*(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 $(v,\tau )$ 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) |