Excursion Measure Away from an Exit Boundary of One-Dimensional Diffusion Processes

Abstract

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 \[8] 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.