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 [Depperschmidt, Greven, Pfaffelhuber, Ann. Appl. Probab. '12] 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.
Highlights
A metric measure spaces is a complete, separable metric space (X, r) together with a finite measure ν on it
The given proof, contains a gap, because it relies on the criterion claimed in [6, Lemma 7.1], which is wrong in general, as we show in Example 4.1. We fill this gap by applying our criteria and showing in Theorem 4.3 that the claim is true and the tree-valued Fleming-Viot process with mutation and selection (TFVMS) admits a mark function at all times, almost surely
With a slight abuse of notation, we identify an mmm-space with its equivalence class and write X = (X, r, μ) ∈ MI for both mmm-spaces and equivalence classes thereof
Summary
A metric (finite) measure spaces (mm-space) is a complete, separable metric space (X, r) together with a finite measure ν on it. The given proof, contains a gap, because it relies on the criterion claimed in [6, Lemma 7.1], which is wrong in general, as we show in Example 4.1 We fill this gap by applying our criteria and showing in Theorem 4.3 that the claim is true and the tree-valued Fleming-Viot process with mutation and selection (TFVMS) admits a mark function at all times, almost surely. Based on the construction of the complete metric and the decomposition of MfIct, we derive in Subsection 3.1 criteria to check if an mmm-space admits a mark function, especially in the case where it is given as a marked Gromov-weak limit. Our criteria are applied in Subsection 4.1 to prove the existence of a mark function for the tree-valued Fleming-Viot dynamics with mutation and selection To this goal, we verify the necessary assumptions for a sequence of approximating tree-valued Moran models. In Subsection 4.3, a future application to evolving phylogenies of trait-dependent branching with mutation and competition is indicated
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