Existence and uniqueness of an invariant probability for a class of Feller Markov chains

We consider the class of Feller Markov chains on a phase spaceX whose kernels mapC₀(X), the space of bounded continuous functions that vanish at infinity, into itself. We provide a necessaryand sufficient condition for the existence of an invariant probability measure using a generalized Farkas Lemma. This condition is a Lyapunov type criterion that can be checked in practice. We also provide a necessaryand sufficient condition for existence of aunique invariant probability measure. When the spaceX is compact, the conditions simplify.