Stochastic Analysis and its application to differential operators

如果我们考虑一个标准布朗运动及其最大过程，其差值与反映布朗运动具有相同的概率规律，最大过程与原始布朗运动的两次差值与三维贝塞尔过程的差值具有相同的规律。这些结果就是众所周知的利维定理和皮特曼定理。首席研究员Matsumoto与Yor合作，通过使用经典拉普拉斯方法，为指数布朗泛函展示了这些定理的类似或扩展。指数布朗泛函也出现在数学金融研究和双曲空间的随机分析中。这一职能的各个方面及其应用都是在这项补助金的支持下进行的。其中，利用指数维纳泛函的结果，给出了由拉普拉斯生成的半群在秩为1的非紧对称空间上的热核的一些显式表达式。其他研究者对上述工作提出了许多建议和意见，并在各自的领域进行了研究。Ihara通过应用大偏差理论来研究信息论。Uemura研究了多维布朗运动的局部时间，将其视为广义的维纳泛函。Ueki研究了随机电磁场下薛定谔算子的谱性质。

If we consider a standard Brownian motion and its maximum process, the difference has a same probability law as reflecting Brownian motion and the difference between twice the maximun process and the original Brownian motion has the same law as a three dimensional Bessel process. These results are well known as Levy's and Pitman's theorems. The head investigator Matsumoto, in joint work with Yor, showed analogus or extensions of these theorems for an exponential Brownian functionals by using the classical Laplace method. The exponential Brownian functional appears also in the studies of mathematical finance and in stochastic analysis on hyperbolic spaces. Various aspects of this functional and their applications have been given under the support of this grant. Among others, some explicit expressions for the heat kernels of the semigroups generated by the Laplacians on non-compact symmetric spaces of rank one have been shown by using the results on the exponential Wiener functionals.Other investigators gave many suggestions and comments on the works mentioned above and also worked on their own fields. Ihara studied information theory by applying the theory of large deviations. Uemura studied on the local time of multi-dimensional Brownian motions, which should be considered as generalized Wiener functionals. Ueki studied on the spectral properties of Schrodinger operators with random electro-magnetic fields.