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Diffusion on irregular sets

不规则集合上的扩散

基本信息

批准号：
281034495
负责人：
Professor Dr. Marc Keßeböhmer
金额：
--
依托单位：
Fachbereich 03: Mathematik und Informatik
依托单位国家：
德国
项目类别：
Research Grants
财政年份：
2015
资助国家：
德国
起止时间：
2014-12-31 至 2019-12-31
项目状态：
已结题

来源：
https://gepris.dfg.de/gepris/projekt/281034495?language=en
关键词：
Diffusion irregular sets

项目摘要

The leading aim of this project is to understand diffusion processes on highly irregular sets. Fractal geometry is a mathematical language and discipline used to describe, study and analyse properties of such structures. A diffusion process models the continuous propagation of a particle or heat conduction in a given medium. The theory of diffusion processes and harmonic structures on homogenous self-similar fractals has been intensively investigated, for example by Barlow, Denker, Hambly, Hattori, Kigami, Lau, Lindstrom, Lapidus and Metz.The results of this project we believe will give a deeper insight into the behaviour of diffusion on highly irregular media in nature. Recent investigations have shown that applications can be found in, for instance, neuronal flows in the brain cortex, oxygen transport in the human lung and gas propagation in rock layers.The first part of the project is to construct diffusion processes on inhomogenous self-similar, self-affine and self-conformal fractals via random walks. The extension of the current known theory to these more general fractals will require a combination of well-established theories and innovative methods, for example, renewal theory, non-stationary multi-type branching processes and thermodynamic formalism. The random walk approach allows for an intuitive and geometric construction yielding new insights, and which is similar to a construction of a Brownian motion on n-dimensional Euclidean space. In our setting, several substantial difficulties arise in that fractals often lack certain regularity conditions, such as symmetry. Some of these difficulties have been overcome on a specific set of self-similar fractals by using an analytic approach due to Kigami, with various further developments by, for instance, Freiberg, Hambly, Strichartz and Teplyaev. In these cases, we are confident that the construction involving random walks will give further information and insight on the behaviour of diffusion processes on fractals and also allow for further generalisations.There is a well-known correspondence between diffusion processes and Laplacians. In the second part of the project, estimates on the transition density of the diffusion processes we will construct will yield the walk dimension and the spectral dimension of the resulting Laplacian. These different notions of dimension shall then be related to a fractal dimension of the set. Namely, we will establish an Einstein-like relation for a wide class of fractals - currently known for self-similar fractals only.The final objective is to establish a connection of diffusion processes to non-commutative geometry; that is, we will compare the resulting Laplacian to the square of Dirac operators proposed by the principle investigator, Falconer, Hinz, Kelleher, Samuel and Teplayev. Further, a class of KMS-states (a generalisation of Gibbs states) motivated by notions in quantum statistical mechanics, will be investigated.

这个项目的主要目标是了解高度不规则集合上的扩散过程。分形几何是一门用来描述、研究和分析这类结构性质的数学语言和学科。扩散过程模拟粒子或热传导在给定介质中的连续传播。均匀自相似分形上的扩散过程和调和结构的理论已经得到了深入的研究，例如Barlow，Denker，Hamble，Hattori，Kigami，Lau，Lindstrom，Lapidus和Metz。我们相信这个项目的结果将使我们对自然界中高度不规则的介质上的扩散行为有更深入的了解。最近的研究表明，它在大脑皮层的神经元流动、人类肺部的氧气传输和岩层中的气体传播等方面都有应用。该项目的第一部分是通过随机行走在非均匀自相似、自仿射和自共形分形上构建扩散过程。将目前已知的理论扩展到这些更一般的分形学将需要成熟的理论和创新方法的结合，例如更新理论、非平稳多类型分支过程和热力学形式论。随机游走方法允许产生新见解的直观和几何构造，并且类似于n维欧氏空间上的布朗运动的构造。在我们的设置中，出现了几个实质性的困难，因为分形图通常缺乏某些规则条件，如对称性。其中一些困难已经通过使用分析方法在一组特定的自相似分形图上克服了，这要归功于Kigami，以及例如Freiberg、Hamble、Strichartz和Teplyaev的各种进一步发展。在这些情况下，我们相信，涉及随机游动的构造将提供关于扩散过程在分形图上的行为的进一步信息和洞察，并允许进一步的推广。扩散过程和拉普拉斯之间存在众所周知的对应。在项目的第二部分，对我们将要构造的扩散过程的跃迁密度的估计将得到所产生的拉普拉斯的步长维度和谱维度。然后，这些不同的维度概念应与集合的分维相关联。也就是说，我们将为一大类分形图建立一个类似爱因斯坦的关系--目前已知的只有自相似分形图。最终目标是建立扩散过程与非对易几何之间的联系；也就是说，我们将把得到的拉普拉斯算符与主要研究者Falconer，Hinz，Kelleher，Samuel和Teplayev提出的Dirac算子的平方进行比较。此外，还将研究由量子统计力学中的概念驱动的一类KMS态(Gibbs态的推广)。