Marked metric measure spaces

TL;DR

This paper introduces the marked Gromov-weak topology for the space of marked metric measure spaces, establishing its topological properties and criteria for convergence, facilitating the study of random genealogical structures.

Contribution

It extends the Gromov-weak topology to marked spaces, proving it forms a Polish space and providing tightness criteria and convergence-determining functions.

Findings

Marked Gromov-weak topology makes the space of mmm-spaces Polish.

Provides tightness criteria for distributions of random mmm-spaces.

Identifies polynomials as convergence-determining algebra.

Abstract

A marked metric measure space (mmm-space) is a triple (X,r,mu), where (X,r) is a complete and separable metric space and mu is a probability measure on XxI for some Polish space I of possible marks. We study the space of all (equivalence classes of) marked metric measure spaces for some fixed I. It arises as state space in the construction of Markov processes which take values in random graphs, e.g. tree-valued dynamics describing randomly evolving genealogical structures in population models. We derive here the topological properties of the space of mmm-spaces needed to study convergence in distribution of random mmm-spaces. Extending the notion of the Gromov-weak topology introduced in (Greven, Pfaffelhuber and Winter, 2009), we define the marked Gromov-weak topology, which turns the set of mmm-spaces into a Polish space. We give a characterization of tightness for families of distributions of random mmm- spaces and identify a convergence determining algebra of functions, called polynomials.