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

Abstract

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 $({\mathit{\sigma }}^2,a,\mathrm{\Lambda })$. 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 $({\mathit{\sigma }}^2,a,\mathrm{\Lambda })$. 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.