Diffusion on irregular sets

该项目的主要目的是了解高度不规则集上的扩散过程。分形几何是一种数学语言和学科，用于描述，研究和分析这种结构的属性。扩散过程模拟粒子在给定介质中的连续传播或热传导。均匀自相似分形上的扩散过程和调和结构理论已经被广泛研究，例如Barlow，Denker，Hambly，Hattori，Kigami，Lau，Lindstrom，Lapidus和梅斯.我们相信这个项目的结果将使我们更深入地了解自然界中高度不规则介质上的扩散行为。最近的研究表明，它可以应用于大脑皮层中的神经元流动、人肺中的氧气传输和岩石层中的气体传播。本项目的第一部分是通过随机游走在非均匀自相似、自仿射和自共形分形上构造扩散过程。将目前已知的理论扩展到这些更一般的分形将需要成熟的理论和创新的方法相结合，例如，更新理论，非稳态多类型分支过程和热力学形式主义。随机游走的方法允许一个直观的和几何的建设产生新的见解，这是类似于一个布朗运动的n维欧氏空间的建设。在我们的设置中，出现了几个实质性的困难，因为分形往往缺乏一定的规则性条件，如对称性。其中一些困难已经克服了一组特定的自相似分形使用的分析方法，由于Kigami，与各种进一步的发展，例如，弗赖贝格，Hambly，Hohartz和Tehartaev。 在这些情况下，我们相信，建设涉及随机游动将提供进一步的信息和洞察力的行为的扩散过程的分形，也允许进一步的generalisations.There是一个众所周知的对应关系之间的扩散过程和拉普拉斯。 在项目的第二部分中，我们将构建的扩散过程的转移密度的估计将产生产生的拉普拉斯算子的行走维数和谱维数。这些不同的维数概念将与集合的分形维数相联系。也就是说，我们将建立一个爱因斯坦般的关系，为广泛的一类分形-目前已知的自相似fractals.The最终目标是建立一个连接的扩散过程的非交换几何，也就是说，我们将比较由此产生的拉普拉斯平方提出的狄拉克运营商的主要调查员，法尔科纳，欣茨，凯莱赫，塞缪尔和Teplayev。此外，一类KMS-状态（吉布斯状态的概括）的动机在量子统计力学的概念，将被调查。

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.