Boundaries, Greens formulae and harmonic functions for graphs and Dirichlet spaces

图和狄利克雷空间的边界、格林公式和调和函数

• 批准号: 339133485
• 负责人: Professor Dr. Matthias Keller
• 金额: --
• 依托单位: Institut für Mathematik
• 依托单位国家: 德国
• 项目类别: Priority Programmes
• 财政年份: 2017
• 资助国家: 德国
• 起止时间: 2016-12-31 至 2022-12-31
• 项目状态: 已结题
• 来源: https://gepris.dfg.de/gepris/projekt/339133485?language=en
• 关键词: Boundaries; Greens; formulae; harmonic; functions

A locally compact separable metric space together with a regular Dirichlet form is called a Dirichlet space. Thus, a Dirichlet space is a metric measure space with an additional structure.Examples for Dirichlet spaces include (weighted) Riemannian manifolds, fractals and graphs. Any Dirichlet form comes with a selfadjoint 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 Brownian motion. Similarly, in the case of fractals the generator is a Laplace type operator and the Markov process is a Brownian motion. In the case of graphs, the generator is the graph Laplacian and the associated Markov process is a jump process. In this way, Dirichlet forms provide an analytic description of (versions of) Brownian motion on the underlying topological space.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 focusing on global properties viz. on properties of the geometry ``far out'' and corresponding spectral and stochastic features.Two approaches will be pursued. One approach is centered around the compactification via the Royden boundary, corresponding boundary terms and Greens formulae. Here, we aim ata) understanding the boundary as a metric boundary,b) describing the selfadjoint Markov extensions for general Dirichlet forms via Dirichlet forms on the boundary,c) describing selfadjoint extensions of the Laplacian on (bundles over) graphs via boundary conditions,d) characterize global properties of the underlying Markov process such as recurrence, stochastic completeness and Markov uniqueness by vanishing boundary terms.In the second approach features of the geometry “far out” are captured via generalized eigenfunctions. With this approach we intend toa) show absence of non-trivial harmonic functions with $L^{p}$ growth conditions on general Dirichlet spaces,b) bound the dimension of the space of polynomially bounded harmonic functions for graphs,c) understand the support of eigenfunctions,d) describe the decay properties of generalized eigenfunctions for graphs.The two approaches are connected and making this explicit is an additional part of the project.The project will focus on the non-smooth non-local situation of graphs. However, we will strive for arguments dealing with general Dirichlet spaces. So, the investigations on graphs can be seen as a first step towards the theory of general Dirichlet spaces.

一个局部紧的可分度量空间和一个正则狄利克雷形式一起称为狄利克雷空间。因此，狄利克雷空间是具有附加结构的度量测度空间。狄利克雷空间的例子包括（加权）黎曼流形，分形和图。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。类似地，在分形的情况下，生成器是一个拉普拉斯型算子，马尔可夫过程是一个布朗运动。在图的情况下，生成器是图拉普拉斯算子，并且相关联的马尔可夫过程是跳跃过程。通过这种方式，狄利克雷形式提供了一个分析描述，在Dirichlet空间的背景下，空间的几何性质之间有很强的相互作用，Dirichlet形式的生成器的谱特征和相关Markov过程的随机特征。在这个项目中，我们研究这种相互作用，重点是全局性质，即几何性质。远”和相应的频谱和随机特征。将采用两种方法。一种方法是通过Royden边界，相应的边界项和格林公式围绕紧化。这里，我们的目标是a）将边界理解为度量边界，B）通过边界上的Dirichlet形式描述一般Dirichlet形式的自伴Markov扩张，c）描述Laplacian在（bundles over）graphs via boundary conditions，d）刻画底层马尔可夫过程的全局性质，例如递归，在第二种方法中，通过广义本征函数来捕获几何“远”的特征。利用这种方法，我们打算a）证明一般Dirichlet空间上不存在满足$L^{p}$增长条件的非平凡调和函数，B）限制图的多项式有界调和函数空间的维数，c）理解特征函数的支撑，d）、描述了图的广义本征函数的衰减特性。这两种方法是相互联系的，明确这一点是该项目的附加部分。项目将集中在非光滑的非局部情况的图。然而，我们将努力讨论一般狄利克雷空间。因此，对图的研究可以看作是迈向一般Dirichlet空间理论的第一步。