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arXiv · 0907.4557

Dynamical sensitivity of recurrence and transience of branching random walks

Sebastian Müller
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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.

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Sebastian Müller. 2009-12-07. Dynamical sensitivity of recurrence and transience of branching random walks. https://arxiv.org/abs/0907.4557

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