A new approach to the construction of multi-dimensional diffusion processes via Dirichlet forms and iso perimetric inequalities

本研究的目的是通过使用首席研究员开发的方法在无限分形上构建扩散，如Sierpinski地毯，构型空间和路径空间。这些空间是非常有趣的，很难用常规方法构造好的扩散过程。对于分形，我们构造了相对于分形上的Hausdroff测度具有自相似和可逆的扩散过程。这里我们用的是奇异时间变化法。对于随机分形，我们引入具有统计自相似性的“气泡”。虽然我们使用基于狄利克雷形式方法的一般理论来构造扩散，但对它们的详细研究是未来的主题。发现了一些关于分形晶格上渗流的新事实。对于无限粒子系统，我们构造的扩散的平稳测度是所谓的“决定论随机点场”。这类概率测度非常有趣，因为它们与随机矩阵理论和特殊函数（如Airy函数）有关。这类概率测度不同于Ruelle的吉布斯测度，我想，它们将在未来得到广泛的研究。对于路径空间，我们构造了路径空间上不变测度为吉布斯测度的扩散，我们在本研究过程中也构造了吉布斯测度。

The purpose of this research is by using a method the head investigator developed to construct diffusion on infinitely ramified fractals such as Sierpinski carpets, configuration spaces and path spaces. These space are outstandingly of interest and hard to construct nice diffusion processes by using usual methods.In case of fractals we construct diffusion processes which are self-similar and reversible with respect to the Hausdroff measures on the fractals. We used here the method of singular time change. As for random fractals we introduce "bubbles" which has a statistical self-similarity. Although we construct diffusion by using our general theory based on Dirichlet form approach, the detailed investigation of them are future's themes.Some new facts about percolation on fractal lattices are discovered. As for infinite particle systems, we construct diffusions whose stationary measures are so-called "determinantal random point fields". This class of probability measures is very interesting because they are related to random matrix theory and special functions such as Airy functions. This class of probability measures Are different from Ruelle's class Gibbs measures and, I suppose, will be studied extensively in future. As for path spaces we construct diffusions whose invariant measures are Gibbs measures on path spaces, which we also constructed in a course of this research.