Strong law of large numbers for branching diffusions

János Engländer; Simon C. Harris; Andreas E. Kyprianou

Annales de l'I.H.P. Probabilités et statistiques (2010)

  • Volume: 46, Issue: 1, page 279-298
  • ISSN: 0246-0203

Abstract

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Let X be the branching particle diffusion corresponding to the operator Lu+β(u2−u) on D⊆ℝd (where β≥0 and β≢0). Let λc denote the generalized principal eigenvalue for the operator L+β on D and assume that it is finite. When λc>0 and L+β−λc satisfies certain spectral theoretical conditions, we prove that the random measure exp{−λct}Xt 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. Poincaré Probab. Statist.42 (2006) 171–185]. We extend significantly the results in [Z. Wahrsch. Verw. Gebiete36 (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.”

How to cite

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Engländer, János, Harris, Simon C., and Kyprianou, Andreas E.. "Strong law of large numbers for branching diffusions." Annales de l'I.H.P. Probabilités et statistiques 46.1 (2010): 279-298. <http://eudml.org/doc/240655>.

@article{Engländer2010,
abstract = {Let X be the branching particle diffusion corresponding to the operator Lu+β(u2−u) on D⊆ℝd (where β≥0 and β≢0). Let λc denote the generalized principal eigenvalue for the operator L+β on D and assume that it is finite. When λc&gt;0 and L+β−λc satisfies certain spectral theoretical conditions, we prove that the random measure exp\{−λct\}Xt 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. Poincaré Probab. Statist.42 (2006) 171–185]. We extend significantly the results in [Z. Wahrsch. Verw. Gebiete36 (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.”},
author = {Engländer, János, Harris, Simon C., Kyprianou, Andreas E.},
journal = {Annales de l'I.H.P. Probabilités et statistiques},
keywords = {law of large numbers; spine decomposition; spatial branching processes; branching diffusions; measure-valued processes; h-transform; criticality; product-criticality; generalized principal eigenvalue; -transform},
language = {eng},
number = {1},
pages = {279-298},
publisher = {Gauthier-Villars},
title = {Strong law of large numbers for branching diffusions},
url = {http://eudml.org/doc/240655},
volume = {46},
year = {2010},
}

TY - JOUR
AU - Engländer, János
AU - Harris, Simon C.
AU - Kyprianou, Andreas E.
TI - Strong law of large numbers for branching diffusions
JO - Annales de l'I.H.P. Probabilités et statistiques
PY - 2010
PB - Gauthier-Villars
VL - 46
IS - 1
SP - 279
EP - 298
AB - Let X be the branching particle diffusion corresponding to the operator Lu+β(u2−u) on D⊆ℝd (where β≥0 and β≢0). Let λc denote the generalized principal eigenvalue for the operator L+β on D and assume that it is finite. When λc&gt;0 and L+β−λc satisfies certain spectral theoretical conditions, we prove that the random measure exp{−λct}Xt 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. Poincaré Probab. Statist.42 (2006) 171–185]. We extend significantly the results in [Z. Wahrsch. Verw. Gebiete36 (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.”
LA - eng
KW - law of large numbers; spine decomposition; spatial branching processes; branching diffusions; measure-valued processes; h-transform; criticality; product-criticality; generalized principal eigenvalue; -transform
UR - http://eudml.org/doc/240655
ER -

References

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