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

Abstract

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 μ and ν are supercritical offspring distributions with means μ¯<ν¯ then every space-time set that is recurrent with respect to the offspring distribution μ is also recurrent with respect to the offspring distribution ν and similarly that every space-time set that is transient with respect to the offspring distribution ν is also transient with respect to the offspring distribution μ. 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 μ is smaller than ν 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.