Infinite Dimensional Stochastic Analysis and its Applications

在科学研究助学金(C)的支持下，我们在基于Levy Laplacian的无限维随机分析方面取得了丰硕的成果。与李维·拉普拉斯有关的拉普拉斯方程等价于量子物理中的杨-米尔斯方程。此外，拉普拉斯量与量子白噪声的平方次方密切相关。研究基于拉普拉斯函数的随机分析对于描述和理解微观现象具有重要作用。在1999年的研究中，我们得到了基于拉普拉斯的随机分析的几个结果：Levy拉普拉斯的对角化，构造拉普拉斯具有连续谱的区域(核空间)，构造区域上的拉普拉斯生成的随机过程和由拉普拉斯生成的量子随机过程，构造由拉普拉斯函数生成的随机过程等。这些随机过程是无限的，但涨落的部分是有限的，因此我们希望将这一理论应用于工程上的其他领域。此外，一些生物现象和经济现象可以用李维-拉普拉斯方程描述为随机模型。如果我们把拉普拉斯算符的定义域中的元素改为某些算符，那么我们就可以研究基于拉普拉斯算符的量子白噪声理论。我们希望这一研究方向能有更大的发展。在助学金的支持下，我们参加了意大利和美国的国际研讨会，就无限维随机分析的最新成果进行了讲座，并在京都大学、九州大学等地进行了演讲。我们邀请了广岛女子大学的伊藤教授对白噪声理论在经济中的应用进行了回顾。

Supported by Grant-in-Aid for Scientific Research(C)we obtained fruitful results on an infinite dimensional stochastic analysis based on the Levy Laplacian. The Laplace equation associated the Levy Laplacian is equivalent to the yang-Mills equation in quantum physics. Moreover, the Laplacian is closely related to the square power of the quantum white noise. To research a stochastic analysis based on the Laplacian has an important role of describing and understanding the micro-phenomena. With our previous researches in 1999 we obtained several results on a stochastic analysis based on the Laplacian in this research : Diagnalization of the Levy Laplacian, Constructing the domain(nuclear space)on which the Laplacian has a continuous spectrum, Constructing a stochastic process and a quantum stochastic process generated by the Laplacian on the domain, Constructing a stochastic process generated by functions of the Laplacian, etc. Those stochastic processes are infinite but the parts of fluctuation are finite and therefore we expect to apply this theory to other fields in Engineering. Moreover we can describe some biological phenomena and economical phenomena as stochastic models using the Levy Laplacian. If we change elements of the domain of the Levy Laplacian to some operators, then we can research the quantum white noise theory based on the Laplacian. We hope that this direction of researches has more furuitful developments.Supported by the Grant we attended at international workshops in Italy and in USA to give lectures on recent results on infinite dimensional stochastic analysis, and also gave talks at Kyoto University, Kyushu University and others. We asked Prof.Ito at Hiroshima Women's University to give a review on applications of white noise theory to the economy.