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

Dublin Core		PKP Metadata Items	Metadata for this Document
1.	Title	Title of document	Equivalence of Gromov-Prohorov- and Gromov's $\underline\Box_\lambda$-metric on the space of metric measure spaces
2.	Creator	Author's name, affiliation, country	Wolfgang Loehr; University of Duisburg-Essen; Germany
3.	Subject	Discipline(s)
3.	Subject	Keyword(s)	space of metric measure spaces, Gromov-Prohorov metric, Gromov's box-metric, Gromov-weak topology, real tree, coding trees by excursions, Lusin topology on excursions
3.	Subject	Subject classification	60b05; 60b10
4.	Description	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.
5.	Publisher	Organizing agency, location
6.	Contributor	Sponsor(s)
7.	Date	(YYYY-MM-DD)	2013-03-02
8.	Type	Status & genre	Peer-reviewed Article
8.	Type	Type
9.	Format	File format	PDF
10.	Identifier	Uniform Resource Identifier	http://ecp.ejpecp.org/article/view/2268
10.	Identifier	Digital Object Identifier	10.1214/ECP.v18-2268
11.	Source	Journal/conference title; vol., no. (year)	Electronic Communications in Probability; Vol 18
12.	Language	English=en	en
14.	Coverage	Geo-spatial location, chronological period, research sample (gender, age, etc.)
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