Biggins’ martingale convergence for branching Lévy processes

Bastien Mallein,Jean Bertoin

doi:10.1214/18-ecp185

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.

Highlights

Introduction and main resultWe start by introducing some notation
For every particle at generation n ≥ 1, say at position x ∈ R, the sequence of positions of the children of that particle is given by x + Y, where Y has the law π, and to different particles correspond independent copies of Y with law π
We find E(ξ1) = κ (θ), and we conclude applying the law of large numbers for Levy processes that ξt ∼ κ (θ)t as t → ∞, a.s

Summary

Introduction

Introduction and main resultWe start by introducing some notation. We denote by x = (xn)n≥1 a generic non-increasing sequence in [−∞, ∞) with limn→∞ xn = −∞. The branching Levy process Z = (Zt)t≥0 is obtained by letting Zt denote the random point measure whose atoms are given by the positions of the particles in the system at time t.

Objectives

Results

Conclusion