## Countable state Markov processes: non-explosiveness and moment function.(English)Zbl 1379.60088

Summary: 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. [W. J. Anderson, Continuous-time Markov chains. An applications-oriented approach. New York etc.: Springer-Verlag (1991; Zbl 0731.60067); S. P. Meyn and R. L. Tweedie, Adv. Appl. Probab. 25, No. 3, 518–548 (1993; Zbl 0781.60053)]), 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 $$\alpha$$-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 positive recurrence of the $$\alpha$$-jump chain implies that all bounded functions satisfy the Kolmogorov integral relation. We present a drift function criterion characterizing positive recurrence of this $$\alpha$$-jump chain. Suppose that to a drift function $$V$$ there corresponds another drift function $$W$$, which is a moment with respect to $$V$$. 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 characterize the $$V$$-bounded functions that satisfy the Kolmogorov forward equation.

### MSC:

 60J27 Continuous-time Markov processes on discrete state spaces

### Citations:

Zbl 0731.60067; Zbl 0781.60053
Full Text:

### References:

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