Strong law of large numbers for branching diffusions
Annales de l'I.H.P. Probabilités et statistiques, Tome 46 (2010) no. 1, pp. 279-298.

Soit X le processus de diffusion avec branchement correspondant à l'operateur Lu+β(u2-u) sur D⊆ℝd (où β≥0 et β≢0). La valeur propre principale généralisée de l'operateur L+β sur D est dénotée par λc et on la suppose finie. Quand λc>0 et L+β-λc satisfait certaines conditions spectrales théoriques, on montre que la mesure aléatoire exp{-λct}Xt converge presque sûrement pour la topologie vague quand t tend vers l'infini. Ce résultat est motivé par un ensemble d'articles par Asmussen et Hering datant du milieu des années soixante-dix, ainsi que par des travaux plus récents [Ann. Probab. 30 (2002) 683-722, Ann. Inst. H. Poincaré Probab. Statist. 42 (2006) 171-185] concernant des résultats analogues pour les superdiffusions. Nous généralisons de manière significative les résultats de [Z. Wahrsch. Verw. Gebiete 36 (1976) 195-212, Math. Scand. 39 (1977) 327-342, J. Funct. Anal. 250 (2007) 374-399] et nous donnons quelques exemples clés de la théorie des processus de branchement. En ce qui concerne les démonstrations, nous faisons appel aux techniques modernes de martingales et aux “spine decompositions” ou “immortal particle pictures.”

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. 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.”

DOI : 10.1214/09-AIHP203
Classification : 60J60, 60J80
Mots clés : law of large numbers, spine decomposition, spatial branching processes, branching diffusions, measure-valued processes, h-transform, criticality, product-criticality, generalized principal eigenvalue
@article{AIHPB_2010__46_1_279_0,
     author = {Engl\"ander, J\'anos and Harris, Simon C. and Kyprianou, Andreas E.},
     title = {Strong law of large numbers for branching diffusions},
     journal = {Annales de l'I.H.P. Probabilit\'es et statistiques},
     pages = {279--298},
     publisher = {Gauthier-Villars},
     volume = {46},
     number = {1},
     year = {2010},
     doi = {10.1214/09-AIHP203},
     mrnumber = {2641779},
     zbl = {1196.60139},
     language = {en},
     url = {http://www.numdam.org/articles/10.1214/09-AIHP203/}
}
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
SP  - 279
EP  - 298
VL  - 46
IS  - 1
PB  - Gauthier-Villars
UR  - http://www.numdam.org/articles/10.1214/09-AIHP203/
DO  - 10.1214/09-AIHP203
LA  - en
ID  - AIHPB_2010__46_1_279_0
ER  - 
%0 Journal Article
%A Engländer, János
%A Harris, Simon C.
%A Kyprianou, Andreas E.
%T Strong law of large numbers for branching diffusions
%J Annales de l'I.H.P. Probabilités et statistiques
%D 2010
%P 279-298
%V 46
%N 1
%I Gauthier-Villars
%U http://www.numdam.org/articles/10.1214/09-AIHP203/
%R 10.1214/09-AIHP203
%G en
%F AIHPB_2010__46_1_279_0
Engländer, János; Harris, Simon C.; Kyprianou, Andreas E. Strong law of large numbers for branching diffusions. Annales de l'I.H.P. Probabilités et statistiques, Tome 46 (2010) no. 1, pp. 279-298. doi : 10.1214/09-AIHP203. http://www.numdam.org/articles/10.1214/09-AIHP203/

[1] S. Asmussen and H. Hering. Strong limit theorems for general supercritical branching processes with applications to branching diffusions. Z. Wahrsch. Verw. Gebiete 36 (1976) 195-212. | MR | Zbl

[2] S. Asmussen and H. Hering. Strong limit theorems for supercritical immigration-branching processes. Math. Scand. 39 (1977) 327-342. | EuDML | MR | Zbl

[3] K. Athreya. Change of measures for Markov chains and the LlogL theorem for branching processes. Bernoulli 6 (2000) 323-338. | MR | Zbl

[4] J. Biggins. Uniform convergence in the branching random walk. Ann. Probab. 20 (1992) 137-151. | MR | Zbl

[5] J. D. Biggins and A. E. Kyprianou. Measure change in multitype branching. Adv. in Appl. Probab. 36 (2004) 544-581. | MR | Zbl

[6] A. Champneys, S. C. Harris, J. Toland, J. Warren and D. Williams. Algebra, analysis and probability for a coupled system of reaction-diffusion equations. Philos. Trans. R. Soc. Lond. Ser. A 350 (1995) 69-112. | MR | Zbl

