Additive functionals and excursions of Kuznetsov processes

Let B be a continuous additive functional for a standard process ( X t ) t ∈ ℠+ and let ( Y t ) t ∈ ℠be a stationary Kuznetsov process with the same semigroup of transition. In this paper, we give the excursion laws of ( X t ) t ∈ ℠+ 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 X g − and X d , where ] g , d [ is an excursion interval of the regenerative set M . We use the ( F D t ) -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 ( Y t ) t ∈ ℠with B , we generalize the laws cited before to ( Y t ) t ∈ ℠, and we express laws of pairs of excursions.