International Journal of Mathematics and Mathematical Sciences
Volume 2005 (2005), Issue 13, Pages 2031-2040

Additive functionals and excursions of Kuznetsov processes

Hacène Boutabia

Département de Mathématiques, Faculté des Sciences, Université Badji Mokhtar, BP 12, Annaba 23000, Algeria

Received 16 September 2004; Revised 28 June 2005

Copyright © 2005 Hacène Boutabia. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.


Let B be a continuous additive functional for a standard process (Xt)t+ and let (Yt)t be a stationary Kuznetsov process with the same semigroup of transition. In this paper, we give the excursion laws of (Xt)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 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 with B, we generalize the laws cited before to (Yt)t, and we express laws of pairs of excursions.