On occupation times of stationary excursions

In this paper excursions of a stationary diffusion in stationary state are studied. In particular, we compute the joint distribution of the occupation times $I^{(+)}_t$ and $I^{(-)}_t$ above and below, respectively, the observed level at time $t$ during an excursion. We consider also the starting time $g_t$ and the ending time $d_t$ of the excursion (straddling $t$) and discuss their relations to the Levy measure of the inverse local time. It is seen that the pairs $(I^{(+)}_t, I^{(-)}_t)$ and $(t-g_t, d_t-t)$ are identically distributed. Moreover, conditionally on $I^{(+)}_t + I^{(-)}_t =v$, the variables $I^{(+)}_t$ and $I^{(-)}_t$ are uniformly distributed on $(0,v)$. Using the theory of the Palm measures, we derive an analoguous result for excursion bridges.