The critical Branching Markov Chain is transient

• DOI: 10.48550/arxiv.math/0510556
• Publication Date: 2005-01-01
• Authors: Nina Gantert; Sebastian Mueller
• Keywords: Irreducibility; Branching (polymer chemistry); Constant (computer programming)

We investigate recurrence and transience of Branching Markov Chains (BMC) in discrete time. are clouds particles which move (according to an irreducible underlying Chain) produce offspring independently. The distribution can depend on the location particle. If is constant for all locations, these Tree-Indexed chains sense \cite{benjamini94}. Starting with one particle at $x$, we denote by $α(x)$ probability that $x$ visited infinitely often cloud. Due irreducibility Chain, there three regimes: either $α(x) = 0$ (transient regime), or $0 < α(x) 1$ (weakly recurrent regime) (strongly regime). give classification results, including a sufficient condition general case. mean constant, criterion involving spectral radius Chain distribution.