Path-properties of the tree-valued Fleming-Viot process

TL;DR

This paper investigates the long-term genealogical properties of the tree-valued Fleming-Viot process, demonstrating pathwise properties such as atomlessness and the existence of mark functions, and extends results from neutral to selective cases.

Contribution

It proves that the entire paths of the process retain properties like atomlessness and mark functions, and answers open questions about the pathwise structure of the process.

Findings

The process's genealogies have no atoms at any positive time.

Each individual in the process can be uniquely assigned a type.

Path properties extend from neutral to selective cases via Girsanov's formula.

Abstract

We consider the tree-valued Fleming-Viot process, $(\mathcal X_t)_{t\geq 0}$, with mutation and selection as studied in Depperschmidt, Greven, Pfaffelhuber (2012). This process models the stochastic evolution of the genealogies and (allelic) types under resampling, mutation and selection in the population currently alive in the limit of infinitely large populations. Genealogies and types are described by (isometry classes of) marked metric measure spaces. The long-time limit of the neutral tree-valued Fleming-Viot dynamics is an equilibrium given via the marked metric measure space associated with the Kingman coalescent. In the present paper we pursue two closely linked goals. First, we show that two well-known properties of the neutral Fleming-Viot genealogies at fixed time $t$ arising from the properties of the dual, namely the Kingman coalescent, hold for the whole path. These properties are related to the geometry of the family tree close to its leaves. In particular we consider the number and the size of subfamilies whose individuals are not further than $\ve$ apart in the limit $\ve\to 0$. Second, we answer two open questions about the sample paths of the tree-valued Fleming-Viot process. We show that for all $t>0$ almost surely the marked metric measure space $\mathcal X_t$ has no atoms and admits a mark function. The latter property means that all individuals in the tree-valued Fleming-Viot process can uniquely be assigned a type. All main results are proven for the neutral case and then carried over to selective cases via Girsanov's formula giving absolute continuity.