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 \(d\rightarrow 4\) 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 \((4-d)T]^{3/2}\) .