Stochastic analysis and its application to analysis of Schrodinger operators

从Poincare上半平面上的布朗运动通过求解随机微分方程给出显式Wiener泛函这一事实出发，给出了质量拉普拉斯的Selberg迹公式，并阐明了它与相应的经典力学在质和量上的关系。这一结果已作为与池田的联合论文发表。遗憾的是，在本文中，我们不能成功地讨论和计算热核的显式。在上述上半平面的随机分析和数学金融理论中，几何布朗运动时由积分定义的维纳泛函起着重要的作用。在与Yor的联合工作中，我们首先指出布朗运动的最大过程可以通过一个极限过程得到，并给出了取极限之前的指数Wiener泛函的反射布朗运动和Bessel过程的Levy和Pitman定理的类似。我们把结果和一些相关的主题一起发表了。

Starting from the fact that the Brownian motion on the Poincare upper half plane is given as an explicit Wiener functional by solving the stochastic differential equation, we showed the Selberg trace formula for the Mass Laplacian and clarified the relation, both in quality and in quantity, with the corresponding classical mechanics. The result has been published as a joint paper with Ikeda. Unfortunately, in this paper, we could not successfully discuss and calculate the explicit form of the heat kernel. As a consequence of further study, we have succeed this in the same framework.In stochastic analysis on the upper half plane mentioned above and the theory of mathematical finance, the Wiener functionals defined by integrals in time of geometric Brownian motions play important roles. In the joint work with Yor, we first remarked that the maximum process of the Brownian motion may be obtained by a limiting procedure and showed analogues of the Levy and Pitman theorems, which gives representations of the reflecting Brownian motion and the Bessel process, for the exponential Wiener functionals before taking the limit. We published the results together with some related topics.