| Co-Investigator(Kenkyū-buntansha) |
YASUDA Kumi Faculty of Mathematics, Kyushu University, Res. Ass., 大学院・数理学研究院, 助手 (40284484)
HAMANA Yuji Faculty of Mathematics, Kyushu University, Ass. Prof., 大学院・数理学研究院, 助教授 (00243923)
SUGITA Hiroshi Faculty of Mathematics, Kyushu University, Ass. Prof., 大学院・数理学研究院, 助教授 (50192125)
MATSUMOTO Hiroyuki Nagoya Univ., Faculty of Information and culture, Ass. Prof., 情報分科学部, 助教授 (00190538)
FUKAI Yasunari Faculty of Mathematics, Kyushu University, Res. Ass., 大学院・数理学研究院, 助手 (00311837)
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| Research Abstract |
In this research, we have made a systematic study on the asymptotic behavior of stochastic oscillatoty integrals. A stochastic oscillatory integral I(a) is, by definition, a integral of exp[iaq(x)]f(x) over the Wiener space X with respect to the Wiener measure on it, where i is the square root of -1, a is a real number, q, f are Wiener functionals on X. Obviously I(a) gives a characteristic function of the distribution of q under f(x)m(dx), and hence it is a basic object in the probability theory. Recalling the theory of Feynman path integrals, one recognizes the real interest of stochastic oscillatory integrals. Namely, a stochastic oscillatory integral is a mathematical counterpart to Feynman path integral, and the study of its asymptotic behavior closely relates to, so called, the WKB approximation, the semi-classical approximation, and so on. In our study, following the well developed theory of statinary phase method on finite dimensional spaces, we made several basic but indispensable researches on the asymptotic behavior of stochastic oscillatory integrals. We established several explicit representation of stochastic oscillatory integrals with quadratic phase functions, and apply them to show a principle of stationary phase for such oscillatory integrals. Moreover, we spelled out the relationship between the decay order of integrals and the quadratic phase functions. We also showed that a localization to stationary points of the main part of the asymptototic behavior occurs for some stochastic oscillatory integrals. We moreover made several concrete observations when the oscillatory integral is defined on the classical Wiener space, the path space.
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