Dynamical sensitivity of recurrence and transience of branching random walks

TL;DR

This paper investigates how recurrence and transience properties of branching random walks on Cayley graphs behave under dynamic conditions, proving stability in certain regimes and leaving some cases open.

Contribution

It establishes the dynamical stability of recurrence and transience for branching random walks in sub- and supercritical regimes on Cayley graphs, using novel combined techniques.

Findings

Recurrence and transience are dynamically stable in sub- and supercritical regimes.

Dynamical stability is proven for a specific class of Cayley graphs.

Critical case stability remains an open problem.

Abstract

Consider a sequence of i.i.d. random variables $X_n$ where each random variable is refreshed independently according to a Poisson clock. At any fixed time $t$ the law of the sequence is the same as for the sequence at time 0 but at random times almost sure properties of the sequence may be violated. If there are such \emph{exceptional times} we say that the property is \emph{dynamically sensitive}, otherwise we call it \emph{dynamically stable}. In this note we consider branching random walks on Cayley graphs and prove that recurrence and transience are dynamically stable in the sub-and supercritical regime. While the critical case is left open in general we prove dynamical stability for a specific class of Cayley graphs. Our proof combines techniques from the theory of ranching random walks with those of dynamical percolation.