A Spine Approach to Branching Diffusions with Applications to Lp-Convergence of Martingales

  • Robert Hardy
  • Simon C. HarrisEmail author
Part of the Lecture Notes in Mathematics book series (LNM, volume 1979)


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, 43, 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 new results for a multi-type branching Brownian motion and discussion of left-most particle speeds.


Brownian Motion Marked Tree Spine Term Conceptual Proof Spine Approach 
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© Springer-Verlag Berlin Heidelberg 2009

Authors and Affiliations

  1. 1.Department of Mathematical SciencesUniversity of BathClaverton DownUK

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