New Aspect of Probabilistic Approach to Chern-Simons Theory

为了在抽象的Wiener空间中给出其全拉格朗日量的Chern-Simons泛函积分的数学意义，我们沿着我的研究同事，德国波恩大学的Albeverio教授和我的大学的Funakubo教授的建议，正则化了用超场方法修正的拉格朗日量中出现的包括无穷行列式在内的第三项。并给出了Wiener泛函的数学定义。我们利用Wiener测度对定义Feynman测度的It^o方法进行了修正，从而对渐近展开式中的余项进行了估计。本文从集值随机过程的观点出发，对包括Chern-Simons理论在内的规范理论中的随机曲面进行了研究。在定义了集值随机积分之后，在一定的限制条件下，得到了一类集值随机微分方程解的存在唯一性。用一个词来比喻这种限制，我们捕捉到了水母的运动，但捕捉不到阿米巴的运动。

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.