STABILITY OF MARKOVIAN PROCESSES .3. FOSTER-LYAPUNOV CRITERIA FOR CONTINUOUS-TIME PROCESSES

In Part I we developed stability concepts for discrete chains, togethe r with Foster-Lyapunov criteria for them to hold. Part II was devoted to developing related stability concepts for continuous-time processes . In this paper we develop criteria for these forms of stability for c ontinuous-parameter Markovian processes on general state spaces, based on Foster-Lyapunov inequalities for the extended generator. Such test function criteria are found for non-explosivity, non-evanescence, Har ris recurrence, and positive Harris recurrence. These results are prov ed by systematic application of Dynkin's formula. We also strengthen k nown ergodic theorems, and especially exponential ergodic results, for continuous-time processes. In particular we are able to show that the test function approach provides a criterion for f-norm convergence, a nd bounding constants for such convergence in the exponential ergodic case. We apply the criteria to several specific processes, including l inear stochastic systems under non-linear feedback, work-modulated que ues, general release storage processes and risk processes.