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Some Path Large-Deviation Results for a Branching Diffusion

  • Robert Hardy
  • Simon C. HarrisEmail author
Conference paper
  • 790 Downloads
Part of the Springer Proceedings in Mathematics & Statistics book series (PROMS, volume 38)

Abstract

We give an intuitive proof of a path large-deviations result for a typed branching diffusion as found in Git, J.Harris and S.C.Harris (Ann. App. Probab. 17(2):609-653, 2007). Our approach involves an application of a change of measure technique involving a distinguished infinite line of descent, or spine, and we follow the spine set-up of Hardy and Harris (Séminaire de Probabilités XLII:281–330, 2009). Our proof combines simple martingale ideas with applications of Varadhan’s lemma and is successful mainly because a “spine decomposition” effectively reduces otherwise difficult calculations on the whole collection of branching diffusion particles down to just a single particle (the spine) whose large-deviations behaviour is well known. A similar approach was used for branching Brownian motion in Hardy and Harris (Stoch. Process. Appl. 116(12):1992–2013, 2006). Importantly, our techniques should be applicable in a much wider class of branching diffusion large-deviations problems.

Keywords

Branching diffusions Spatial branching process Path large deviations Spine decomposition Spine change of measure Additive martingales 

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Copyright information

© Springer Science+Business Media New York 2013

Authors and Affiliations

  1. 1.Department of Mathematical SciencesUniversity of BathBathUK
  2. 2.Currently at VTB CapitalLondonUK

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