(Reference retrieved automatically from Web of Science through information on FAPESP grant and its corresponding number as mentioned in the publication by the authors.)
Global survival of branching random walks and tree-like branching random walks
Coletti, Cristian F.
Total Authors: 3
ALEA-LATIN AMERICAN JOURNAL OF PROBABILITY AND MATHEMATICAL STATISTICS;
Web of Science Citations:
The reproduction speed of a continuous-time branching random walk is proportional to a positive parameter lambda. There is a threshold for lambda, which is called lambda(w), that separates almost sure global extinction from global survival. Analogously, there exists another threshold lambda(s) 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 lambda(s) 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 lambda(w) is the inverse of a certain function of the reproduction rates, which we denote by K-w. We provide here new sufficient conditions which guarantee that the global critical parameter equals 1/K-w. 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 lambda(w) = 1/K-w were known; here we provide an example where lambda(w) > 1/K-w. (AU)