Existence of mark functions in marked metric measure spaces

TL;DR

This paper establishes criteria for the existence of mark functions in marked metric measure spaces, proves their stability under limits, and applies these results to demonstrate properties of tree-valued Fleming-Viot processes.

Contribution

It provides new criteria for the existence and stability of mark functions in mmm-spaces and shows that the space of fmm-spaces is a Polish space, filling gaps in previous proofs.

Findings

Criteria for deterministic and random mmm-spaces to admit mark functions.

The space of fmm-spaces is a dense, complete Polish space.

Application to tree-valued Fleming-Viot dynamics with mutation and selection.

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.