Partial orders on metric measure spaces

TL;DR

This paper introduces a new partial order on metric measure spaces, generalizing Gromov's Lipschitz order, and explores its properties and applications in probabilistic models like Fleming-Viot processes and branching processes.

Contribution

It defines a novel partial order on metric measure spaces, proves its topological closure, and provides new characterizations and applications in probability theory.

Findings

The partial order is closed under the Gromov-weak topology.

A new characterization of the Lipschitz order is provided.

Applications to Fleming-Viot processes and branching processes are demonstrated.

Abstract

A partial order on the set of metric measure spaces is defined; it generalizes the Lipschitz order of Gromov. We show that our partial order is closed when metric measure spaces are equipped with the Gromov-weak topology and give a new characterization for the Lipschitz order. We will then consider some probabilistic applications. The main importance is given to the study of Fleming-Viot processes with different resampling rates. Besides that application we also consider tree-valued branching processes and two semigroups on metric measure spaces.