A generalization of the excursion measure away from an exit boundary is deﬁned for a one-dimensional diffusion process. It is constructed through the disintegration formula with respect to the lifetime. The counterpart of the Williams description, the disintegration formula with respect to the maximum, is also established. This generalized excursion measure is applied to explain and generalize the convergence theorem of Kasahara and Watanabe  in terms of the Poisson point ﬁelds, where the inverse local time processes of regular diﬀusion processes converge in the sense of probability law to some Lévy process, which is closely related to a diffusion process with an exit boundary.
Cite this article
Kouji Yano, Excursion Measure Away from an Exit Boundary of One-Dimensional Diffusion Processes. Publ. Res. Inst. Math. Sci. 42 (2006), no. 3, pp. 837–878DOI 10.2977/PRIMS/1166642162