Marked metric measure spaces

Abstract

A marked metric measure space (mmm-space) is a triple (X, r,μ), where (X, r) is a complete and separable metric space and μ is a probability measure on X × I 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 a 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 andWinter, 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. © 2011 Association for Symbolic Logic.