Equivalence of Gromov-Prohorov- and Gromov's $\underline\Box_\lambda$-metric on the space of metric measure spaces

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 $d_{\mathrm{GP}}$ 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 $\underline\Box_1$ metric.  Here, we show that this is indeed true, and the metrics are even bi-Lipschitz equivalent.  More precisely, $d_{\mathrm{GP}}=\frac12\underline\Box_{\frac12}$, and hence $d_{\mathrm{GP}}\le \underline\Box_1 \le 2d_{\mathrm{GP}}$.  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 $\mathbb{R}$-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.