The Poisson Equation for Countable Markov Chains: Probabilistic Methods and Interpretations

Abstract

This paper considers the Poisson equation associated with time-homogeneous Markov chains on a countable state space. The discussion emphasizes probabilistic arguments and focuses on three separate issues, namely (i) the existence and uniqueness of solutions to the Poisson equation, (ii) growth estimates and bounds on these solutions and (iii) their parametric dependence. Answers to these questions are obtained under a variety of recurrence conditions.