In this paper we consider $\psi$-irreducible Markov processes evolving in discrete or continuous time on a general state space. We develop a Liapounov function criterion that permits one to obtain explicit bounds on the solution to the Poisson equation and, in particular, obtain conditions under which the solution is square integrable.
These results are applied to obtain sufficient conditions that guarantee the validity of a functional central limit theorem for the Markov process. As a second consequence of the bounds obtained, a perturbation theory for Markov processes is developed which gives conditions under which both the solution to the Poisson equation and the invariant probability for the process are continuous functions of its transition kernel. The techniques are illustrated with applications to queueing theory and autoregressive processes.
"A Liapounov bound for solutions of the Poisson equation." Ann. Probab. 24 (2) 916 - 931, April 1996. https://doi.org/10.1214/aop/1039639370