New Aspect of Probabilistic Approach to Chern-Simons Theory

为了在抽象的Wiener空间中赋予Chern-Simons泛函积分其总拉格朗日量的数学意义，我们按照我的研究同事德国波恩大学的Albario教授和我的大学的Funakubo教授的建议，正则化了包括由超场方法修正的拉格朗日中出现的无穷行列式在内的第三项，并给出了作为Wiener泛函的数学定义。通过修正用Wiener测度定义Feynman测度的方法，给出了渐近展开式中余项的估计。对于包括Chern-Simons理论在内的规范理论，我们从集值随机过程的一个视点出发研究了随机曲面。在定义集值随机积分的基础上，得到了一类集值随机微分方程解的存在唯一性。比喻地说，在一个词中的限制，我们抓住了水母的运动，但不是阿米巴的运动。

To give a mathematical meaning to the Chern-Simons functional integral of its total Lagrangian in an abstract Wiener space setting, we regularize the 3rd terms including the infinite determinant appeared in the Lagrangian modified by the method of super fields, along the suggestions of my research colleague, Prof. Albeverio atthe University of Bonn in Germany and Prof. Funakubo in my University, and give themathematical definition as a Wiener functional. We did the estimate of the remainder terms in the asymptotic expansion by modifying the It^o method of defining the Feynman measure by using the Wiener measure. Concerning the Gauge theory including the Chern-Simons theory, we study the random surfaces from a viw point of set-valued stochastic processes. After defining the set-valued stochastic integral, we get the existence and uniqueness for the solutions of a set-valued stochastin differential equation undersome restriction. Metaphorically speaking the restriction in a word, we catch the motion of a jellyfish but not of an ameba.