STABILITY OF MARKOVIAN PROCESSES .2. CONTINUOUS-TIME PROCESSES AND SAMPLED CHAINS

In this paper we extend the results of Meyn and Tweedie (1992b) from d iscrete-time parameter to continuous-parameter Markovian processes PHI evolving on a topological space. We consider a number of stability co ncepts for such processes in terms of the topology of the space, and p rove connections between these and standard probabilistic recurrence c oncepts. We show that these structural results hold for a major class of processes (processes with continuous components) in a manner analog ous to discrete-time results, and that complex operations research mod els such as storage models with state-dependent release rules, or diff usion models such as those with hypoelliptic generators, have this pro perty. Also analogous to discrete time, 'petite sets', which are known to provide test sets for stability, arc here also shown to provide co nditions for continuous components to exist. New ergodic theorems for processes with irreducible and countably reducible skeleton chains are derived, and we show that when these conditions do not hold, then the process may be decomposed into an uncountable orbit of skeleton chain s.