Strong law of large numbers for branching diffusions

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

Annales de l'I.H.P. Probabilités et statistiques, Tome 46 (2010), p. 279-298 / Harvested from Numdam

Résumé

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.”

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>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.”

Détail

Publié le : 2010-01-01

MR 2641779 | Zbl 1196.60139 | 1 citation dans Numdam

DOI : https://doi.org/10.1214/09-AIHP203
Classification:  60J60,  60J80

Bibliographie

[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 420889 | Zbl 0325.60081

[2] S. Asmussen and H. Hering. Strong limit theorems for supercritical immigration-branching processes. Math. Scand. 39 (1977) 327-342. | MR 438498 | Zbl 0348.60117

[3] K. Athreya. Change of measures for Markov chains and the LlogL theorem for branching processes. Bernoulli 6 (2000) 323-338. | MR 1748724 | Zbl 0969.60076

[4] J. Biggins. Uniform convergence in the branching random walk. Ann. Probab. 20 (1992) 137-151. | MR 1143415 | Zbl 0748.60080

[5] J. D. Biggins and A. E. Kyprianou. Measure change in multitype branching. Adv. in Appl. Probab. 36 (2004) 544-581. | MR 2058149 | Zbl 1056.60082

[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 1325205 | Zbl 0824.60070

[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 968823 | Zbl 0653.60077

[8] Z.-Q. Chen and Y. Shiozawa. Limit theorems for branching Markov processes. J. Funct. Anal. 250 (2007) 374-399. | MR 2352485 | Zbl 1125.60087

[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 2397881 | Zbl 1138.60054

[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 1242575 | Zbl 0799.60080

[11] E. B. Dynkin. An Introduction to Branching Measure-Valued Processes. CRM Monograph Series 6. Amer. Math. Soc., Providence, RI, 1994. | MR 1280712 | Zbl 0824.60001

[12] J. Engländer. Branching diffusions, superdiffusions and random media. Probab. Surv. 4 (2007) 303-364. | MR 2368953 | Zbl 1189.60143

[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 2500226 | Zbl 1172.60022

[14] J. Engländer and A. Kyprianou. Local extinction versus local exponential growth for spatial branching processes. Ann. Probab. 32 (2003) 78-99. | MR 2040776 | Zbl 1056.60083

[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 1698955 | Zbl 0979.60078

[16] J. Engländer and D. Turaev. A scaling limit theorem for a class of superdiffusions. Ann. Probab. 30 (2002) 683-722. | MR 1905855 | Zbl 1014.60080

[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 2199796 | Zbl 1093.60058

[18] A. Etheridge. An Introduction to Superprocesses. University Lecture Series 20. Amer. Math. Soc., Providence, RI, 2000. | MR 1779100 | Zbl 0971.60053

[19] S. N. Evans. Two representations of a superprocess. Proc. Roy. Soc. Edinburgh Sect. A 123 (1993) 959-971. | MR 1249698 | Zbl 0784.60052

[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 2308337 | Zbl 1131.60077

[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 2307069 | Zbl 1114.60065

[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 2599214 | Zbl 1193.60100

[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 1768067 | Zbl 0985.60053

[24] J. Jacod and A. N. Shiryaev. Limit Theorems for Stochastic Processes, 2nd edition. Grundlehren der Mathematischen Wissenschaften 288. Springer, Berlin, 2003. | MR 1943877 | Zbl 1018.60002

[25] O. Kallenberg. Stability of critical cluster fields. Math. Nachr. 77 (1977) 7-43. | MR 443078 | Zbl 0361.60058

[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 1349164 | Zbl 0840.60077

[27] R. G. Pinsky. Positive Harmonic Functions and Diffusion. Cambridge Univ. Press, Cambridge, 1995. | MR 1326606 | Zbl 0858.31001

[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 1387634 | Zbl 0854.60087

[29] S. Watanabe. A limit theorem of branching processes and continuous state branching processes. J. Math. Kyoto Univ. 8 (1968) 141-167. | MR 237008 | Zbl 0159.46201