Dual constructions for pure-jump Markov processes

Mark Bebbington, Phil Pollett and Xiaogu Zheng

Abstract: In this paper we shall present a systematic treatment of the construction of the dual of a pure-jump Markov process on a general state space, specifically an inner-regular, measurable topological space with a countable basis. Firstly, we prove the existence of a dual q-pair with respect to an invariant or subinvariant measure and then study its structure, highlighting those properties which are peculiar to the uncountable-state setting. Next, we deal with the construction of dual transition functions, giving particular attention to the minimal process. This enables us to prove several results on invariant measures for pure-jump processes which are analogous to those for Markov chains. Finally, we shall explain how our results can be used in understanding the relationship between invariant and subinvariant measures for a transition function and those of the associated q-pair. Various examples will be used to illustrate our results, and an application given for a stochastic version of a slip-predictable model for earthquake occurrences.

AMS 1991 Subject Classification: 60J75.

Keywords: Markov processes; q-processes; invariant measures.

Acknowledgement: This worked was funded by the Australian Research Council, the University of Queensland and Victoria University of Wellington.

The authors:

• Mark Bebbington, Department of Statistics, School of Mathematical Information Sciences, Massey University. M.Bebbington@massey.ac.nz
• Phil Pollett, Discipline of Mathematics, The University of Queensland. pkp@maths.uq.edu.au
• Xiaogu Zheng, National Institute of Water and Atmospheric Research, Crown Research Institute, Wellington, NEW ZEALAND. zheng@award1.kelburn.cri.nz

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Last modified: 26th December 1995