Stability of Markovian Processes III: Foster-Lyapunov Criteria for Continuous-Time Processes

In Part I we developed stability concepts for discrete chains, together 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 continuous-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, Harris recurrence, and positive Harris recurrence. These results are proved by systematic application of Dynkin's formula. We also strengthen known 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, and bounding constants for such convergence in the exponential ergodic case. We apply the criteria to several specific processes, including linear stochastic systems under non-linear feedback, work-modulated queues, general release storage processes and risk processes. FOSTER'S CRITERION; IRREDUCIBLE MARKOV PROCESSES; STOCHASTIC LYAPUNOV FUNCTIONS; ERGODICITY; EXPONENTIAL ERGODICITY; RECURRENCE; STORAGE MODELS; RISK MODELS; OUEUES; HYPOELLIPTIC DIFFUSION AMS 1991 SUBJECT CLASSIFICATION: PRIMARY 60J10