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A conceptual approach to a path result for branching Brownian motion

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  • Hardy, Robert
  • Harris, Simon C.

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 -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 LlogL 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].

Suggested Citation

  • Hardy, Robert & Harris, Simon C., 2006. "A conceptual approach to a path result for branching Brownian motion," Stochastic Processes and their Applications, Elsevier, vol. 116(12), pages 1992-2013, December.
    • Handle: RePEc:eee:spapps:v:116:y:2006:i:12:p:1992-2013
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    Cited by:

    1. Berestycki, Julien & Brunet, Éric & Harris, John W. & Harris, Simon C. & Roberts, Matthew I., 2015. "Growth rates of the population in a branching Brownian motion with an inhomogeneous breeding potential," Stochastic Processes and their Applications, Elsevier, vol. 125(5), pages 2096-2145.
    2. Krell, N. & Rouault, A., 2011. "Martingales and rates of presence in homogeneous fragmentations," Stochastic Processes and their Applications, Elsevier, vol. 121(1), pages 135-154, January.

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