Dynamical sensitivity of recurrence and transience of branching random walks

2009/07/27 by Sebastian Müller, Müller, Sebastian
Biochemistry, Genetics and Molecular Biology · Mathematics · #60J25 #60J80 #Diffusion and Search Dynamics #FOS: Mathematics #Mathematical Dynamics and Fractals #Probability (math.PR) #Stochastic processes and statistical mechanics #math.PR #msc:60J25 #msc:60J80

paper · pdf · doi:10.48550/arxiv.0907.4557

v2: proofs and presentation revised, corrected the proof for dynamical stability of transience, behaviour in critical case is left open in general but covered for a certain class of Cayley graphs

openalex publication_date 2009/07/27 · arxiv created 2009/12/07 · arxiv updated 2009/12/08 · openalex created_date 2016/06/24 · openalex updated_date 2026/07/31

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 exceptional times we say that the property is dynamically sensitive, otherwise we call it 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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