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A Study of asymptotic behaviors of stochastic oscillatory integrals

随机振荡积分渐近行为的研究

基本信息

批准号：
11440051
负责人：
TANIGUCHI Setsuo
金额：
$ 5.63万
依托单位：
KYUSHU UNIVERSITY
依托单位国家：
日本
项目类别：
Grant-in-Aid for Scientific Research (B)
财政年份：
1999
资助国家：
日本
起止时间：
1999 至 2001
项目状态：
已结题

项目摘要

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.

在本研究中，我们对随机振荡积分的渐近行为进行了系统的研究。根据定义，随机振荡积分 I(a) 是 exp[iaq(x)]f(x) 在维纳空间 X 上相对于维纳测度的积分，其中 i 是 -1 的平方根，a 是实数，q、f 是 X 上的维纳泛函。显然，I(a) 给出了 f(x)m(dx) 下 q 分布的特征函数，因此它是概率论。回顾一下费曼路径积分理论，我们就会认识到随机振荡积分的真正意义。即，随机振荡积分是费曼路径积分的数学对应物，其渐近行为的研究与所谓的WKB近似、半经典近似等密切相关。在我们的研究中，遵循有限维空间上成熟的静态相位法理论，我们对随机振荡积分的渐近行为进行了一些基础但不可或缺的研究。我们建立了带有二次相位函数的随机振荡积分的几种显式表示，并应用它们来展示此类振荡积分的平稳相位原理。此外，我们还阐明了积分的衰减阶数与二次相函数之间的关系。我们还表明，对于某些随机振荡积分，渐近行为的主要部分会定位于驻点。此外，当在经典维纳空间（路径空间）上定义振荡积分时，我们还进行了一些具体的观察。