We develop potential-theoretical methods in the construction of measure-valued branching processes.We complete results of P. J. Fitzsimmons and E. B. Dynkin on the construction, regularity and other properties of the superprocess associated with a given right process and a branching mechanism.
On définit de nouveaux processus de naissance à temps discret; la population est, à chaque instant, organisée en graphe. Pour obtenir la -ième génération on remplace aléatoirement les sommets de la -ième génération par des graphes que l’on accroche convenablement les uns aux autres. On autorise une certaine dépendance entre les substitutions de sommets voisins. On étudie, pour certains processus surcritiques, la croissance de la population et la structure des graphes générés : sous des hypothèses...
We present a new pruning procedure on discrete trees by adding marks on the nodes of trees. This procedure allows us to construct and study a tree-valued Markov process by pruning Galton–Watson trees and an analogous process by pruning a critical or subcritical Galton–Watson tree conditioned to be infinite. Under a mild condition on offspring distributions, we show that the process run until its ascension time has a representation in terms of . A similar result was obtained by Aldous and...
We study a spatial branching model, where the underlying motion is d-dimensional (d≥1) brownian motion and the branching rate is affected by a random collection of reproduction suppressing sets dubbed mild obstacles. The main result of this paper is the quenched law of large numbers for the population for all d≥1. We also show that the branching brownian motion with mild obstacles spreads less quickly than ordinary branching brownian motion by giving an upper estimate on its speed. When the underlying...
This paper presents a new model of asymmetric bifurcating autoregressive process with random coefficients. We couple this model with a Galton−Watson tree to take into account possibly missing observations. We propose least-squares estimators for the various parameters of the model and prove their consistency, with a convergence rate, and asymptotic normality. We use both the bifurcating Markov chain and martingale approaches and derive new results in both these frameworks.
We survey recent developments about random real trees, whose prototype is the Continuum Random Tree (CRT) introduced by Aldous in 1991. We briefly explain the formalism of real trees, which yields a neat presentation of the theory and in particular of the relations between discrete Galton-Watson trees and continuous random trees. We then discuss the particular class of self-similar random real trees called stable trees, which generalize the CRT. We review several important results concerning stable...