A new formulation of the spine approach to branching diffusions

Abstract

We present a formalization of the spine change of measure approach for branching diffusions that improves on the scheme laid out for branching Brownian motion in Kyprianou (2004) ["Travelling wave solutions to the KPP equation, Ann. Inst. H. Poincare Probab. Statist. 40, no.1, pp53-72] which itself made use of earlier works of Lyons et al (1997) ["A conceptual proof of the Kesten-Stigum theorem for multi-type branching processes", Classical and modern branching processes, IMA Vol. Math. Appl., vol.84, Springer, New York, pp181-185]. We use our new formulation to interpret certain `Gibbs-Boltzmann' weightings of particles and use this to give a new, intuitive and proof of a more general `Many-to-One' result which enables expectations of sums over particles in the branching diffusion to be calculated purely in terms of an expectation of one particle. Significantly, our formalization has provided the foundations that facilitate a variety of new, greatly simplified and more intuitive proofs in branching diffusions: see, for example, the L^p convergence of additive martingales in Hardy and Harris (2006) ["Spine proofs for L^p-convergence of branching-diffusion martingales", arXiv:math.PR/0611056], the path large deviation results for branching Brownian motion in Hardy and Harris (2006) ["A conceptual approach to a path result for branching Brownian motion", Stochastic Processes and their Applications, doi:10.1016/j.spa.2006.05.010] and the large deviations for a continuous-typed branching diffusion in Git et al (2006) ["Exponential growth rates in a typed branching diffusion", Annals Applied Prob., (under revision)] and Hardy and Harris (2004) ["A spine proof of a lower-bound for a typed branching diffusion", no.0408, Mathematics Preprint, University of Bath].