Transience and recurrence of sets for branching random walk via non-standard stochastic orders

We study how the recurrence and transience of space-time sets for a branching random walk on a graph depends on the offspring distribution. Here, we say that a space-time set $A$ is recurrent if it is visited infinitely often almost surely on the event that the branching random walk survives forever, and say that $A$ is transient if it is visited at most finitely often almost surely. We prove that if $\mu$ and $\nu$ are supercritical offspring distributions with means $\bar \mu < \bar \nu$ then every space-time set that is recurrent with respect to the offspring distribution $\mu$ is also recurrent with respect to the offspring distribution $\nu$ and similarly that every space-time set that is transient with respect to the offspring distribution $\nu$ is also transient with respect to the offspring distribution $\mu$. To prove this, we introduce a new order on probability measures that we call the germ order and prove more generally that the same result holds whenever $\mu$ is smaller than $\nu$ in the germ order. Our work is inspired by the work of Johnson and Junge (AIHP 2018), who used related stochastic orders to study the frog model.