A Study of asymptotic behaviors of stochastic oscillatory integrals

本文系统地研究了随机可积积分的渐近性态。根据定义，随机振荡积分I（a）是关于Wiener空间X上的Wiener测度的exp[iaq（x）]f（x）的积分，其中i是-1的平方根，a是真实的数，q，f是X上的Wiener泛函。显然，I（a）给出了f（x）m（dx）下q分布的特征函数，因此它是概率论中的一个基本对象。回顾费曼路径积分理论，人们认识到随机振荡积分的真实的意义。即随机振荡积分是Feynman路径积分的数学对应物，其渐近性态的研究与所谓的WKB近似、半经典近似等密切相关.本文根据有限维空间上已发展成熟的定相方法理论，我们对随机振荡积分的渐近性做了一些基本的但又是必不可少的研究。建立了随机振荡积分的二次相函数的几种显式表示，并应用它们给出了这类振荡积分的一个稳相原理。此外，还阐明了积分的衰减阶与二次相函数的关系。我们还表明，局部化的稳定点的渐近行为的主要部分发生的随机振荡积分。此外，我们提出了几个具体的意见时，振荡积分定义在经典的维纳空间，路径空间。

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.