Additive functionals and excursions of Kuznetsov processes

Abstract

Let B be a continuous additive functional for a standard process and let be a stationary Kuznetsov process with the same semigroup of transition. In this paper, we give the excursion laws of conditioned on the strict past and future without duality hypothesis. We study excursions of a general regenerative system and of a regenerative system consisting of the closure of the set of times the regular points of B are visited. In both cases, those conditioned excursion laws depend only on two points and Xd, where ]g, d[ is an excursion interval of the regenerative set M. We use the ‐predictable exit system to bring together the isolated points of M and its perfect part and replace the classical optional exit system. This has been a subject in literature before (e.g., Kaspi (1988)) under the classical duality hypothesis. We define an “additive functional” for with B, we generalize the laws cited before to , and we express laws of pairs of excursions.