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

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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.