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 
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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. https://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

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  • Foxall, Eric; Labossi, Jen Takeover, fixation and identifiability in finite neutral genealogy models, Electronic Journal of Probability, Volume 29 (2024) no. none | DOI:10.1214/24-ejp1219
  • Medous, Charles Spinal constructions for continuous type-space branching processes with interactions, Electronic Journal of Probability, Volume 29 (2024) no. none | DOI:10.1214/24-ejp1227
  • Öz, Mehmet Branching Brownian motion under soft killing, Latin American Journal of Probability and Mathematical Statistics, Volume 21 (2024) no. 1, p. 491 | DOI:10.30757/alea.v21-20
  • Horton, Emma; Kyprianou, Andreas E. Martingale Convergence and Laws of Large Numbers, Stochastic Neutron Transport (2023), p. 243 | DOI:10.1007/978-3-031-39546-8_12
  • Bansaye, Vincent; Gu, Chenlin; Yuan, Linglong A growth-fragmentation-isolation process on random recursive trees and contact tracing, The Annals of Applied Probability, Volume 33 (2023) no. 6B | DOI:10.1214/23-aap1947
  • Bansaye, Vincent; Cloez, Bertrand; Gabriel, Pierre; Marguet, Aline A non‐conservative Harris ergodic theorem, Journal of the London Mathematical Society, Volume 106 (2022) no. 3, p. 2459 | DOI:10.1112/jlms.12639
  • NISHIMORI, Yasuhito; SHIOZAWA, Yuichi Limiting distributions for the maximal displacement of branching Brownian motions, Journal of the Mathematical Society of Japan, Volume 74 (2022) no. 1 | DOI:10.2969/jmsj/85158515
  • Horton, Emma; Watson, Alexander R. Strong laws of large numbers for a growth-fragmentation process with bounded cell sizes, Latin American Journal of Probability and Mathematical Statistics, Volume 19 (2022) no. 2, p. 1799 | DOI:10.30757/alea.v19-68
  • Gonzalez, Isaac; Horton, Emma; Kyprianou, Andreas E. Asymptotic moments of spatial branching processes, Probability Theory and Related Fields, Volume 184 (2022) no. 3-4, p. 805 | DOI:10.1007/s00440-022-01131-2
  • Harris, Simon C.; Horton, Emma; Kyprianou, Andreas E.; Wang, Minmin Yaglom limit for critical nonlocal branching Markov processes, The Annals of Probability, Volume 50 (2022) no. 6 | DOI:10.1214/22-aop1585
  • Bansaye, Vincent; Cloez, Bertrand; Gabriel, Pierre Ergodic Behavior of Non-conservative Semigroups via Generalized Doeblin’s Conditions, Acta Applicandae Mathematicae, Volume 166 (2020) no. 1, p. 29 | DOI:10.1007/s10440-019-00253-5
  • Bertoin, Jean; Watson, Alexander R. The strong Malthusian behavior of growth-fragmentation processes, Annales Henri Lebesgue, Volume 3 (2020), p. 795 | DOI:10.5802/ahl.46
  • Shi, Quan A growth-fragmentation model related to Ornstein–Uhlenbeck type processes, Annales de l'Institut Henri Poincaré, Probabilités et Statistiques, Volume 56 (2020) no. 1 | DOI:10.1214/19-aihp974
  • Jonckheere, Matthieu; Saglietti, Santiago On laws of large numbers in L2 for supercritical branching Markov processes beyond λ-positivity, Annales de l'Institut Henri Poincaré, Probabilités et Statistiques, Volume 56 (2020) no. 1 | DOI:10.1214/19-aihp961
  • Hebbar, Pratima; Koralov, Leonid; Nolen, James Asymptotic behavior of branching diffusion processes in periodic media, Electronic Journal of Probability, Volume 25 (2020) no. none | DOI:10.1214/20-ejp527
  • Louidor, Oren; Saglietti, Santiago A Strong Law of Large Numbers for Super-Critical Branching Brownian Motion with Absorption, Journal of Statistical Physics, Volume 181 (2020) no. 4, p. 1112 | DOI:10.1007/s10955-020-02620-1
  • Öz, Mehmet Large deviations for local mass of branching Brownian motion, Latin American Journal of Probability and Mathematical Statistics, Volume 17 (2020) no. 2, p. 711 | DOI:10.30757/alea.v17-27
  • Wang, Li; Zong, Guowei Supercritical branching Brownian motion with catalytic branching at the origin, Science China Mathematics, Volume 63 (2020) no. 3, p. 595 | DOI:10.1007/s11425-017-9267-7
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  • Marguet, Aline A law of large numbers for branching Markov processes by the ergodicity of ancestral lineages, ESAIM: Probability and Statistics, Volume 23 (2019), p. 638 | DOI:10.1051/ps/2018029
  • Shiozawa, Yuichi Maximal displacement and population growth for branching Brownian motions, Illinois Journal of Mathematics, Volume 63 (2019) no. 3 | DOI:10.1215/00192082-7854864
  • Bertoin, Jean On a Feynman-Kac approach to growth-fragmentation semigroups and their asymptotic behaviors, Journal of Functional Analysis, Volume 277 (2019) no. 11, p. 108270 | DOI:10.1016/j.jfa.2019.06.012
