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

这项研究的目的是使用首席研究员开发的一种方法来构造无限分支分形上的扩散，例如Sierpinski地毯、配置空间和路径空间。这些空间是非常有趣的，而且很难用通常的方法来构造良好的扩散过程。对于分形图，我们构造了关于分形图上的Hausdroff测度的自相似且可逆的扩散过程。我们在这里使用了奇异时变法。对于随机分形，我们引入了具有统计自相似性的“气泡”。虽然我们使用基于Dirichlet形式方法的一般理论来构造扩散，但对它们的详细研究仍是未来的主题。对于无限粒子系统，我们构造了其定态度量为所谓的“行列式随机点场”的扩散。这类概率度量非常有趣，因为它们与随机矩阵理论和特殊函数(如Ary函数)有关。这类概率度量不同于Ruelle的Gibbs度量，我认为它将在未来得到广泛的研究。对于路径空间，我们在路径空间上构造了以Gibbs测度为不变测度的扩散，这也是我们在研究过程中所构造的。

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.