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 satisfying the branching property and an exponential integrability condition is called a branching Lévy process, and its law is characterized by a triplet (σ2, a,Λ). We obtain a necessary and sufficient condition for the convergence of the derivative martingale of such a process to a nontrivial limit in terms of (σ2, a,Λ). 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.