Kolmogorov forward equation and explosiveness in countable state Markov processes

In countable state non-explosive minimal Markov processes the Kolmogorov forward equations hold under sufficiently weak conditions. However, a precise description of the functions that one may integrate with respect to these equations seems to be absent in the literature. This problem arises for instance when studying the Poisson equation, as well as the average cost optimality equation in a Markov decision process. We will show that the class of non-negative functions for which an associated transformed Markov process is non-explosive do have this desirable property. This characterisation easily allows to construct counter-examples of functions for which the functional form of the Kolmogorov forward equations does not hold. Another approach of the problem is to study the transition operator as a transition semi-group on Banach space. The domain of the generator is a collection of functions that can be integrated with respect to the Kolmogorov forward equations. We focus on Banach spaces equipped with a weighted supremum norm, and we identify subsets of the domain of the generator.