Persistence of competing systems of branching random walks

We consider a system of independent branching random walks on $\R$ which start off a Poisson point process with intensity of the form $e_{\lambda}(du)=e^{-\lambda u}du$, where $\lambda\in\R$ is chosen in such a way that the overall intensity of particles is preserved. Denote by $\chi$ the cluster distribution and let $\phi$ be the log-Laplace transform of the intensity of $\chi$. If $\lambda\phi'(\lambda)>0$, we show that the system is persistent (stable) meaning that the point process formed by the particles in the $n$-th generation converges as $n\to\infty$ to a non-trivial point process $\Pi_{e_{\lambda}}^{\chi}$ with intensity $e_{\lambda}$. If $\lambda\phi'(\lambda)<0$, then the branching population suffers local extinction meaning that the limiting point process is empty. We characterize (generally, non-stationary) point processes on $\R$ which are cluster-invariant with respect to the cluster distribution $\chi$ as mixtures of the point processes $\Pi_{ce_{\lambda}}^{\chi}$ over $c>0$ and $\lambda\in K_{\text{st}}$, where $K_{\text{st}}=\{\lambda\in\R: \phi(\lambda)=0, \lambda\phi'(\lambda)>0\}$.