Biggins' Martingale Convergence for Branching Lévy Processes

Jean Bertoin (1) , Bastien Mallein (2)

1 Institute of Mathematics University of Zurich
2 LAGA - Laboratoire Analyse, Géométrie et Applications

A branching Lévy process can be seen as the continuous-time version of a branching random walk; see [BM17]. It describes a particle system on the real line in which particles move and reproduce independently one of the others, in a Poissonian manner. Just as for Lévy processes, the law of a branching Lévy process is determined by its characteristic triplet (σ 2 , a, Λ), where the Lévy measure Λ describes the intensity of the Poisson point process of births and jumps. We establish a version of Biggins' theorem [Big77] in this framework, that is we provide necessary and sufficient conditions in terms of the characteristic triplet (σ 2 , a, Λ) for additive martingales of branching Lévy processes to have a non-degenerate limit. The proof is adapted from Lyons [Lyo97].

Mots clés: uniform integrability; additive martingale; Branching Lévy process; spinal decomposition; 60J80; spinal decomposition AMS subject classifications: 60G44

Domaines: Probabilités [math.PR]

Jean Bertoin, Bastien Mallein. Biggins' Martingale Convergence for Branching Lévy Processes. Electronic Communications in Probability, 2018, 23, ⟨10.1214/18-ECP185⟩. ⟨hal-01689648v2⟩