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) 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.”

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