Infinite Dimensional Analysis and its Applications

无限维分析及其应用

基本信息

批准号：
09640300
负责人：
SAITO Kimiaki
金额：
$ 0.96万
依托单位：
Meijo University
依托单位国家：
日本
项目类别：
Grant-in-Aid for Scientific Research (C)
财政年份：
1997
资助国家：
日本
起止时间：
1997 至 1998
项目状态：
已结题

来源：
https://kaken.nii.ac.jp/grant/KAKENHI-PROJECT-09640300/
关键词：
Infinite Dimensional Analysis White Noise Analysis The Levy Laplacian Infinite Dimensional Stochastic Process Poisson Noise Analysis ホワイトノイズ無限次元ラプラシアン

项目摘要

We have researched the infinite dimensional stochastic analysis based on the white noise on the theme 'Infinite Dimensional Analysis and its Applications' from April 1997 to March 1998. We organized several symposiums and seminars, and discussed with co-researchers each other. In this research, we obtained fundamental progress on infinite dimensional Laplacians, in particular the Levy Laplacian. The self-adjoint operator is very important in Quantum theory. So far the Levy Laplacian has been considered as non self-adjoint operator. But in this research we succeeded in extending the Levy Laplacian to a self-adjoint operator densely defined on some Hilbert space in the space of generalized white noise functionals. Based on this result we expect to get many applications to describe phenomena. As applications we have already obtained a relation between the Levy Laplacian and the number operator and also a relation between a semi-group generated by the Laplacian and an infinite dimensional Ornstein-Uhlenbeck process. Moreover if we consider the Levy Laplacian acting on Poisson noise functionals, then more fruitful results can be obtained and those results are closely related to the mathematical finance. Those are also developments on Hida calculus and can be applied to Matsuzawa's researches on partial differential equations in terms of the infnite dimensional method. An analogue to the p-adic white noise analysis can be easily obtained by those results. Poisson noise analysis is useful to find many examples in Nishi's researches. The Levy Laplacian can be characterized by the ergodic property. This is closely related to Mimachi's researches on the ergodic theory. The self-adjointness of the Levy Laplacian made joint projects between co-researchers.

我们研究了从1997年4月至1998年3月的主题“无限尺寸分析及其应用”主题的无限尺寸随机分析。我们组织了几个研讨会和研讨会，并与共同研究者进行了讨论。在这项研究中，我们在无限的尺寸拉普拉斯人（尤其是征税拉普拉斯主义者）方面取得了基本进展。自伴操作员在量子理论中非常重要。到目前为止，Laplacian被认为是非自我伴侣操作员。但是在这项研究中，我们成功地将征费的拉普拉斯式扩展到了在普遍的白噪声功能空间中在某些希尔伯特空间上密集定义的自动接合操作员。基于这个结果，我们希望得到许多应用程序来描述现象。作为应用程序，我们已经获得了Laplacian和数字操作员之间的关系，以及Laplacian生成的半组与无限尺寸Ornstein-Ornstein-uhlenbeck过程之间的关系。此外，如果我们考虑作用于泊松噪声功能的征费拉普拉斯，那么可以获得更富有成果的结果，并且这些结果与数学金融密切相关。这些也是HIDA微积分的发展，可以根据Infnite尺寸方法应用于Matsuzawa对部分微分方程的研究。这些结果可以很容易地获得对P-Adic白噪声分析的类似物。泊松噪声分析对于在Nishi的研究中找到许多示例很有用。征税拉普拉斯人的特征是以崇高的特性为特征。这与Mimachi对厄运理论的研究密切相关。 Laplacian的自我伴侣在共同研究者之间进行了联合项目。