Totally Ordered Measured Trees and Splitting Trees with Infinite Variation

TL;DR

This paper introduces totally ordered measured trees (TOM trees) as a continuum analogue of genealogical trees, characterizes their structure via contour functions, and links their splitting properties to spectrally positive Lévy processes.

Contribution

It defines the space of TOM trees, establishes a unique contour function representation, and connects splitting properties to Lévy processes, extending previous work on splitting trees.

Findings

Every compact TOM tree has a unique contour function representation.

Contour functions of splitting TOM trees relate to spectrally positive Lévy processes.

The framework applies to both compact and locally compact TOM trees.

Abstract

Combinatorial trees can be used to represent genealogies of asexual individuals. These individuals can be endowed with birth and death times, to obtain a so-called `chronological tree'. In this work, we are interested in the continuum analogue of chronological trees in the setting of real trees. This leads us to consider totally ordered and measured trees, abbreviated as TOM trees. First, we define an adequate space of TOM trees and prove that under some mild conditions, every compact TOM tree can be represented in a unique way by a so-called contour function, which is right-continuous, admits limits from the left and has non-negative jumps. The appropriate notion of contour function is also studied in the case of locally compact TOM trees. Then we study the splitting property of (measures on) TOM trees which extends the notion of `splitting tree' studied in \cite{MR2599603}, where during her lifetime, each individual gives birth at constant rate to independent and identically distributed copies of herself. We prove that the contour function of a TOM tree satisfying the splitting property is associated to a spectrally positive L\'evy process that is not a subordinator, both in the critical and subcritical cases of compact trees as well as in the supercritical case of locally compact trees.