A necessary and sufficient condition for the convergence of the derivative martingale in a branching L\'evy process

TL;DR

This paper establishes a precise criterion for the convergence of the derivative martingale in branching Le9vy processes, extending known results from branching Brownian motion and random walks using advanced probabilistic techniques.

Contribution

It provides a necessary and sufficient condition for martingale convergence in branching Le9vy processes based on their defining triplet, generalizing previous specific cases.

Findings

Derived a complete criterion for martingale convergence.

Extended results from Brownian motion and random walks.

Introduced a novel zero-one law for Le9vy processes.

Abstract

A continuous-time particle system on the real line satisfying the branching property and an exponential integrability condition is called a branching L\'evy 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 L\'evy processes conditioned to stay positive.