Stochastic evolution of genealogies of spatial populations: state description, characterization of dynamics and properties

TL;DR

This paper surveys the mathematical modeling of evolving genealogies in spatial populations using tree-valued Markov processes, focusing on state space, dynamics, and long-term behavior.

Contribution

It provides a comprehensive overview of the state spaces, dynamics, and analytical techniques for genealogical models of spatial populations, including Fleming-Viot and branching processes.

Findings

Description of state spaces and topologies for genealogical processes

Well-posed martingale problems for population models

Analysis of long-term behavior and duality techniques

Abstract

We survey results on the description of stochastically evolving genealogies of populations and marked genealogies of multitype populations or spatial populations via tree-valued Markov processes on (marked) ultrametric measure spaces. In particular we explain the choice of state spaces and their topologies, describe the dynamics of genealogical Fleming-Viot and branching models by well-posed martingale problems, and formulate the typical results on the longtime behavior. Furthermore we discuss the basic techniques of proofs and sketch as two key tools of analysis the different forms of duality and the Girsanov transformation.