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 $e_λ(du)=e^{-λu}du$, where $λ\in\R$ is chosen in such a way that the overall intensity of particles is preserved. Denote by $χ$ the cluster distribution and let $ϕ$ be the log-Laplace transform of the intensity of $χ$. If $λϕ'(λ)>0$, 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 $Π_{e_λ}^χ$ with intensity $e_λ$. If $λϕ'(λ)<0$, 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 $χ$ as mixtures of the point processes $Π_{ce_λ}^χ$ over $c>0$ and $λ\in K_{\text{st}}$, where $K_{\text{st}}=\{λ\in\R: ϕ(λ)=0, λϕ'(λ)>0\}$.

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