The Hartman-Watson distribution with density $f_r(t)$ is a probability distribution defined on $t \geq 0$ which appears in several problems of applied probability. The density of this distribution is expressed in terms of an integral $\theta(r,t)$ which is difficult to evaluate numerically for small $t\to 0$. Using saddle point methods, we obtain the first two terms of the $t\to 0$ expansion of $\theta(\rho/t,t)$ at fixed $\rho >0$. An error bound is obtained by numerical estimates of the integrand, which is furthermore uniform in $\rho$. As an application we obtain the leading asymptotics of the density of the time average of the geometric Brownian motion as $t\to 0$. This has the form $\mathbb{P}(\frac{1}{t} \int_0^t e^{2(B_s+\mu s)} ds \in da) \sim (2\pi t)^{-1/2} g(a,\mu) e^{-\frac{1}{t} J(a)} da/a$, with an exponent $J(a)$ which reproduces the known result obtained previously using Large Deviations theory.

翻译：Hartman-Watson 分布是一个定义在 $t \geq 0$ 上的概率分布，其密度函数为 $f_r(t)$，在多个应用概率问题中出现。该分布的密度函数通过一个积分 $\theta(r,t)$ 表示，当 $t\to 0$ 时，该积分难以进行数值计算。利用鞍点方法，我们得到了在固定 $\rho >0$ 条件下，$\theta(\rho/t,t)$ 在 $t\to 0$ 时的前两项展开式。通过对被积函数进行数值估计，我们获得了一个误差界，该误差界在 $\rho$ 上是一致的。作为一个应用，我们得到了几何布朗运动的时间平均密度在 $t\to 0$ 时的主导渐近行为。其形式为 $\mathbb{P}(\frac{1}{t} \int_0^t e^{2(B_s+\mu s)} ds \in da) \sim (2\pi t)^{-1/2} g(a,\mu) e^{-\frac{1}{t} J(a)} da/a$，其中指数 $J(a)$ 再现了先前使用大偏差理论获得的已知结果。