The Critical Branching Markov Chain Is Transient

N. Gantert, Sebastian Mueller

2006, v.12, Issue 4, 805-814

ABSTRACT

We study recurrence and transience of Branching Markov Chains (BMC) in discrete time. Branching Markov Chains are clouds of particles which move (according to an irreducible underlying Markov chain) and produce offspring independently. The offspring distribution can depend on the location of the particle. If the offspring distribution is constant for all locations, these are tree-indexed Markov chains in the sense of [I. Benjamini and Y. Peres, Markov chains indexed by trees. Ann. Prob., 1994, v.22, N1, 219-243]. Starting with one particle at location $x$, we denote by $\alpha(x)$ the probability that $x$ is visited infinitely often by the cloud. Due to the irreducibility of the underlying Markov chain, there are three regimes: either $\alpha(x) = 0$ for all $x$ (transient regime), or $0 < \alpha(x) < 1$ for all $x$ (weakly recurrent regime) or $\alpha(x) = 1$ for all $x$ (strongly recurrent regime). We give classification results, including a sufficient condition for transience in the general case. If the mean of the offspring distribution is constant, we give a criterion for transience involving the spectral radius of the underlying Markov chain and the mean of the offspring distribution. In particular, the critical BMC is transient. Examples for the classification are provided.

Keywords: Branching Markov Chains,recurrence and transience,Lyapunov function,spectral radius

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