Countable state Markov processes: non-explosiveness and moment function

The existence of a moment function satisfying a drift function condition is well-known to guarantee non-explosiveness of the associated minimal Markov process (cf.[1, 6]), under standard technical conditions. Surprisingly, the reverse is true as well for a countable space Markov process. We prove this result by showing that recurrence of an associated jump process, that we call the α-jump process, is equivalent to non-explosiveness. Non-explosiveness corresponds in a natural way to the validity of the Kolmogorov integral relation for the function identically equal to 1. In particular, we show that the α-jump chain is positive recurrent, all bounded functions satisfy the Kolmogorov integral relation. Positive recurrence can be characterised by a drift function criterion as well. If to a drift function V , there corresponds another drift function W , which is a moment with respect to V , then via a transformation argument, the above relations hold for the transformed process with respect to V . Transferring the results back to the original process, allows to characterise which V -bounded functions satisfy the Kolmogorov forward equation.