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
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.
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.