A spine approach to branching diffusions with applications to L-p-convergence of martingales

We present a modified formalization of the 'spine' change of measure approach for branching diffusions in the spirit, of those found in Kyprianou [40] and Lyons et al. [44, 437 41]. We use our formulation to interpret certain 'Gibbs-Boltzmann' weightings of particles and use this to give an intuitive proof of a 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 'spine' particle. We also exemplify spine proofs of the L-p-convergence (p >= 1) of some key 'additive' martingales for three distinct models of branching diffusions including flew results for a multi-type branching Brownian motion and discussion of left-most particle speeds.