Biggins' Martingale Convergence for Branching Lévy Processes

Abstract

A branching Lévy process can be seen as the continuous-time version of a branching random walk. It describes a particle system on the real line in which particles move and reproduce independently in a Poissonian manner. Just as for Lévy processes, the law of a branching Lévy process is determined by its characteristic triplet $({\sigma }^2,a,\Lambda )$, where the branching Lévy measure $\Lambda$ describes the intensity of the Poisson point process of births and jumps. We establish a version of Biggins' theorem in this framework, that is we provide necessary and sufficient conditions in terms of the characteristic triplet $({\sigma }^2,a,\Lambda )$ for additive martingales to have a non-degenerate limit.

To appear in Electronic Communications in Probability

References in corpus (2)

• Asymptotics of self-similar growth-fragmentation processes
• Infinitely ramified point measures and branching Lévy processes

Cited by in corpus (3)

• Probability tilting of compensated fragmentations
• Branching-stable point measures and processes
• A result on power moments of Lévy-type perpetuities and its application to the $L_p$-convergence of Biggins' martingales in branching Lévy processes