Strong law of large numbers for branching diffusions

J Englander, Simon C Harris, Andreas E Kyprianou

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Let X be the branching particle diffusion corresponding to the operator Lu + beta (u(2) - u) on D subset of R-d (where beta >= 0 and beta not equivalent to 0). Let lambda(c) denote the generalized principal eigenvalue for the operator L + beta on D and assume that it is finite. When lambda(c) > 0 and L + beta - lambda(c) satisfies certain spectral theoretical conditions, we prove that the random measure exp{-lambda(c)t}X-t converges almost surely in the vague topology as t tends to infinity. This result is motivated by a cluster of articles due to Asmussen and Hering dating from the mid-seventies as well as the more recent work concerning analogous results for superdiffusions of [Ann. Probab. 30 (2002) 683-722, Ann. Inst. H. Poincare Probab. Statist. 42 (2006) 171-185]. We extend significantly the results in [Z. Wahrsch. Verw. Gebiete 36 (1976) 195-212, Math. Scand. 39 (1977) 327-342, J. Funct. Anal. 250 (2007) 374-399] and include some key examples of the branching process literature. As far as the proofs are concerned, we appeal to modern techniques concerning martingales and "spine" decompositions or "immortal particle pictures."
Original languageEnglish
Pages (from-to)279-298
Number of pages20
JournalAnnales de l'Institut Henri Poincaré, Probabilités et Statistiques
Issue number1
Publication statusPublished - Feb 2010


  • spine decomposition
  • criticality
  • Law of Large Numbers
  • generalized principal eigenvalue
  • branching diffusions
  • h-transform
  • spatial branching processes
  • product-criticality
  • measure-valued processes


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