## Electronic Journal of Probability

### Existence of mark functions in marked metric measure spaces

#### Abstract

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.

#### Article information

Source
Electron. J. Probab. Volume 20 (2015), paper no. 73, 24 pp.

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

http://projecteuclid.org/euclid.ejp/1465067179

Digital Object Identifier
doi:10.1214/EJP.v20-3969

Mathematical Reviews number (MathSciNet)
MR3371432

Zentralblatt MATH identifier
1350.60103

Rights

#### Citation

Kliem, Sandra; Loehr, Wolfgang. Existence of mark functions in marked metric measure spaces. Electron. J. Probab. 20 (2015), paper no. 73, 24 pp. doi:10.1214/EJP.v20-3969. http://projecteuclid.org/euclid.ejp/1465067179.

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