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 (mu) over bar and (nu) over bar 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.