Stochastic analysis and its application to analysis of differential operators

本文明确求解了庞加莱上半平面（双曲平面）上布朗运动的随机微分方程，得到了布朗运动作为维纳泛函的具体表示。在本研究中，作为这一已知事实的推广，我们证明了布朗运动到标准正交坐标系束的水平寿命也有一个维纳泛函的表达式。在此基础上，我们给出了拉普拉斯热核作用于微分形式的概率表示，并给出了紧黎曼曲面上微分1-形式的Selberg迹公式的证明，该证明可由等距群的双曲离散子群作为上半平面的商空间给出。这是一个解析和/或几何证明，我们不需要谐波分析。此外，我们还得到了塞尔伯格迹公式的一种更合理的形式，因为我们把自己限制在二维的情况下。对一般维数情况的推广研究仍在继续。在二维情况下，我们可以用辅助的一维布朗运动来表示水平升力的旋转部分，这种表示起着至关重要的作用。我们还没有找到对应的表示，这应该是下一个任务。如果我们找到这样的表示，我们将能够根据McKean的思想给出Selberg迹公式的证明，这是本研究的基础。当我们将概率论应用于上半平面的分析时，出现了几何布朗运动积分的指数维纳泛函。由于在数学金融理论中也出现了同类型的维纳泛函，以及对随机环境中某些扩散过程的研究，对这些泛函的研究一直在继续。在与国外同事YOR教授的合作项目中，我们收集了这些指数维纳泛函的一些结果和应用，并从分析的角度给出了一些见解。研究结果已发表在杂志上。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