Biggins' Martingale Convergence for Branching L\'evy Processes

TL;DR

This paper extends Biggins' martingale convergence theorem to branching Le9vy processes, characterizing conditions for the convergence of additive martingales based on the process's characteristic triplet.

Contribution

It provides necessary and sufficient conditions for martingale convergence in branching Le9vy processes, generalizing previous results from branching random walks.

Findings

Established a version of Biggins' theorem for branching Le9vy processes.

Derived conditions for non-degenerate limits of additive martingales.

Linked martingale convergence to the characteristic triplet (c3^2,a,bLambda).

Abstract

A branching L\'evy process can be seen as the continuous-time version of a branching random walk. It describes a particle system on the real line in which particles move and reproduce independently in a Poissonian manner. Just as for L\'evy processes, the law of a branching L\'evy process is determined by its characteristic triplet $(\sigma^2,a,\Lambda)$, where the branching L\'evy measure $\Lambda$ describes the intensity of the Poisson point process of births and jumps. We establish a version of Biggins' theorem in this framework, that is we provide necessary and sufficient conditions in terms of the characteristic triplet $(\sigma^2,a,\Lambda)$ for additive martingales to have a non-degenerate limit.