带状复形上的热半群与布朗运动
The heat semigroup and Brownian motion on strip complexes
摘要：
我们引入了带状复形的概念。带状复形是一种特殊类型的复形，它是根据给定的图结构沿着其自然边界粘合“条”而得到的。最熟悉的例子是与图经典相关的一维复形，在这种情况下，条带只是单位区间的副本（我们的设置实际上允许可变的边长）。一个主要的关键例子是树曲空间，一个在最近的一些文章中研究的几何对象，它作为度量树与双曲平面的horocyclic乘积而出现。在这种情况下，图是一个正则树，带是封闭的单位区间乘以实直线，每个带都具有上半平面中特定带的双曲几何。我们考虑了一般带状复形上的自然Dirichlet型族，并证明了相应的热核和调和函数具有很强的光滑性。我们研究的问题，如基本的微分算子作用在一个合适的空间的光滑函数满足基尔霍夫型条件的带复杂的分叉点的基本自伴。与适当的群作用引起的预测的兼容性也被考虑。
英文摘要：
We introduce the notion of strip complex. A strip complex is a special type of complex obtained by gluing "strips" along their natural boundaries according to a given graph structure. The most familiar example is the one dimensional complex classically associated with a graph, in which case the strips are simply copies of the unit interval (our setup actually allows for variable edge length). A leading key example is treebolic space, a geometric object studied in a number of recent articles, which arises as a horocyclic product of a metric tree with the hyperbolic plane. In this case, the graph is a regular tree, the strips are the closed unit interval times the real line, and each strip is equipped with the hyperbolic geometry of a specific strip in upper half plane. We consider natural families of Dirichlet forms on a general strip complex and show that the associated heat kernels and harmonic functions have very strong smoothness properties. We study questions such as essential selfadjointness of the underlying differential operator acting on a suitable space of smooth functions satisfying a Kirchoff type condition at points where the strip complex bifurcates. Compatibility with projections that arise from proper group actions is also considered.
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