The Distribution of the Area Under a Bessel Excursion and its Moments

A Bessel excursion is a Bessel process that begins at the origin and first returns there at some given time . We study the distribution of the area under such an excursion, which recently found application in the context of laser cooling. The area scales with the time as , independent of the dimension, , but the functional form of the distribution does depend on . We demonstrate that for , the distribution reduces as expected to the distribution for the area under a Brownian excursion, known as the Airy distribution, deriving a new expression for the Airy distribution in the process. We show that the distribution is symmetric in , with nonanalytic behavior at . We calculate the first and second moments of the distribution, as well as a particular fractional moment. We also analyze the analytic continuation from to . In the limit where from below, this analytically continued distribution is described by a one-sided L,vy -stable distribution with index and a scale factor proportional to .