A conceptual approach to a path result for branching Brownian motion

Abstract

We give a new, intuitive and relatively straightforward proof of a path large-deviations result for branching Brownian motion (BBM) that can be thought of as extending Schilder’s theorem for a single Brownian motion. Our conceptual approach provides an elegant and striking new application of a change of measure technique that induces a ‘spine’ decomposition and builds on the new foundations for the use of spines in branching diffusions recently developed in Hardy and Harris [Robert Hardy, Simon C. Harris, A new formulation of the spine approach to branching diffusions, 2004, no. 0404, Mathematics Preprint, University of Bath. http://www.bath.ac.uk/~massch/Research/Papers/spine-foundations.pdf; Robert Hardy, Simon C. Harris, Spine proofs for L p -convergence of branching-diffusion martingales, 2004, no. 0405, Mathematics Preprint, University of Bath. http://www.bath.ac.uk/~massch/Research/Papers/spine-Lp-cgce.pdf], itself inspired by related works of Kyprianou [Andreas Kyprianou, Travelling wave solutions to the K–P–P equation: alternatives to Simon Harris’s probabilistic analysis, Ann. Inst. H. Poincaré Probab. Statist. 40 (1) (2004) 53–72] and Lyons et al. [Russell Lyons, A simple path to Biggins’ martingale convergence for branching random walk, in: Classical and Modern Branching Processes (Minneapolis, MN, 1994), IMA Vol. Math. Appl., vol. 84, Springer, New York, 1997, pp. 217–221; Thomas Kurtz, Russell Lyons, Robin Pemantle, Yuval Peres, A conceptual proof of the Kesten–Stigum theorem for multi-type branching processes, in: Classical and Modern Branching Processes (Minneapolis, MN, 1994), IMA Vol. Math. Appl., vol. 84, Springer, New York, 1997, pp. 181–185; Russell Lyons, Robin Pemantle, Yuval Peres, Conceptual proofs of L log L criteria for mean behavior of branching processes, Ann. Probab. 23 (3) (1995) 1125–1138]. Some of the techniques developed here will also apply in more general branching Markov processes, for example, see Hardy and Harris [Robert Hardy, Simon C. Harris, A spine proof of a lower bound for a typed branching diffusion, 2004, no. 0408, Mathematics Preprint, University of Bath. http://www.bath.ac.uk/~massch/Research/Papers/spine-oubbm.pdf].