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

Abstract

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 $d\to 4$ 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 $[(4-d)T{]}^{3/2}$.

References in corpus (2)

• Exact and explicit probability densities for one-sided Levy stable distributions
• From the area under the Bessel excursion to anomalous diffusion of cold atoms

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• First-passage functionals of Brownian motion in logarithmic potentials and heterogeneous diffusion
• Fluctuations of ring polymers
• Dimensionality-induced dynamical phase transition in the large deviation of local time density for Brownian motion