The critical Branching Markov Chain is transient

We investigate 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 \cite{benjamini94}. 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.