A necessary and sufficient condition for the convergence of the derivative martingale in a branching Lévy process
A continuous-time particle system on the real line verifying the branching property and an exponential integrability condition is called a branching Lévy process, and its law is characterized by a triplet $(\sigma^2,a,\Lambda)$. 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 $(\sigma^2,a,\Lambda)$. 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.
Submitted to Bernoulli A necessary and sufficient condition for the convergence of the derivative martingale in a branching Lévy process arXiv:2105.07919v2 [math.PR] 24 Feb 2022 BASTIEN MALLEIN1 and QUAN SHI2,* 1 Université Sorbonne Paris Nord, LAGA, UMR 7539, F-93430, Villetaneuse, France,…
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