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.

在这项研究中，我们对随机振荡积分的渐近行为进行了系统的研究。从定义上讲，随机振荡的积分I（a）是exp [iaq（x）] f（x）在维纳尔空间x上相对于维也纳措施的积分，其中i是-1的平方根，a是一个实数，q，f是X上的wiener函数。显然，i（a）给出了f（x）m（dx）下q分布的特征函数，因此它是概率中的基本对象理论。回忆起Feynman Path积分的理论，人们认识到随机振荡积分的真正兴趣。也就是说，随机振荡的积分是Feynman Path积分的数学对应物，其渐近行为的研究与所谓的WKB近似，半古典近似等等密切相关。在我们的研究中，遵循有限维空间的统计学方法的良好发展理论，我们就随机振荡性积分的渐近行为进行了几项基本但不可或缺的研究。我们建立了具有二次相函数的随机振荡积分的几个明确表示，并将它们应用它们显示了此类振荡积分的固定相原理。此外，我们阐明了积分的衰减顺序与二次相函数之间的关系。我们还表明，对于某些随机振荡性积分，发生了渐近行为主要部分的固定点的定位。此外，当在经典的维纳空间（路径空间）上定义振荡性积分时，我们进行了几个具体的观察。