Quenched law of large numbers for branching brownian motion in a random medium

Engländer, János

Annales de l'I.H.P. Probabilités et statistiques, Tome 44 (2008) no. 3, pp. 490-518.

Nous étudions un modèle de branchement spatial où le mouvement de base est Brownien $d$-dimensionnel ($d\ge 1$) et le taux de branchement est modifié par une collection aléatoire d’ensembles sur lesquels la reproduction n’a pas lieu (obstacles moux). Le résultat principal de cet article est l’asymptotique (en probabilité) des taux de croissance globaux «quenchés» pour tout $d\ge 1$, et nous identifions les termes de correction sous-exponentielle. Nous montrons aussi que le branchement Brownien avec obstacles moux diffuse moins vite que le branchement Brownien classique en donnant une borne supérieure de sa vitesse. Dans le cas où le mouvement de base est un processus de diffusion arbitraire nous obtenons une dichotomie pour la croissance locale «quenchée» qui est indépendante de l'intensité Poissonnienne. Le cas de distributions plus générales du nombre de descendants (autre que le cas dyadique considéré dans les théorème principaux), ainsi que des modèles d'obstacles moux pour des superprocessus, sont aussi discutés.

We study a spatial branching model, where the underlying motion is $d$-dimensional ($d\ge 1$) brownian motion and the branching rate is affected by a random collection of reproduction suppressing sets dubbed mild obstacles. The main result of this paper is the quenched law of large numbers for the population for all $d\ge 1$. We also show that the branching brownian motion with mild obstacles spreads less quickly than ordinary branching brownian motion by giving an upper estimate on its speed. When the underlying motion is an arbitrary diffusion process, we obtain a dichotomy for the quenched local growth that is independent of the poissonian intensity. More general offspring distributions (beyond the dyadic one considered in the main theorems) as well as mild obstacle models for superprocesses are also discussed.

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