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The contour of splitting trees is a Lévy process. (English) Zbl 1190.60083

Author’s abstract: Splitting trees are those random trees where individuals give birth at a constant rate during a lifetime with general distribution, to i.i.d. copies of themselves. The width process of a splitting tree is then a binary, homogeneous Crump-Mode-Jagers (CMJ) process, and is not Markovian unless the lifetime distribution is exponential (or a Dirac mass at $\infty$). Here, we allow the birth rate to be infinite, that is, pairs of birth times and life spans of newborns form a Poisson point process along the lifetime of their mother, with possibly infinite intensity measure.

A splitting tree is a random (so-called) chronological tree. Each element of a chronological tree is a (so-called) existence point $\left(v,\tau \right)$ of some individual $v$ (vertex) in a discrete tree where $\tau$ is a nonnegative real number called chronological level (time). We introduce a total order on existence points, called linear order, and a mapping $\varphi$ from the tree into the real line which preserves this order. The inverse of $\varphi$ is called the exploration process, and the projection of this inverse on chronological levels the contour process.

For splitting trees truncated up to level $\tau$, we prove that a thus defined contour process is a Lévy process reflected below $\tau$ and killed upon hitting 0. This allows one to derive properties of (i) splitting trees: conceptual proof of Le Gall-Le Jan’s theorem in the finite variation case, exceptional points, coalescent point process and age distribution; (ii) CMJ processes: one-dimensional marginals, conditionings, limit theorems and asymptotic numbers of individuals with infinite versus finite descendances.

##### MSC:
 60J80 Branching processes 37E25 Maps of trees and graphs 60G51 Processes with independent increments; Lévy processes 60G55 Point processes 60G70 Extreme value theory; extremal processes (probability theory) 60J55 Local time, additive functionals 60J75 Jump processes 60J85 Applications of branching processes 92D25 Population dynamics (general)