  • Kouritzin, Michael A.; Lê, Khoa; Sezer, Deniz Laws of large numbers for supercritical branching Gaussian processes, Stochastic Processes and their Applications, Volume 129 (2019) no. 9, p. 3463 | DOI:10.1016/j.spa.2018.09.011
  • Shiozawa, Yuichi Spread Rate of Branching Brownian Motions, Acta Applicandae Mathematicae, Volume 155 (2018) no. 1, p. 113 | DOI:10.1007/s10440-017-0148-8
  • Wang, Li Central Limit Theorems for Supercritical Superprocesses with Immigration, Journal of Theoretical Probability, Volume 31 (2018) no. 2, p. 984 | DOI:10.1007/s10959-016-0736-y
  • Cloez, Bertrand Limit theorems for some branching measure-valued processes, Advances in Applied Probability, Volume 49 (2017) no. 2, p. 549 | DOI:10.1017/apr.2017.12
  • Chen, Zhen-Qing; Ren, Yan-Xia; Yang, Ting Law of Large Numbers for Branching Symmetric Hunt Processes with Measure-Valued Branching Rates, Journal of Theoretical Probability, Volume 30 (2017) no. 3, p. 898 | DOI:10.1007/s10959-016-0671-y
  • Getan, A.; Molchanov, S.; Vainberg, B. Intermittency for branching walks with heavy tails, Stochastics and Dynamics, Volume 17 (2017) no. 06, p. 1750044 | DOI:10.1142/s0219493717500447
  • Engländer, János; Zhang, Liang Branching diffusion with particle interactions, Electronic Journal of Probability, Volume 21 (2016) no. none | DOI:10.1214/16-ejp4782
  • Madaule, Thomas First order transition for the branching random walk at the critical parameter, Stochastic Processes and their Applications, Volume 126 (2016) no. 2, p. 470 | DOI:10.1016/j.spa.2015.09.008
  • Harris, S. C.; Hesse, M.; Kyprianou, A. E. Branching Brownian motion in a strip: Survival near criticality, The Annals of Probability, Volume 44 (2016) no. 1 | DOI:10.1214/14-aop972
  • Wang, Li Strong law of large number for branching hunt processes, Acta Mathematica Sinica, English Series, Volume 31 (2015) no. 7, p. 1189 | DOI:10.1007/s10114-015-3413-7
  • Adamczak, Radosław; Miłoś, Piotr CLT for Ornstein-Uhlenbeck branching particle system, Electronic Journal of Probability, Volume 20 (2015) no. none | DOI:10.1214/ejp.v20-4233
  • Chen, Zhen-Qing; Ren, Yan-Xia; Song, Renming; Zhang, Rui Strong law of large numbers for supercritical superprocesses under second moment condition, Frontiers of Mathematics in China, Volume 10 (2015) no. 4, p. 807 | DOI:10.1007/s11464-015-0482-y
  • Eckhoff, Maren; Kyprianou, Andreas E.; Winkel, Matthias Spines, skeletons and the strong law of large numbers for superdiffusions, The Annals of Probability, Volume 43 (2015) no. 5 | DOI:10.1214/14-aop944
  • Ren, Yan-Xia; Song, Renming; Zhang, Rui Central Limit Theorems for Super Ornstein-Uhlenbeck Processes, Acta Applicandae Mathematicae, Volume 130 (2014) no. 1, p. 9 | DOI:10.1007/s10440-013-9837-0
  • Bocharov, Sergey; Harris, Simon C. Branching Brownian Motion with Catalytic Branching at the Origin, Acta Applicandae Mathematicae, Volume 134 (2014) no. 1, p. 201 | DOI:10.1007/s10440-014-9879-y
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  • Ren, Yan-Xia; Song, Renming; Zhang, Rui Central limit theorems for supercritical branching Markov processes, Journal of Functional Analysis, Volume 266 (2014) no. 3, p. 1716 | DOI:10.1016/j.jfa.2013.10.015
  • Adamczak, Radosław; Miłoś, Piotr U U -Statistics of Ornstein–Uhlenbeck Branching Particle System, Journal of Theoretical Probability, Volume 27 (2014) no. 4, p. 1071 | DOI:10.1007/s10959-013-0503-2
  • Kouritzin, Michael A.; Ren, Yan-Xia A strong law of large numbers for super-stable processes, Stochastic Processes and their Applications, Volume 124 (2014) no. 1, p. 505 | DOI:10.1016/j.spa.2013.08.009
  • Liu, Rong-Li; Ren, Yan-Xia; Song, Renming Strong Law of Large Numbers for a Class of Superdiffusions, Acta Applicandae Mathematicae, Volume 123 (2013) no. 1, p. 73 | DOI:10.1007/s10440-012-9715-1
  • Koralov, L.; Molchanov, S. Structure of Population Inside Propagating Front, Journal of Mathematical Sciences, Volume 189 (2013) no. 4, p. 637 | DOI:10.1007/s10958-013-1212-1
  • Grummt, Robert; Kolb, Martin Law of large numbers for super-Brownian motions with a single point source, Stochastic Processes and their Applications, Volume 123 (2013) no. 4, p. 1183 | DOI:10.1016/j.spa.2012.12.002
  • Harris, S. C.; Knobloch, R.; Kyprianou, A. E. Strong law of large numbers for fragmentation processes, Annales de l'Institut Henri Poincaré, Probabilités et Statistiques, Volume 46 (2010) no. 1 | DOI:10.1214/09-aihp311
  • Englander, Janos The Center of Mass for Spatial Branching Processes and an Application for Self-Interaction, Electronic Journal of Probability, Volume 15 (2010) no. none | DOI:10.1214/ejp.v15-822
  • Hardy, Robert; Harris, Simon C. A Spine Approach to Branching Diffusions with Applications to L p -Convergence of Martingales, Séminaire de Probabilités XLII, Volume 1979 (2009), p. 281 | DOI:10.1007/978-3-642-01763-6_11

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