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

TL;DR

This paper investigates how the recurrence and transience of space-time sets in branching random walks depend on offspring distributions, introducing a new stochastic order called the germ order to establish monotonicity results.

Contribution

It introduces the germ order for probability measures and proves that recurrence and transience properties are monotone with respect to this order for branching random walks.

Findings

Recurrence with respect to a smaller offspring distribution implies recurrence for larger distributions.

Transience with respect to a larger offspring distribution implies transience for smaller distributions.

The germ order generalizes previous stochastic orders used in related models.

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 $\mu$ 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.