Bessel游程下的面积分布及其矩
The Distribution of the Area under a Bessel Excursion and its Moments
摘要：
贝塞尔偏移是从原点开始并在某个给定时间$T$首先返回原点的贝塞尔过程。我们研究了这种偏移下的区域的分布，最近发现应用在激光冷却的背景下。面积$A$与时间成比例为$A \sim T^{3/2}$，与维度$d$无关，但分布的函数形式确实取决于$d$。我们证明，对于$d=1$，分布减少预期的分布下的布朗漂移，被称为艾里分布，推导出一个新的表达式的艾里分布的过程中的区域。我们发现，分布是对称的$d-2$，与非解析行为在$d=2$。我们计算分布的一阶和二阶矩，以及特定的分数阶矩。我们还分析了从$d<2$到$d>2$的解析延拓。在$d\to 4$的极限下，该解析连续分布由单侧Lévy $α$ -稳定分布描述，其指数为$2/3$，比例因子与$[(4-d)T]^{3/2}$成比例。
英文摘要：
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 $α$-stable distribution with index $2/3$ and a scale factor proportional to $[(4-d)T]^{3/2}$.
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