Two Results on Dual Excursions

Abstract

This paper is a sequel to the recent work [5]. As in that paper, it is supposed for the two principal results that one is given a pair X,$\widehat{X}$ of standard Markov processes on a common state space E having a dual density relative to some σ-finite measure on E. This condition is considerably stronger than the usual duality of resolvents. In fact, it was shown in [5] that duality of densities is equivalent to classical duality of certain space-time processes associated with X,$\widehat{X}$. The main result of [5] concerned the construction and properties of families of measures P ^x,ℓ,y which were shown to govern the distribution of excursions of X from a given closed homogeneous optional set M, conditional on the excursion starting at x, ending at y and having length ℓ. The precise meaning of “govern” in the statement above was laid out in two situations in [5]. One considers the excursion straddling a stopping time T. In the first case, one considers the stopping times defined by Maisonneuve [8]. See §3 for a formal description. Roughly speaking, such stopping times correspond to rules where the decision to stop during an excursion may be based only on information available before the excursion began, and on the age process of the excursion.