Infinite Dimensional Analysis and its Applications

1997年4月至1998年3月，我们以“无限维分析及其应用”为主题，对基于白噪声的无限维随机分析进行了研究。我们组织了几次座谈会和研讨会，并与共同研究人员相互讨论。在这一研究中，我们取得了无限维拉普拉斯算子，特别是利维拉普拉斯算子的基本进展。自伴随算子在量子理论中是非常重要的。到目前为止，Levy拉普拉斯算子一直被认为是非自伴随算子。但在本研究中，我们成功地将Levy拉普拉斯算子推广到广义白噪声泛函空间中Hilbert空间上的自伴随算子。基于这一结果，我们期望得到许多描述现象的应用。作为应用，我们已经得到了利维拉普拉斯算子与数算子之间的关系，以及由拉普拉斯算子生成的半群与无限维Ornstein-Uhlenbeck过程之间的关系。此外，如果我们考虑作用于泊松噪声泛函的Levy Laplacian，则可以得到更丰富的结果，这些结果与数学金融学密切相关。这些也是Hida微积分的发展，可以应用于Matsuzawa在无限维方法下对偏微分方程的研究。由这些结果可以很容易地得到与p进白噪声分析类似的结果。泊松噪声分析有助于在Nishi的研究中找到许多例子。利维拉普拉斯算子可以用遍历性来表征。这与弥町对遍历论的研究密切相关。列维拉普拉斯算子的自伴随性使得共同研究人员之间有了联合项目。

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.