Occupation time distributions for Lévy bridges and excursions

Let X be a one-dimensional Lévy process. It is shown that under the bridge law for X starting from 0 and ending at 0 at time t, the amount of time X spends positive has a uniform distribution on [0, t]. When 0 is a regular point, this uniform distribution result leads to an explicit expression for the Laplace transform of the joint distribution of the pair (R, AR), where R is the length of an excursion of X from 0, and AR is the total time X spends positive during the excursion. More concrete expressions are obtained for stable processes by specialization. In particular, a formula determining the distribution of ARR is given in the stable case.