Research on differential operators in infinite dimensional spaces with symmetry

本文的研究对象是无限维空间中的微分算子，特别是具有对称性的微分算子。我们从随机分析的角度研究了复Wiener空间上的全纯函数，研究了带边界条件的热方程基本解的渐近性，并对复Wiener空间上的全纯函数进行了一些研究。但它们都只处理定义在整个空间中的全纯函数。因此，这些工作并没有明确地说明全纯是一个局部性质。由于这项研究的目的是建立对Wiener流形的分析，我们确实需要概念的本地化。我们在全纯和Skelton概念的本地化方面取得了很大进展。全纯函数的Skelton是与原始全纯函数相关的Cameron-Martin空间中的全纯函数。定义在全c-…上的全纯函数的Skelton存在性九州大学的Sugita教授展示了更复杂的Wiener空间。但对定义在子域上的全纯函数的Skelton却知之甚少。在这项研究中，我们证明了如果子域是由一个好函数的正域给出的，则存在定义在该域上的全纯函数的Skelton，并且Skelton函数与全纯函数的对应是一一对应的。将这一结果与已有结果相结合，可以看出Dolbeault上同调与区域上全纯函数束的Cech上同调一致。我们希望这一结果能在无限维空间的复分析中起到重要的作用。我们还得到了具有Dirichlet或Neumann条件的热方程的基本解在具有光滑边界的严格凸域外的渐近行为的精确估计。我们通过用Wiener测度表示基本解并利用对称性得到了这一结果。我们没有想到会有这样的结果，但我们是作为副产品获得的。较少

The object of this research was differential operators in infinite dimensional spaces, especially ones with symmetry. We had a progess in the research on holomorphic functions on complex Wiener spaces from viewpoint of stochastic analysis and in the research on asymptotics of fundamental solutions of heat equations with boundary conditions.There have been some works on holomorphic functions in complex Wiener spaces. But all of them handled holomorphic functions defined in the whole space only. So these works did not make clear the holomorphicity as a local property. Since the aim of this research is to establish analysis on Wiener manifolds, we really need the localization of the concepts. We had a big progress in the localization of the notion of holomorphicity and skelton. The skelton of holomorphic functions is holomorphic functions in the Cameron-Martin space associated the original holomorphic functions. The existence of the skelton for holomorphic functions defined on the whole c … More omplex Wiener space was shown by Prof.Sugita in Kyushu University. But nothing is known about the skelton for holomorphic functions defined in subdomains. In this research we showed that if the subdomain is given by a positive domain of a good function, then there exists the skelton of holomorphic functions defined in the domain and the corespondance of skelton and holomorphic functions is one-to-one. Combining this result with known results, we can see that the Dolbeault cohomology coincides with the Cech cohomology of the sheaf of holomorphic functions in the domain. We hope that this result plays an important role in complex analysis in infinite dimensional spaces.We also obtained the precise estimates on the asymptotic behavior of the fundamental solution for the heat equation with Dirichlet or Neumann conditions in the outside of strictly convex domains with smooth boundaries. We obtained this result by representing the fundamental solutions by the Wiener measure and by using the symmetry. We did not expect such a result, but we obtained it as a by-product. Less