric Measure Spaces

TL;DR

This paper proves the equivalence and bi-Lipschitz relationship between the Gromov-Prohorov and Gromov's Box metrics on the space of metric measure spaces, confirming they induce the same topology.

Contribution

It establishes the bi-Lipschitz equivalence of two key metrics on metric measure spaces, confirming their topological compatibility and introducing a new separable topology for excursions.

Findings

Metrics are bi-Lipschitz equivalent: d_{GPW} = 1/2 Box_{1/2}

Different metrics induce the same Gromov-weak topology

Introduces a new separable topology on excursions

Abstract

The space of metric measure spaces (complete separable metric spaces with a probability measure) is becoming more and more important as state space for stochastic processes. Of particular interest is the subspace of (continuum) metric measure trees. Greven, Pfaffelhuber and Winter introduced the Gromov-Prohorov metric d_{GPW} on the space of metric measure spaces and showed that it induces the Gromov-weak topology. They also conjectured that this topology coincides with the topology induced by Gromov's Box_1 metric. Here, we show that this is indeed true, and the metrics are even bi-Lipschitz equivalent. More precisely, d_{GPW}= 1/2 Box_{1/2}, and hence d_{GPW} <= Box_1 <= 2d_{GPW}. The fact that different approaches lead to equivalent metrics underlines their importance and also that of the induced Gromov-weak topology. As an application, we give an easy proof of the known fact that the map associating to a lower semi-continuous excursion the coded R-tree is Lipschitz continuous when the excursions are endowed with the (non-separable) uniform metric. We also introduce a new, weaker, metric topology on excursions, which has the advantage of being separable and making the space of bounded excursions a Lusin space. We obtain continuity also for this new topology.