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arXiv · 1712.04769

Biggins' Martingale Convergence for Branching Lévy Processes

Jean Bertoin·Bastien Mallein
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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 $(σ^2,a,Λ)$, 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 $(σ^2,a,Λ)$ for additive martingales to have a non-degenerate limit.

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Jean Bertoin, Bastien Mallein. 2018-10-19. Biggins' Martingale Convergence for Branching Lévy Processes. https://doi.org/10.1214/18-ecp185

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