Research on phenomenon caused by simultaneous change of smooth measure and energy measure associated with Dirichlet forms

1.将光滑度和能量度同时变化引起的现象看作是度量的密度函数序列的变化。当两个度量的密度函数彼此重合且参数被变换为合适的密度函数时，密度函数的一系列图形成一个图的长度在上面有界的函数族。因此，我们可以取密度函数的一个子序列，它一致收敛于一个极限函数。在极限函数的图形很简单的情况下，我们可以找到参数空间的划分。这个除法将我们引向一族一维广义扩散过程和一族谱测度族。极限过程的谱由这样的族表示。一般来说，要得到这样一个参数空间的划分并不容易。我们刻画了一个正规化的度量空间，因此我们可以控制度量的行为。2.对于离散谱的广义扩散过程，我们证明了与击中时间相关的条件分布存在一个非平凡的极限分布。我们得到了不命中边界条件下转移概率随时间的渐近行为。我们的结果表明，渐近行为受边界附近样本路径的渐近行为的影响。3.我们定义了Sierpinski地毯晶格及其族的渗流模型，并证明了存在一个不存在相变的模型。

1.We treated phenomenon caused by simultaneous change of smooth measure and energy measure as a change of sequence of density functions of measures. When density functions of two measures coincide with each other and the parameter is transformed to a suitable one, a sequence of graphs of density function forms a family of functions whose length of graphs is bounded above. Therefore we can take a subsequence of density functions which converges to a limit function uniformly. In the case that the graph of the limit function is simple, we can find a division of a space of parameters. The division leads us to a family of one-dimensional generalized diffusion processes and a family of spectral measures. The spectral of limit process is represented by such family. In general, it is not easy to get such division of a space of parameters. We characterized a space of normalized measures, and hence we could control the behavior of measures. Thus we obtained a limit process.2.For generalized diffusion processes with discrete spectrum, we showed that there exists a nontrivial limit distribution of conditional distributions related to hitting times. We obtained an asymptotic behavior of transition probability conditioned by no hitting to the boundaries as time goes to infinity. Our results say that the asymptotic behavior is affected by the asymptotic behavior of sample paths near the boundaries. Further the asymptotic behavior drastically changes according to discrete or continuous spectrum.3.We defined models of percolation for Sierpinski carpet lattices and its family and showed that there exits a model for which there does not exist phase transition.