Stochastic analysis and its application to differential operators

最重要的结果是关于基本随机过程之一的Bessel过程的首中时的结果。 当起始点离原点远于终点时，我们用修正贝塞尔函数的零点给出了概率密度的相当简单的表达式。 作为应用，我们研究了概率密度和尾概率的渐近行为。我们还证明了布朗运动的tubar邻域的期望体积的渐近性。 另一方面，我们找到了修正贝塞尔函数零点所满足的代数方程。

The most important result is the result on the first hitting times of Bessel processes which is one of the fundamental stochastic processes. When the starting point is far from the arigin than the terminal point, we have shown fairly simple expressions for the probability densities by means of the zeros of the modified Bessel functions. As applications, we have investigated the asymptotic behavior of the probability densities and the tail probabilities. We have also shown the asymptotics of the expected volume of the tubar neighborhood of Brownian motions. On the other hand, we have found some algebraic equations which are satisfied by the zeros of the Modified Bessel functions.