Excursions of dual processes

There has been considerable interest in the excursions of a Markov process in recent years. Perhaps the most important general result is that of Maisonneuve [13], in which the existence of an "exit system" is established. This general result was applied to the study of specific excursions by Getoor in [5]. Then Maisonneuve recast and generalized these results in [14] where he showed how to discuss the excursion straddling a certain class of stopping times T. These included the excursion straddling a fixed time and the first excursion whose length exceeds a given value which were discussed in [5]. The purpose of this paper is to study excursions for processes which satisfy duality conditions. A start in this direction was made in Section 10 of [5] under assumptions that were much too strong. It turns out that the appropriate duality hypotheses are stronger than those of "classical" duality as described in [I]. We assume the existence of dual transition densities rather than the weaker assumption of dual potential kernel densities. Our precise assumptions are described in Section 2 and especially in Section 3. In particular it is shown in these sections that a systematic use of space-time processes obviates the need for any regularity assumptions on the density over and above the Chapman-Kolmogorov equation. Section 4 summarizes the special properties of additive functionals under our hypotheses that are needed in later sections. In Theorem 4.5 we show that if M is a closed homogeneous optional set, then, under duality assumptions, there exists a Borel set F c E × E such that Me3 ]0, ~[ and {t > 0: (Xt_, Xt) ~ F} are indistinguishable. In Section 5 we begin the discussion of excursions from a closed homogeneous optional set M with R i n f { t > 0 : t C M } . If F ~ E × E corresponds to M as above, R----inf{t: (Xt_,Art) ~ F } almost surely. Especially important is (5.17) which gives the joint distribution of R, X R, and X R . This is obtained by applying a reversal argument to the excursion