Stochastic analysis and its application to analysis of Schrodinger operators

从求解随机微分方程将Poincare上半平面上的布朗运动作为显式Wiener泛函给出出发，给出了Mass Laplacian的Selberg迹公式，阐明了它与相应的经典力学在质和量上的关系.研究结果已作为与池田的联合论文发表。不幸的是，在本文中，我们不能成功地讨论和计算显式形式的热核。在上述上半平面上的随机分析和数理金融理论中，由几何布朗运动的时间积分定义的Wiener泛函起着重要的作用。在联合工作与约，我们首先指出，最大过程的布朗运动可能会得到一个限制程序，并显示类似的利维和皮特曼定理，这使代表的反射布朗运动和贝塞尔过程，指数维纳泛函之前采取的限制。我们将结果与一些相关主题一起发表。

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.