International Journal of Mathematics and Mathematical Sciences
Volume 2005 (2005), Issue 13, Pages 2031-2040
doi:10.1155/IJMMS.2005.2031

Additive functionals and excursions of Kuznetsov processes

Abstract

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