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

Sean P. Meyn and R. L. Tweedie
Advances in Applied Probability
Vol. 25, No. 3 (Sep., 1993), pp. 518-548
DOI: 10.2307/1427522
Stable URL: http://www.jstor.org/stable/1427522
Page Count: 31
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Stability of Markovian Processes III: Foster-Lyapunov Criteria for Continuous-Time Processes
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Abstract

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.

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