Stochastic analysis and its application to differential operators

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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.