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来构建扩散的无限分歧分形，如谢尔宾斯基地毯，配置空间和路径空间。在分形空间中，我们构造了关于分形上的Hausdroff测度是自相似的、可逆的扩散过程。我们在这里使用奇异时变的方法。对于随机分形，我们引入了具有统计自相似性的“气泡”。尽管我们是使用基于狄利克雷形式方法的一般理论来构造扩散的，但对其进行详细的研究是未来的主题，并发现了有关分形晶格上渗透的一些新事实。对于无限粒子系统，我们构造了其定常测度为“行列式随机点场”的扩散。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类Gibbs测度，我想，这类测度将在将来得到广泛的研究。对于路径空间，我们构造了其不变测度为路径空间上的Gibbs测度的扩散，这也是我们在研究过程中构造的。

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.