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

## TL;DR

This paper investigates the thresholds for global and local survival in continuous-time branching random walks, providing new conditions under which the global critical parameter equals a known inverse function, with applications to tree-like structures.

## Contribution

The paper introduces new sufficient conditions ensuring the global critical parameter equals the inverse of a reproduction rate function, extending previous results to more complex structures.

## Key findings

- Sufficient conditions for $oldsymbol{ ext{global critical parameter} = 1/K_w}$
- Application of conditions to periodic tree-like branching random walks
- Example where $oldsymbol{ ext{global critical parameter} > 1/K_w}$

## Abstract

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 $\lambda_w>1/K_w$.