Séminaire de Probabilités XLVI pp 1-32

Part of the Lecture Notes in Mathematics book series (LNM, volume 2123) | Cite as

Branching Random Walk in an Inhomogeneous Breeding Potential

Chapter

Abstract

We consider a continuous-time branching random walk in the inhomogeneous breeding potential β | ⋅ | p, where β > 0, p ≥ 0. We prove that the population almost surely explodes in finite time if p > 1 and doesn’t explode if p ≤ 1. In the non-explosive cases, we determine the asymptotic behaviour of the rightmost particle.

References

  1. 1.
    J. Berestycki, É. Brunet, J. Harris, S. Harris, The almost-sure population growth rate in branching Brownian motion with a quadratic breeding potential. Stat. Probab. Lett. 80, 1442–1446 (2010)CrossRefMathSciNetMATHGoogle Scholar
  2. 2.
    J. Berestycki, É. Brunet, J. Harris, S. Harris, M. Roberts, Growth rates of the population in a branching Brownian motion with an inhomogeneous breeding potential. arXiv:1203.0513Google Scholar
  3. 3.
    J.D. Biggins, The growth and spread of the general branching random walk. Ann. Appl. Probab. 5(4), 1008–1024 (1995)CrossRefMathSciNetMATHGoogle Scholar
  4. 4.
    J.D. Biggins, How Fast Does a General Branching Random Walk Spread? Classical and Modern Branching Processes. IMA Vol. Math. Appl., vol. 84 (Springer, New York, 1996), pp. 19–39Google Scholar
  5. 5.
    R. Durrett, Probability: Theory and Examples, 2nd edn. (Duxbury Press, Belmont, 1996)Google Scholar
  6. 6.
    J. Feng, T.G. Kurtz, Large Deviations for Stochastic Processes (American Mathematical Society, Providence, 2006)CrossRefMATHGoogle Scholar
  7. 7.
    R. Hardy, S.C. Harris, A Spine Approach to Branching Diffusions with Applications to L p -Convergence of Martingales. Séminaire de Probabilités, XLII, Lecture Notes in Math., vol. 1979 (Springer, Berlin, 2009)Google Scholar
  8. 8.
    J.W. Harris, S.C. Harris, Branching Brownian motion with an inhomogeneous breeding potential. Ann. de l’Institut Henri Poincaré (B) Probab. Stat. 45(3), 793–801 (2009)Google Scholar
  9. 9.
    K. Itô, H.P. McKean, Diffusion Processes and Their Sample Paths, 2nd edn. (Springer, Berlin, 1974)MATHGoogle Scholar

Copyright information

© Springer International Publishing Switzerland 2014

Authors and Affiliations

  1. 1.Department of Mathematical SciencesUniversity of BathBathUK

Personalised recommendations