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 (σ2, a, Λ), where the branching Lévy measure Λ 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 (σ2, a, Λ) for additive martingales to have a non-degenerate limit.