Law of large numbers for superdiffusions : the non-ergodic case
Annales de l'I.H.P. Probabilités et statistiques, Volume 45 (2009) no. 1, p. 1-6

In previous work of D. Turaev, A. Winter and the author, the Law of Large Numbers for the local mass of certain superdiffusions was proved under an ergodicity assumption. In this paper we go beyond ergodicity, that is we consider cases when the scaling for the expectation of the local mass is not purely exponential. Inter alia, we prove the analog of the Watanabe-Biggins LLN for super-brownian motion.

Dans un travail précédent, l'auteur, D. Turaev et A. Winter, ont prouvé la Loi des Grand Nombres pour la masse locale de certaines diffusions sous une hypothèse d'ergodicité. Dans cet article nous allons au delà de l'ergodicité, plus précisement nous considérons des cas où le scaling de l'espérance de la masse locale n'est pas purement exponentiel. Entre autres, nous prouvons l'analogue de la LGN de Watanabe-Biggins pour le super mouvement brownien.

DOI : https://doi.org/10.1214/07-AIHP156
Classification:  60J60,  60J80
Keywords: super-brownian motion, superdiffusion, superprocess, law of large numbers, H-transform, weighted superprocess, scaling limit, local extinction
@article{AIHPB_2009__45_1_1_0,
     author = {Engl\"ander, J\'anos},
     title = {Law of large numbers for superdiffusions : the non-ergodic case},
     journal = {Annales de l'I.H.P. Probabilit\'es et statistiques},
     publisher = {Gauthier-Villars},
     volume = {45},
     number = {1},
     year = {2009},
     pages = {1-6},
     doi = {10.1214/07-AIHP156},
     zbl = {1172.60022},
     mrnumber = {2500226},
     language = {en},
     url = {http://www.numdam.org/item/AIHPB_2009__45_1_1_0}
}
Engländer, János. Law of large numbers for superdiffusions : the non-ergodic case. Annales de l'I.H.P. Probabilités et statistiques, Volume 45 (2009) no. 1, pp. 1-6. doi : 10.1214/07-AIHP156. http://www.numdam.org/item/AIHPB_2009__45_1_1_0/

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

[2] J. Engländer and R. G. 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

[3] J. Engländer and R. G. Pinsky. The compact support property for measure-valued processes. Ann. Inst. H. Poincaré Probab. Statist. 42 (2006) 535-552. | Numdam | MR 2259973 | Zbl 1104.60049

[4] 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

[5] J. Engländer and A. Winter. Law of large numbers for a class of superdiffusions. Ann. Inst. H. Poincaré Probab. Statist. 42 (2006) 171-185. | Numdam | MR 2199796 | Zbl 1093.60058

[6] R. G. Pinsky. On the large time growth rate of the support of supercritical super-Brownian motion. Ann. Probab. 23 (1995) 1748-1754. | MR 1379166 | Zbl 0852.60094

[7] S. Watanabe. Limit theorems for a class of branching processes. In Markov Processes and Potential Theory 205-232. J. Chover, Ed. Wiley, New York, 1967. | MR 237007 | Zbl 0253.60072