# Measure representation of evolving genealogies

## TL;DR

This paper introduces a measure representation framework for evolving genealogies, providing a tightness criterion and applying it to analyze finite system schemes in tree-valued interacting Fleming-Viot processes.

## Contribution

It develops a novel measure representation approach for evolving genealogies and establishes a tightness criterion, advancing the analysis of complex genealogical processes.

## Key findings

- Established a tightness criterion for evolving genealogies.
- Applied the theory to finite system schemes in Fleming-Viot processes.
- Provided a new perspective on genealogical distance and ancestor-descendant relationships.

## Abstract

We study evolving genealogies, i.e. processes that take values in the space of (marked) ultra-metric measure spaces and satisfy some sort of "consistency" condition. This condition is based on the observation that the genealogical distance of two individuals who do not have common ancestors up to a time $h$ in the past is completely determined by the genealogical distance of the respective ancestors at that time $h$ in the past. Now the idea is to color all possible ancestors at time $h$ in the past and measure the relative number of their descendants. The resulting collection of measure-valued processes (the construction is possible for all $h$) is called a measure representation. As a main result we give a tightness criterion of evolving genealogies in terms of their measure representation. We then apply our theory to study a finite system scheme for tree-valued interacting Fleming-Viot processes.