Electronic Journal of Probability

Existence of mark functions in marked metric measure spaces

Sandra Kliem and Wolfgang Loehr

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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

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

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

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

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

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

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


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

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  • Aldous, David. The continuum random tree. III. Ann. Probab. 21 (1993), no. 1, 248–289.
  • Billingsley, Patrick. Convergence of probability measures. John Wiley & Sons, Inc., New York-London-Sydney 1968 xii+253 pp.
  • Bogachev, V. I. Measure theory. Vol. I, II. Springer-Verlag, Berlin, 2007. Vol. I: xviii+500 pp., Vol. II: xiv+575 pp. ISBN: 978-3-540-34513-8; 3-540-34513-2.
  • Depperschmidt, Andrej; Greven, Andreas; Pfaffelhuber, Peter. Marked metric measure spaces. Electron. Commun. Probab. 16 (2011), 174–188.
  • Depperschmidt, Andrej; Greven, Andreas; Pfaffelhuber, Peter. Tree-valued Fleming-Viot dynamics with mutation and selection. Ann. Appl. Probab. 22 (2012), no. 6, 2560–2615.
  • Depperschmidt, Andrej; Greven, Andreas; Pfaffelhuber, Peter. Path-properties of the tree-valued Fleming-Viot process. Electron. J. Probab. 18 (2013), no. 84, 47 pp.
  • Ethier, Stewart N.; Kurtz, Thomas G. Markov processes. Characterization and convergence. Wiley Series in Probability and Mathematical Statistics: Probability and Mathematical Statistics. John Wiley & Sons, Inc., New York, 1986. x+534 pp. ISBN: 0-471-08186-8.
  • Greven, Andreas; Klimovsky, Anton; Winter, Anita. Evolving genealogies for spatial Λ-Fleming-Viot processes with mutation. In preparation, 2015.
  • Greven, Andreas; Pfaffelhuber, Peter; Winter, Anita. Convergence in distribution of random metric measure spaces ($\Lambda$-coalescent measure trees). Probab. Theory Related Fields 145 (2009), no. 1-2, 285–322.
  • Greven, Andreas; Pfaffelhuber, Peter; Winter, Anita. Tree-valued resampling dynamics martingale problems and applications. Probab. Theory Related Fields 155 (2013), no. 3-4, 789–838.
  • Gromov, Misha. Metric structures for Riemannian and non-Riemannian spaces. Based on the 1981 French original [ (85e:53051)]. With appendices by M. Katz, P. Pansu and S. Semmes. Translated from the French by Sean Michael Bates. Progress in Mathematics, 152. Birkh�user Boston, Inc., Boston, MA, 1999. xx+585 pp. ISBN: 0-8176-3898-9.
  • Klenke, Achim. Probability theory. A comprehensive course. Second edition. Translation from the German edition. Universitext. Springer, London, 2014. xii+638 pp. ISBN: 978-1-4471-5360-3; 978-1-4471-5361-0.
  • Kliem, Sandra. A compact containment result for nonlinear historical superprocess approximations for population models with trait-dependence. Electron. J. Probab. 19 (2014), no. 97, 13 pp.
  • Kliem, Sandra; Winter, Anita. Evolving phylogenies of trait-dependent branching with mutation and competition. In preparation, 2015.
  • Löhr, Wolfgang. Equivalence of Gromov-Prohorov- and Gromov's $\underline\square_ \lambda$-metric on the space of metric measure spaces. Electron. Commun. Probab. 18 (2013), no. 17, 10 pp.
  • Löhr, Wolfgang; Voisin, Guillaume; Winter, Anita. Convergence of bi-measure $\mathbb{R}$-trees and the pruning process. Ann. Inst. H. Poincaré Probab. Statist., in press, 2014. arXiv:1304.6035.
  • Piotrowiak, Sven. Dynamics of Genealogical Trees for Type- and State-dependent Resampling Models. PhD thesis, University of Erlangen-Nuremberg, 2011.
  • Pitman, Jim. Coalescents with multiple collisions. Ann. Probab. 27 (1999), no. 4, 1870–1902.
  • Rogers, L. C. G.; Williams, D. Diffusions, Markov Processes and Martingales, Volume II. Cambridge University Press, second edition, 2000.