A continuous-time particle system on the real line satisfying the branching property and an exponential integrability condition is called a branching Lévy process, and its law is characterized by a triplet . We obtain a necessary and sufficient condition for the convergence of the derivative martingale of such a process to a non-trivial limit in terms of . This extends previously known results on branching Brownian motions and branching random walks. To obtain this result, we rely on the spinal decomposition and establish a novel zero-one law on the perpetual integrals of centred Lévy processes conditioned to stay positive.
Part of this work was supported by a PEPS JCJC grant. BM is partially supported by the Projet ANR-16-CE93-0003 (ANR MALIN). QS was partially supported by the SNSF Grant P2ZHP2_171955.
The authors would like to thank Samuel Baguley, Loïc Chaumont, Leif Döring and Andreas Kyprianou for valuable inputs in the proof of Proposition 3.3. We also thank Haojie Hou and an anonymous referee for their helpful comments.
"A necessary and sufficient condition for the convergence of the derivative martingale in a branching Lévy process." Bernoulli 29 (1) 597 - 624, February 2023. https://doi.org/10.3150/22-BEJ1470