Stochastic Analysis and Asymptotics

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1. The Van Vleck formula for Feynman's path integrals with qusdratic phase functionals plays an important role in the study of the mathematical science. By using the method of stochastic analysis, N.Ikeda proved an analogous formula for Weiner integrals with quadratic phase functionals to the Van Vleck formula mentioned above. By using transformations described in terms of Jacpbi fields, we can rewrite quadratic Wiener functionals associated with quadratic Lagrangians in the canonical form on the Wiener space. Then the functional determinants appeared in the Van Vleck formula can be regarded as the Jacobian of the transformation above as in case of finite dimensional manifolds.2. Without using the exact formula of the fundamental solution of the heat equation on the Poincare upper half plane, N.Ikeda obtained a proof of Selberg trace formula based on the integral representation of the fundamental solution of the heat equation on the Wiener space.3. T.Yamada has studied principal values of Brownian local times and representations of Brownian additive functionals of zero energy. M.Arai obtained remarkable results about the growth order of eigenfuctions of Schrodinger operators with potentials admitting some integral conditions.T.Natsume has studied several index theorems by using the method of functional analysis.Y.Takayama and Y.Sato have studied several problems of mathematical science from the point of view of the computer science.