Electronic Journal of Probability

Existence of mark functions in marked metric measure spaces

Sandra Kliem and Wolfgang Loehr

Full-text: Open access

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

Permanent link to this document
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

Subjects
Primary: 60K35: Interacting random processes; statistical mechanics type models; percolation theory [See also 82B43, 82C43]
Secondary: 60J25: Continuous-time Markov processes on general state spaces 60G17: Sample path properties 60G57: Random measures

Keywords
mark function tree-valued Fleming-Viot process mutation marked metric measure space Gromov-weak topology Prohorov metric Lusin's theorem

Rights
This work is licensed under a Creative Commons Attribution 3.0 License.

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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