  This paper is concerned with ergodic properties of inhomogeneous Markov
processes. Since the transition probabilities depend on initial times, the
existing methods to obtain invariant measures for homogeneous Markov processes
are not applicable straightforwardly. We impose some appropriate conditions
under which invariant measure families for inhomogeneous Markov processes can
be studied. Specifically, the existence of invariant measure families is
established by either a generalization of the classical Krylov-Bogolyubov
method or a Lyapunov criterion. Moreover, the uniqueness and exponential
ergodicity are demonstrated under a contraction assumption of the transition
probabilities on a large set. Finally, three examples, including Markov chains,
diffusion processes and storage processes, are analyzed to illustrate the
practicality of our method.
