Some Path Large-Deviation Results for a Branching Diffusion

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


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.


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


  1. 1.
    Athreya, K.B. Change of measures for Markov chains and the Llog L theorem for branching processes, Bernoulli 6 (2000), no. 2, 323–338.MathSciNetzbMATHCrossRefGoogle Scholar
  2. 2.
    Dembo, A. and Zeitouni, O. Large deviations techniques and applications, Springer, 1998.Google Scholar
  3. 3.
    Engländer, J. and Kyprianou, A.E. Local extinction versus local exponential growth for spatial branching processes, Ann. Probab. 32 (2004), no. 1A, 78–99.MathSciNetzbMATHCrossRefGoogle Scholar
  4. 4.
    Git, Y., Harris, J.W. and Harris, S.C Exponential growth rates in a typed branching diffusion, Ann. App. Probab., 17 (2007), no. 2, 609-653.Google Scholar
  5. 5.
    Hardy, R., and Harris, S. C. A spine approach to branching diffusions with applications to L p -convergence of martingales, Séminaire de Probabilités, XLII, (2009), 281–330.Google Scholar
  6. 6.
    Hardy, R., and Harris, S. C. A conceptual approach to a path result for branching Brownian motion, Stochastic Processes and their Applications, 116 (2006), no. 12, 1992–2013.MathSciNetzbMATHCrossRefGoogle Scholar
  7. 7.
    Harris, S. C. and Williams, D. Large deviations and martingales for a typed branching diffusion. I, Astérisque (1996), no. 236, 133–154, Hommage à P. A. Meyer et J. Neveu.Google Scholar
  8. 8.
    Harris, S.C. Convergence of a ‘Gibbs-Boltzmann’ random measure for a typed branching diffusion, Séminaire de Probabilités, XXXIV, Lecture Notes in mathematics, vol. 1729, Springer, (2000), 133–154.Google Scholar
  9. 9.
    Iksanov, A.M. Elementary fixed points of the BRW smoothing transforms with infinite number of summands, Stochastic Process. Appl. 114 (2004), no. 1, 27–50.MathSciNetzbMATHCrossRefGoogle Scholar
  10. 10.
    Kallenberg, O. Foundations of modern probability, Springer-Verlag, 2002.Google Scholar
  11. 11.
    Kurtz, T., Lyons, R., Pemantle, R. and Peres, Y. A conceptual proof of the Kesten-Stigum theorem for multi-type branching processes, Classical and modern branching processes (Minneapolis, MN, 1994), IMA Vol. Math. Appl., vol. 84, Springer, New York, 1997, pp. 181–185.MathSciNetCrossRefGoogle Scholar
  12. 12.
    Kyprianou, A.E. Travelling wave solutions to the K-P-P equation: alternatives to Simon Harris’s probabilistic analysis, 40 (2004), no. 1, 53–72.Google Scholar
  13. 13.
    Kyprianou, A.E. and Rahimzadeh Sani, A. Martingale convergence and the functional equation in the multi-type branching random walk, Bernoulli 7 (2001), no. 4, 593–604.MathSciNetzbMATHCrossRefGoogle Scholar
  14. 14.
    Lee, T. Some large-deviation theorems for branching diffusions, Ann. Probab. 20 (1992), no. 3, 1288–1309.MathSciNetzbMATHCrossRefGoogle Scholar
  15. 15.
    Lyons, R. A simple path to Biggins’ martingale convergence for branching random walk, Classical and modern branching processes (Minneapolis, MN, 1994), IMA Vol. Math. Appl., vol. 84, Springer, New York, 1997, pp. 217–221.MathSciNetCrossRefGoogle Scholar
  16. 16.
    Lyons, R., Pemantle, R. and Peres, Y. Conceptual proofs of Llog L criteria for mean behavior of branching processes, Ann. Probab. 23 (1995), no. 3, 1125–1138.MathSciNetzbMATHCrossRefGoogle Scholar
  17. 17.
    Øksendal, B. Stochastic differential equations, fifth ed., Springer, 2000.Google Scholar
  18. 18.
    Olofsson, P. The xlog x condition for general branching processes, J. Appl. Probab. 35 (1998), no. 3, 537–544.MathSciNetzbMATHCrossRefGoogle Scholar

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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