Dual Constructions for Pure-Jump Markov Processes

#### M. Bebbington, P. Pollett, X. Zheng

1995, v.1, №4, 513-558

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.

Keywords: Markov processes,q-processes,invariant measures