Spine proofs for Lp-convergence of branching-diffusion martingales∗

Using the foundations laid down in Hardy and Harris [17], we present new spine proofs of the Lp-convergence (p ≥ 1) of some key ‘additive’ martingales for three distinct models of branching diffusions, including new results for a multi-type branching Brownian motion and discussion of left-most particle speeds. The spine techniques we develop give clear and simple arguments in the spirit of the conceptual spine proofs found in Kyprianou [28] and Lyons et al [31, 30, 27], and they should also extend to more general classes of branching diffusions. Importantly, the techniques in this paper also pave the way for the path largedeviation results for branching diffusions found in Hardy and Harris [16, 15]. 1 Overview In this article we use a change of measure together with spine techniques to analyze the Lpconvergence properties (for p ≥ 1) of the strictly-positive ‘additive’ martingales for three different models of branching diffusions. It is a common feature of these diffusion models, where there is actually a family of such martingales { Zλ : λ ∈ R } , that for all λ within an open interval about the origin the martingale Zλ is convergent in Lp for some p ≥ 1 subject to a suitable pth−moment (p > 1) or L logL (p = 1) condition on the offspring distribution; for λ outside of this interval, or if the L logL condition fails for the offspring distribution, the limit of Zλ is almost surely null. The first model we consider is a branching Brownian motion (BBM) with random family sizes. After introducing the fundamental ‘additive’ Zλ martingales and describing a ‘spine’ construction for the BBM under a change of measure using Zλ, we will recall Kyprianou’s [28] L1-convergence result before stating necessary and sufficient conditions for Lp-convergence of the Zλ martingales. Our new proof of martingale Lp-convergence for BBM uses ‘spine’ change of measure techniques and, early on in Section 2, we will include a summary of the underlying space and filtrations that we shall use for our spine techniques throughout this paper; a foundation article [17] contains full details, but we have tried to keep this article reasonably self-contained. Note that for BBM, the spine construction first appeared in Chauvin and Rouault [5], whilst Kyprianou [28] really exploited ‘spine’ methods in his proofs, however our spine approach does possess some significant differences from others. Neveu[32] used classical techniques for Lpconvergence in the special case of binary branching. Also see Harris [21] for further discussion of martingale convergence in BBM and applications. This arXiv article is an revision of Spine proofs for L-convergence of branching-diffusion martingales, (2004), no. 0405, Mathematics Preprint, University of Bath. Email: [email protected], Web: http://people.bath.ac.uk/massch