Fields of Dirichlet spaces: continuity, approximation, and stability

本项目重点发展了不同空间上Dirichlet空间连续域的新概念，并将其应用于不同类型的样例类。狄利克雷空间是合适的，因为它们的众多应用以及它们的方法多样性：狄利克雷形式或狄利克雷空间以一种自然的方式结合了解析结构、几何结构和概率结构。因此，狄利克雷空间场的连续性、逼近性和稳定性也涉及到由狄利克雷形式给出的几何和谱理论数据，它们的控制在许多应用领域中尤为重要。在两个主要子项目中，抽象概念被应用于更具体的情况。一方面，通过考虑奇异的例子类，如狄利克雷到诺伊曼算子和狄利克雷形式的更一般的轨迹。另一方面，对于黎曼流形的域和更一般的度量度量空间，这些域有时是由流定义的。

The project focuses on the development of new concepts for continuous fields of Dirichlet spaces over varying spaces and the application to different types of example classes. Dirichlet spaces are suitable because of their numerous applications as well as because of their methodological diversity: Dirichlet forms or Dirichlet spaces combine analytic, geometric and probabilistic structures in a natural way. The aspects of continuity, approximation and stability for fields of Dirichlet spaces therefore also concern geometric and spectral theoretical data, which are given by the Dirichlet forms and whose control is particularly important in many application areas. In two major subprojects the abstract concepts are applied to more specific situations. On the one hand, by considering singular example classes, such as Dirichlet-to-Neumann operators and more general traces of Dirichlet forms. On the other hand, to fields of Riemannian manifolds and more general metric measure spaces, where the fields are sometimes defined by flows.