Totally Ordered Measured Trees and Splitting Trees with Infinite Variation II: Prolific Skeleton Decomposition

TL;DR

This paper extends the theory of splitting trees and TOM trees by decomposing them via their prolific skeletons and explores the genealogical structures using height processes, including a new Ray-Knight theorem for supercritical cases.

Contribution

It introduces a decomposition of TOM trees based on their prolific skeletons and extends the Ray-Knight theorem to supercritical Lévy trees.

Findings

Decomposition of TOM trees into prolific skeletons and grafted subtrees.

Construction of supercritical splitting trees from subcritical ones.

Extension of Ray-Knight theorem to supercritical Lévy trees.

Abstract

The first part of this paper ( arXiv:1607.02114 ) introduced splitting trees, those chronological trees admitting the self-similarity property where individuals give birth, at constant rate, to iid copies of themselves. It also established the intimate relationship between splitting trees and L\'evy processes. The chronological trees involved were formalized as Totally Ordered Measured (TOM) trees. The aim of this paper is to continue this line of research in two directions: we first decompose locally compact TOM trees in terms of their prolific skeleton (consisting of its infinite lines of descent). When applied to splitting trees, this implies the construction of the supercritical ones (which are locally compact) in terms of the subcritical ones (which are compact) grafted onto a Yule tree (which corresponds to the prolific skeleton). As a second (related) direction, we study the genealogical tree associated to our chronological construction. This is done through the technology of the height process introduced by Duquesne and Le Gall. In particular we prove a Ray-Knight type theorem which extends the one for (sub)critical L\'evy trees to the supercritical case.