A description of the long-term behaviour of absorbing continuous-time Markov chains using a centre manifold

We use the notion of an invariant manifold to describe the long-term behaviour of absorbing continuous-time Markov processes with a denumerable infinity of states. We show that there exists an invariant manifold for the forward differential equations and we are able to describe the evolution of the state probabilities on this manifold. Our approach gives rise to a new method for calculating conditional limiting distributions, one which is also appropriate for dealing with processes whose transition probabilities satisfy a system of non-linear differential equations.