The reproduction speed of a continuous-time branching random walk is proportional to a positive parameter λ. There is a threshold for λ, which is called λw, that separates almost sure global extinction from global survival. Analogously, there exists another threshold λs below which any site is visited almost surely a finite number of times (i.e. local extinction) while above it there is a positive probability of visiting every site infinitely many times. The local critical parameter λs is completely understood and can be computed as a function of the reproduction rates. On the other hand, only for some classes of branching random walks it is known that the global critical parameter λw is the inverse of a certain function of the reproduction rates, which we denote by Kw. We provide here new sufficient conditions which guarantee that the global critical parameter equals 1/Kw. This result extends previously known results for branching random walks on multigraphs and general branching random walks. We show that these sufficient conditions are satisfied by periodic tree-like branching random walks. We also discuss the critical parameter and the critical behaviour of continuous-time branching processes in varying environment. So far, only examples where λw = 1/Kw were known; here we provide an example where λw > 1/Kw.
Bertacchi, D., Coletti, C., & Zucca, F. (2017). Global survival of branching random walks and tree-like branching random walks. ALEA, 14(1), 381-402 [10.30757/alea.v14-21].
|Citazione:||Bertacchi, D., Coletti, C., & Zucca, F. (2017). Global survival of branching random walks and tree-like branching random walks. ALEA, 14(1), 381-402 [10.30757/alea.v14-21].|
|Tipo:||Articolo in rivista - Articolo scientifico|
|Carattere della pubblicazione:||Scientifica|
|Presenza di un coautore afferente ad Istituzioni straniere:||Si|
|Titolo:||Global survival of branching random walks and tree-like branching random walks|
|Autori:||Bertacchi, D; Coletti, C; Zucca, F|
|Data di pubblicazione:||2017|
|Digital Object Identifier (DOI):||http://dx.doi.org/10.30757/alea.v14-21|
|Appare nelle tipologie:||01 - Articolo su rivista|