Boundaries, Green's formulae and harmonic functions for graphs and Dirichlet spaces - follow up

一个局部紧可分度量 空间与正则狄利克雷形式一起称为狄利克雷空间。 狄利克雷空间的例子包括（加权）黎曼流形，分形和图形。任何狄利克雷形式都有一个自伴算子，称为生成元，以及一个相关的马尔可夫过程。在黎曼流形的情况下，生成元是拉普拉斯-贝尔特拉米算子，而所讨论的马尔可夫过程是布朗运动。 在Dirichlet空间的背景下，空间的几何性质、Dirichlet形式生成元的谱特征和相应的Markov过程的随机特征之间存在着很强的相互作用，在本项目中，我们用两种方法研究这种相互作用。一种方法是通过Royden边界、边界项和Greens公式进行紧化，研究几何的整体性质以及相应的谱和随机特征;另一种方法是通过调和函数和（广义）特征函数进行局部化，研究几何的局部特征。这些方法密切相关，探索它们的关系将带来更多的见解。该项目将集中在图的非光滑，非局部情况。然而，我们将努力讨论一般狄利克雷空间。

A locally compact separable metric space together with a regular Dirichlet form is called a Dirichlet space. Examples for Dirichlet spaces include (weighted) Riemannian manifolds, fractals and graphs. Any Dirichlet form comes with a self-adjoint operator, called the generator, and an associated Markov process. In the case of Riemannian manifolds, the generator is the Laplace-Beltrami operator and the Markov process in question is the Brownian motion. In the setting of Dirichlet spaces there is a strong interplay between geometric properties of the space, spectral features of the generator of the Dirichlet form and stochastic features of the associated Markov process.In this project we study this interplay by two approaches. One approach, focusing on global properties of the geometry and the corresponding spectral and stochastic features, is centered around compactification via the Royden boundary, boundary terms and Greens formulae.The other approach, focusing on more local features of the geometry, is centered around harmonic functions and (generalized) eigenfunctions. These approaches are strongly related and exploring their relationship will lead to additional insights. The project will focus on the non-smooth, non-local situation of graphs. However, we will strive for arguments dealing with general Dirichlet spaces.