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 T. We study the distribution of the area under such an excursion, which recently found application in the context of laser cooling. The area A scales with the time as A ~ T^3/2, independent of the dimension, d, but the functional form of the distribution does depend on d. We demonstrate that for d = 1, 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 d - 2, with nonanalytic behavior at d = 2. We calculate the first and second moments of the distribution, as well as a particular fractional moment. We also analyze the analytic continuation from d < 2 to d > 2. In the limit where d → 4 from below, this analytically continued distribution is described by a one-sided Lévy α-stable distribution with index 2/3 and a scale factor proportional to [(4 - d)T]^3/2.