Stochastic analysis and its application to analysis of Schrodinger operators

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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.