A necessary and sufficient condition for the convergence of the derivative martingale in a branching Levy process

A continuous-time particle system on the real line verifying the branching property and an exponential integrability condition is called a branching Levy 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 Levy processes conditioned to stay positive.