Growth of Levy trees

TL;DR

This paper introduces Levy trees, a new class of random real trees representing genealogies of continuous-state branching processes, constructed via limits of discrete Galton-Watson trees with exponential edge lengths.

Contribution

It provides an elementary construction of Levy trees as limits of discrete trees, including supercritical cases, without relying on the height process framework.

Findings

Levy trees are constructed as Gromov-Hausdorff limits of discrete Galton-Watson trees.

The mass measure of Levy trees is explicitly constructed.

A decomposition along ancestral subtrees is established.

Abstract

We construct random locally compact real trees called Levy trees that are the genealogical trees associated with continuous-state branching processes. More precisely, we define a growing family of discrete Galton-Watson trees with i.i.d. exponential branch lengths that is consistent under Bernoulli percolation on leaves; we define the Levy tree as the limit of this growing family with respect to the Gromov-Hausdorff topology on metric spaces. This elementary approach notably includes supercritical trees and does not make use of the height process introduced by Le Gall and Le Jan to code the genealogy of (sub)critical continuous-state branching processes. We construct the mass measure of Levy trees and we give a decomposition along the ancestral subtree of a Poisson sampling directed by the mass measure.