Branching random walks in random environment and super-brownian motion in random environment

Nakashima, Makoto

Annales de l'I.H.P. Probabilités et statistiques, Tome 51 (2015) no. 4, pp. 1251-1289.

Nous étudions l’existence et la caractérisation de la limite de marches branchantes critiques dans un environnement spatio-temporel aléatoire en dimension 1 introduit par Birkner, Geiger and Kersting dans (In Interacting Stochastic Systems (2005) 269–291 Springer). Chaque particule effectue une marche aléatoire simple sur $\mathbb{Z}$ et le mécanisme de branchement dépend du site indexé par l’espace et le temps. La limite de ce processus à valeur mesure est caractérisée comme l’unique solution d’un problème de martingale non-trivial et correspond au super mouvement Brownien en environnement aléatoire par Mytnik dans (Ann. Probab. 24 (1996) 1953–1978).

We focus on the existence and characterization of the limit for a certain critical branching random walks in time–space random environment in one dimension which was introduced by Birkner, Geiger and Kersting in (In Interacting Stochastic Systems (2005) 269–291 Springer). Each particle performs simple random walk on $\mathbb{Z}$ and branching mechanism depends on the time–space site. The limit of this measure-valued processes is characterized as the unique solution to the non-trivial martingale problem and called super-Brownian motion in a random environment by Mytnik in (Ann. Probab. 24 (1996) 1953–1978).

Mots clés : superprocesses in a random environment, branching random walks in a random environment, stochastic heat equations, uniqueness

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