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2015 Existence of mark functions in marked metric measure spaces
Sandra Kliem, Wolfgang Loehr
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Electron. J. Probab. 20: 1-24 (2015). DOI: 10.1214/EJP.v20-3969


We give criteria on the existence of a so-called mark function in the context of marked metric measure spaces (mmm-spaces). If an mmm-space admits a mark function, we call it functionally-marked metric measure space (fmm-space). This is not a closed property in the usual marked Gromov-weak topology, and thus we put particular emphasis on the question under which conditions it carries over to a limit. We obtain criteria for deterministic mmm-spaces as well as random mmm-spaces and mmm-space-valued processes. As an example, our criteria are applied to prove that the tree-valued Fleming-Viot dynamics with mutation and selection from previous works admits a mark function at all times, almost surely. Thereby, we fill a gap in a former proof of this fact, which used a wrong criterion. Furthermore, the subspace of fmm-spaces, which is dense and not closed, is investigated in detail. We show that there exists a metric that induces the marked Gromov-weak topology on this subspace and is complete. Therefore, the space of fmm-spaces is a Polish space. We also construct a decomposition into closed sets which are related to the case of uniformly equicontinuous mark functions.


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Sandra Kliem. Wolfgang Loehr. "Existence of mark functions in marked metric measure spaces." Electron. J. Probab. 20 1 - 24, 2015.


Accepted: 27 June 2015; Published: 2015
First available in Project Euclid: 4 June 2016

zbMATH: 1350.60103
MathSciNet: MR3371432
Digital Object Identifier: 10.1214/EJP.v20-3969

Primary: 60K35
Secondary: 60G17, 60G57, 60J25


Vol.20 • 2015
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