[7] B. Chauvin and A. Rouault. KPP equation and supercritical branching Brownian motion in the subcritical speed area. Application to spatial trees. Probab. Theory Related Fields 80 (1988) 299-314. | MR | Zbl

[8] Z.-Q. Chen and Y. Shiozawa. Limit theorems for branching Markov processes. J. Funct. Anal. 250 (2007) 374-399. | MR | Zbl

[9] Z.-Q. Chen, Y. Ren and H. Wang. An almost sure scaling limit theorem for Dawson-Watanabe superprocesses J. Funct. Anal. 254 (2008) 1988-2019. | MR | Zbl

[10] D. A. Dawson. Measure-valued Markov processes. In Ecole d'Eté Probabilités de Saint Flour XXI 1-260. Lecture Notes in Math. 1541. Springer, Berlin, 1993. | MR | Zbl

[11] E. B. Dynkin. An Introduction to Branching Measure-Valued Processes. CRM Monograph Series 6. Amer. Math. Soc., Providence, RI, 1994. | MR | Zbl

[12] J. Engländer. Branching diffusions, superdiffusions and random media. Probab. Surv. 4 (2007) 303-364. | MR | Zbl

[13] J. Engländer. Law of large numbers for superdiffusions: The non-ergodic case. Ann. Inst. H. Poincare Probab. Statist. 45 (2009) 1-6. | Numdam | MR | Zbl

[14] J. Engländer and A. Kyprianou. Local extinction versus local exponential growth for spatial branching processes. Ann. Probab. 32 (2003) 78-99. | MR | Zbl

[15] J. Engländer and R. Pinsky. On the construction and support properties of measure-valued diffusions on D⊆Rd with spatially dependent branching. Ann. Probab. 27 (1999) 684-730. | MR | Zbl

[16] J. Engländer and D. Turaev. A scaling limit theorem for a class of superdiffusions. Ann. Probab. 30 (2002) 683-722. | MR | Zbl

[17] J. Engländer and A. Winter. Law of large numbers for a class of superdiffusions. Ann. Inst. H. Poincare Probab. Statist. 42 (2006) 171-185. | Numdam | MR | Zbl

[18] A. Etheridge. An Introduction to Superprocesses. University Lecture Series 20. Amer. Math. Soc., Providence, RI, 2000. | MR | Zbl

[19] S. N. Evans. Two representations of a superprocess. Proc. Roy. Soc. Edinburgh Sect. A 123 (1993) 959-971. | MR | Zbl

[20] Y. Git, J. W. Harris and S. C. Harris. Exponential growth rates in a typed branching diffusion. Ann. Appl. Probab. 17 (2007) 609-653. | MR | Zbl

[21] R. Hardy and S. C. Harris. A conceptual approach to a path result for branching Brownian motion. Stochastic Process Appl. 116 (2006) 1992-2013. | MR | Zbl

[22] R. Hardy and S. C. Harris. A spine approach to branching diffusions with applications to Lp-convergence of martingales. In Séminaire de Probabilités XLII. C. Donati-Martin, M. Émery, A. Rouault and C. Stricker (Eds). 1979, 2009. | MR | Zbl

[23] S. C. Harris. Convergence of a “Gibbs-Boltzman” random measure for a typed branching diffusion. In Séminaire de Probabilités XXXIV 239-256. Lecture Notes in Math. 1729. Springer, Berlin, 2000. | Numdam | MR | Zbl

[24] J. Jacod and A. N. Shiryaev. Limit Theorems for Stochastic Processes, 2nd edition. Grundlehren der Mathematischen Wissenschaften 288. Springer, Berlin, 2003. | MR | Zbl

[25] O. Kallenberg. Stability of critical cluster fields. Math. Nachr. 77 (1977) 7-43. | MR | Zbl

[26] R. Lyons, R. Pemantle and Y. Peres. Conceptual proofs of L log L criteria for mean behaviour of branching processes. Ann. Probab. 23 (1995) 1125-1138. | MR | Zbl

[27] R. G. Pinsky. Positive Harmonic Functions and Diffusion. Cambridge Univ. Press, Cambridge, 1995. | MR | Zbl

[28] R. G. Pinsky. Transience, recurrence and local extinction properties of the support for supercritical finite measure-valued diffusions. Ann. Probab. 24 (1996) 237-267. | MR | Zbl

[29] S. Watanabe. A limit theorem of branching processes and continuous state branching processes. J. Math. Kyoto Univ. 8 (1968) 141-167. | MR | Zbl

Cité par Sources :