  Point processes often have a natural interpretation with respect to a
continuous process. We propose a point process construction that describes
arrival time observations in terms of the state of a latent diffusion process.
In this framework, we relate the return times of a diffusion in a continuous
path space to new arrivals of the point process. This leads to a continuous
sample path that is used to describe the underlying mechanism generating the
arrival distribution. These models arise in many disciplines, such as financial
settings where actions in a market are determined by a hidden continuous price
or in neuroscience where a latent stimulus generates spike trains. Based on the
developments in It\^o's excursion theory, we propose methods for inferring and
sampling from the point process derived from the latent diffusion process. We
illustrate the approach with numerical examples using both simulated and real
data. The proposed methods and framework provide a basis for interpreting point
processes through the lens of diffusions.
