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

通过狄利克雷形式和等周长不等式构建多维扩散过程的新方法

• 批准号: 11440029
• 负责人: OSADA Hirofumi
• 金额: $ 8.26万
• 依托单位: Graduate School of Mathematics, Nagoya University
• 依托单位国家: 日本
• 项目类别: Grant-in-Aid for Scientific Research (B)
• 财政年份: 1999
• 资助国家: 日本
• 起止时间: 1999 至 2002
• 项目状态: 已结题
• 来源: https://kaken.nii.ac.jp/en/grant/KAKENHI-PROJECT-11440029/
• 关键词: diffusion processes; Dirichlet forms; Fractal; Sierpinski Carpet; Iso perimetic inequality; Marka process; Random Matrix; determinantal randomu-point field; Dirichlet form; heat kernel; ディリクレ形式; パーコレーション

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.

这项研究的目的是通过使用一种方法，负责人调查员开发的方法是在无限分支的分形上构建扩散，例如sierpinski地毯，配置空间和路径空间。这些空间令人感兴趣，很难通过使用通常的方法来构建不错的扩散过程。在分形的情况下，我们构建了相对于分形的Hausdroff措施，它们是自相似和可逆的。我们在这里使用了奇异时间变化的方法。至于随机分形，我们引入了具有统计自相似性的“气泡”。尽管我们通过基于Dirichlet形式方法的一般理论来构建扩散，但对它们的详细研究是Future的主题。发现了一些有关分形晶格的渗透的新事实。至于无限粒子系统，我们构建了固定度量的扩散，其固定度量是所谓的“决定性随机点”场。这类概率度量非常有趣，因为它们与随机矩阵理论和特殊功能（例如通风函数）有关。这类概率措施与Ruelle的类Gibbs度量不同，我想将来将进行广泛的研究。至于路径空间，我们构建了扩散的扩散，其不变的度量是吉布斯在路径空间上的测量，我们也在本研究的过程中构建了gibbs。

项目成果

Hariya,Yuu, Osada,Hirofumi: "Diffusion processes on path spaces with Interactions"Rev.Math.Phys.. 13,no.2. 199-220 (2001)

Hariya，Yuu，Osada，Hirofumi：“具有相互作用的路径空间上的扩散过程”Rev.Math.Phys.. 13，no.2。

Hattori, Tetsuya: "Renormalization group analysis of the self-avoiding paths on the $d$-dimensional SierpiY'nski gaskets"J. Statist. Phys. 109. 39-66 (2002)

Hattori，Tetsuya：“$d$ 维 SierpiYnski 垫片上自回避路径的重正化群分析”J.

Hattori,Tetsuya, Tsuda,Toshiro: "Renormalization group analysis of the self-avoiding paths on the $d$-dimensional SierpiY'nski gaskets"J.Statist.Phys.. 109,no.1-2. 39-66 (2002)

Hattori、Tetsuya、Tsuda、Toshiro：“$d$ 维 SierpiYnski 垫片上自回避路径的重正化群分析”J.Statist.Phys.. 109，no.1-2。

Ichihara, Kanji: "Long time asymptotic properties of heat kernels on negatively curved Riemannian manifolds"Infin. Dimens. Anal. Quantum Probab. Relat. Top. 4. 377-400 (2001)

Ichihara，Kanji：“负弯曲黎曼流形上热核的长期渐近性质”Infin。

Ichihara, Kanji: "Large deviation for pinned covering diffusion"Bull. Sci. Math.. 125. 529-551 (2001)

Ichihara, Kanji：“固定覆盖扩散偏差较大”公牛。