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 sim 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 (drightarrow 4) from below, this analytically continued distribution is described by a one-sided Lévy (alpha ) -stable distribution with index (2/3) and a scale factor proportional to ([(4-d)T]^{3/2}) .