  We present a number of important identities related to the excursion theory
of linear diffusions. In particular, excursions straddling an independent
exponential time are studied in detail. Letting the parameter of the
exponential time tend to zero it is seen that these results connect to the
corresponding results for excursions of stationary diffusions (in stationary
state). We characterize also the laws of the diffusion prior and posterior to
the last zero before the exponential time. It is proved using Krein's
representations that, e.g., the law of the length of the excursion straddling
an exponential time is infinitely divisible. As an illustration of the results
we discuss Ornstein-Uhlenbeck processes.
