Stochastic analysis and its application to analysis of differential operators

的随机微分方程的布朗运动的庞加莱上半平面（双曲平面），一个扩散过程所产生的一半的拉普拉斯算子，明确解决，我们有一个具体的表示作为一个维纳功能的布朗运动。作为这一已知事实的推广，我们证明了标准正交标架丛的布朗运动的水平寿命也有一个Wiener泛函的表达式。在此基础上，给出了作用于微分形式的Laplacian算子的热核的概率表示，并证明了微分1-形式在紧致黎曼曲面上的Selberg迹公式，其中紧致黎曼曲面可以由等距群的双曲离散子群给出为上半平面的商空间.这是一个解析和/或几何证明，我们不需要调和分析。此外，我们还得到了Selberg迹公式的一个非常精确的表达式 关于我们 合法形式，因为我们已经限制了我们自己的二维情况下。扩展到一般维的情况下的研究一直在继续。在二维情况下，我们可以通过使用辅助一维布朗运动来表示水平升力的旋转部分，并且这种表示起着至关重要的作用。我们还没有找到相应的代表，这应该是下一个任务。如果我们找到这样一个表示，我们就能够按照McKean的思想证明Selberg迹公式，这是本研究的基础。当我们把概率论应用于上半平面的分析时，出现了指数Wiener泛函，它是几何布朗运动的积分。自从Wiener泛函出现在金融数学理论和随机环境中某些扩散过程的研究中以来，对这些泛函的一些研究一直在继续。在与YOR教授的一个合作项目中，我们收集了这些指数Wiener泛函的一些结果和应用，并从分析的角度给出了一些见解。研究结果已发表在一份期刊上。少

The stochastic differential equation for the Brownian motion on the Poincare upper half plane (the hyperbolic plane), a diffusion process generated by a half of the Laplacian, is explicitly solved and we have an concrete representation for the Brownian motion as a Wiener functional. In this research, as an extension of this known fact, we showed that the horizontal life of the Brownian motion to the bundle of orthonormal frames also has an expression as a Wiener functional. Based on this representation, we may show a probabilistic representation for the heat kernel of the Laplacian acting on the differential forms and give a proof of the Selberg trace formula for the differential 1-forms on a compact Riemannian surface, which may be given as a quotient space of the upper half plane by a hyperbolic discrete subgroup of the isometry group. This is an analytic and/or geometric proof and we do not need the harmonic analysis. Moreover we have obtained the Selberg trace formula in a very exp … More licit form, since we have restrict ourselves to the two-dimensional case.The research for an extension to the general dimension case has been continued. In the two dimensional case, we can represent the rotation part of the horizontal lift by using an auxiliary one-dimensional Brownian motion and this representation plays a crucial role. We have not found a corresponding representation and this should be the next task. If we find such a representation, we will be able to give a proof for the Selberg trace formula following the idea of McKean which has been the basis of this research.When we apply probability theory to the analysis on the upper half plane, the exponential Wiener functionals which is an integral of a geometric Brownian motion appear. Some studies on these functionals has been continued since the Wiener functionals of the same type also appear in the theory of Mathematical Finance and a study for some diffusion processes in random environments. In a joint project with Professot YOR, who is a foreign co-worker in thie research, we gathered some results and applications of these exponential Wiener functionals and gave an insight from analytic point of view. The results have been published in a journal. Less