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

Amaury Lambert, Gerónimo Uribe Bravo

TL;DR

The paper extends the theory of splitting trees to locally compact TOM trees, unveiling a prolific skeleton that enables decomposing supercritical splitting trees into a Yule-like backbone with subcritical grafts. It builds a genealogical tree via height processes, and proves a Ray-Knight type theorem for supercritical Lévy trees, describing the frontier of prolific versus nonprolific mass through a two-type (CB/CBI) framework. Central tools include the prolific skeleton, grafting mechanisms, and a height-process coding that links chronological trees to continuous-state branching dynamics. The results unify backbone decompositions with continuous genealogical structures, providing a rigorous continuous analogue of classical discrete supercritical branching results and offering a robust framework for studying the genealogy of supercritical growth in Lévy-type populations.

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 Lévy 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 Lévy trees to the supercritical case.

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

TL;DR

The paper extends the theory of splitting trees to locally compact TOM trees, unveiling a prolific skeleton that enables decomposing supercritical splitting trees into a Yule-like backbone with subcritical grafts. It builds a genealogical tree via height processes, and proves a Ray-Knight type theorem for supercritical Lévy trees, describing the frontier of prolific versus nonprolific mass through a two-type (CB/CBI) framework. Central tools include the prolific skeleton, grafting mechanisms, and a height-process coding that links chronological trees to continuous-state branching dynamics. The results unify backbone decompositions with continuous genealogical structures, providing a rigorous continuous analogue of classical discrete supercritical branching results and offering a robust framework for studying the genealogy of supercritical growth in Lévy-type populations.

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 Lévy 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 Lévy trees to the supercritical case.

Paper Structure

This paper contains 16 sections, 14 theorems, 83 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Motivation
  3. Preliminaries
  4. Spectrally positive Lévy processes
  5. The compact tree coded by a càdlàg function
  6. Locally compact trees and their coding sequence
  7. Statement of the results
  8. Organization
  9. The prolific skeleton on a locally compact TOM tree
  10. Backbone decomposition of supercritical splitting trees
  11. Infinite lines of descent under $\Upsilon$
  12. Yule trees and the prolific skeleton decomposition
  13. The height processes and the genealogical tree associated to supercritical splitting trees
  14. Ray-Knight type theorems for supercritical Lévy trees
  15. A grafting construction for the genealogy under $\Upsilon$ and the corresponding Ray-Knight theorem
  16. ...and 1 more sections

Key Result

Proposition 1

Let $\mathscr{I}$ be the collection of individuals with infinite lines of descent. Then $\mathscr{I}=\emptyset$ if and only if $\tau$ is compact. If $\tau$ is non-compact, $\mathscr{I}$ is a non-compact connected subset of $\tau$ containing the root which can be given the structure of a locally comp Furthermore, if $\alpha_u=d\!\left( \rho,\sigma_u\right)$ for $u\in\tau_I$, then $\mathscr{I}$ can

Figures (2)

  • Figure 1: Left: the first 7 generations of an infinite plane tree. Generations increase from left to right. On each generation, labels (lexicographically) increase from bottom to top. (Hence, the tree is $\left\{ \emptyset,1,2,3,11,12,21,22,31,32,33,\ldots\right\}$). Right: the prolific individuals are identified by black disks. Notice that the root has three subtrees above it: two finite ones and an infinite one.
  • Figure 2: Trajectory of a killed compound Poisson process with drift -1 (A) and of the tree it codes (B). The jumps in (A) correspond to the vertical segments in (B) to be joined through the horizontal dashed lines.

Theorems & Definitions (32)

  • Definition
  • Definition
  • Definition : From MR1397084 and MR2221786
  • Definition
  • Definition
  • Proposition 1
  • Definition
  • Theorem 2
  • Corollary 3
  • Theorem 4
  • ...and 22 more