A Study of asymptotic behaviors of stochastic oscillatory integrals

在本研究中，我们系统地研究了随机振荡积分的渐近行为。根据定义，随机振荡积分I(A)是Wiener空间X上关于其上Wiener测度的exp[iaq(X)]f(X)的积分，其中i是-1的平方根，a是实数，q，f是X上的Wiener泛函。显然，I(A)给出了q在f(X)m(Dx)下的分布的特征函数，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正意义。也就是说，随机振荡积分是Feynman路径积分的数学对应，对其渐近性的研究与所谓的WKB逼近、半经典逼近等密切相关。在我们的研究中，遵循有限维空间上成熟的定相法理论，我们对随机振荡积分的渐近行为做了几个基本而又不可或缺的研究。我们建立了随机振荡积分与二次相函数的几种显式表示，并应用它们证明了这类振荡积分的定相原理。此外，我们还阐明了积分的衰减级与二次相函数之间的关系。对于某些随机振荡积分，我们还证明了渐近行为的主要部分的局部局部化。此外，我们还在经典Wiener空间，即路径空间上定义了振荡积分，并做了几个具体的观察。

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.