任意维Laakso和Barlow-Evans分形上的Dirichlet型
Dirichlet Forms on Laakso and Barlow-Evans Fractals of Arbitrary Dimension
摘要：
本文研究同一度量测度空间族的两种构造。第一个建设介绍了Laakso在2000年，他用它作为一个例子，庞加莱不等式可以举行的空间任意豪斯多夫维数。这是用最小广义上梯度证明的。根据Cheeger的工作，这些上梯度可以用来定义索伯列夫空间。我们表明，这导致一个狄利克雷形式。第二种构造是由Barlow和Evans在2004年提出的，作为一种从简单的空间和过程产生奇异空间和马尔可夫过程的方法。证明了对于Barlow Evans构造中的正确基过程，这个马尔可夫过程对应于由最小广义上梯度导出的Dirichlet形式.
英文摘要：
In this paper we explore two constructions of the same family of metric measure spaces. The first construction was introduced by Laakso in 2000 where he used it as an example that Poincaré inequalities can hold on spaces of arbitrary Hausdorff dimension. This was proved using minimal generalized upper gradients. Following Cheeger's work these upper gradients can be used to define a Sobolev space. We show that this leads to a Dirichlet form. The second construction was introduced by Barlow and Evans in 2004 as a way of producing exotic spaces along with Markov processes from simpler spaces and processes. We show that for the correct base process in the Barlow Evans construction that this Markov process corresponds to the Dirichlet form derived from the minimal generalized upper gradients.
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