The spread of a catalytic branching random walk

Carmona, Philippe; Hu, Yueyun

Annales de l'I.H.P. Probabilités et statistiques, Tome 50 (2014) no. 2, pp. 327-351.

Nous considérons une marche aléatoire branchant catalytique sur $\mathbb{Z}$ qui ne branche qu’à l’origine. Dans le cas surcritique, nous établissons une loi des grands nombres pour la position maximale $M_n$ : Il existe une constante $\alpha$ explicite telle que $\frac{M_n}{n}\to \alpha$ presque sûrement sur l’ensemble des trajectoires pour lesquelles l’origine est visitée une infinité de fois. Ensuite, nous déterminons toutes les lois limites possibles, lorsque $n\to +\infty$, pour la suite $M_n-\alpha n$.

We consider a catalytic branching random walk on $\mathbb{Z}$ that branches at the origin only. In the supercritical regime we establish a law of large number for the maximal position $M_n$: For some constant $\alpha$, $\frac{M_n}{n}\to \alpha$ almost surely on the set of infinite number of visits of the origin. Then we determine all possible limiting laws for $M_n-\alpha n$ as $n$ goes to infinity.

Classification : 60K37
Mots clés : branching processes, catalytic branching random walk

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