Global survival of branching random walks and tree-like branching random walks

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 $\lambda_w>1/K_w$.