Biggins' Martingale Convergence for Branching L\'evy Processes

A branching L'evy 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 L'evy processes, the law of a branching L'evy process is determined by its characteristic triplet $({\sigma }^2,a,\mathrm{\Lambda })$, where the branching L'evy measure $\mathrm{\Lambda }$ 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 $({\sigma }^2,a,\mathrm{\Lambda })$ for additive martingales to have a non-degenerate limit.