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Project Details 
Funding Scheme :  General Research Fund  
Project Number :  14316816  
Project Title(English) :  Induced Dirichlet forms on Selfsimilar sets  
Project Title(Chinese) :  自相似集上的狄克雷型  
Principal Investigator(English) :  Prof Feng, DeJun  
Principal Investigator(Chinese) :  
Department :  Dept of Mathematics  
Institution :  The Chinese University of Hong Kong  
Email Address :  djfeng@math.cuhk.edu.hk  
Tel :  26097965  
Co  Investigator(s) : 


Panel :  Physical Sciences  
Subject Area :  Mathematics  
Exercise Year :  2016 / 17  
Fund Approved :  727,647  
Project Status :  Completed  
Completion Date :  31122019  
Project Objectives : 


Abstract as per original application (English/Chinese): 
分形是一類不規則，複雜的集合或現象，它們具有自相似性質 及分數維數。分形在自然界及人類生活領域中經常出現，例如 海岸綫形狀，地層的組成，天體星雲的分布，DNA樣本，微血 管及肺的結構以及股票的上落數據曲綫。科學家及工程師們利 用分形作圖像壓縮，手機上天綫的設計，石油地質的探測。 醫學上，分形維數的計算，亦可用來顯示異常細胞的出現。 在這個建議書中，我們提出研究分形上的狄克雷型（或稱能量 型）。這套理論包括重要的拉普拉斯算子。我們知道，在經典 分析，熱傳導及波傳播以及許多在偏微分方程上的問題，都依 賴於拉氏算子。這類算子由微分定義，但分形是絶對非光滑的， 微分在上面沒有定義。因此所有經典上的概念及理論，都需要 有適當的調整。在分形上，拉氏算子的存在性，成為一個中心 的問題。在這個計劃上，我們將會利用分析及概率上的工具， 對這個問題作一個深入的研究。 

Realisation of objectives:  In a previous project ``Tree structure and boundary theories on fractals"(Project 1430221), we proved that on the tree of symbolic space of a selfsimilar set, we can add new edges to the tree according the neighboring cells to obtain an augmented tree; the augmented tree is hyperbolic, and the selfsimilar set is Holder equivalent to the hyperbolic boundary. The importance of this result is that if we run a reversible transient random walk on a hyperbolic graph, then the limits of the sample paths, called Martin boundary, is homeomorphic to the hyperbolic boundary, and hence also homeomorphic to the selfsimilar sets under consideration. It is wellknown that in the study of Markov chains, there is a discrete potential theory to investigate the harmonic structure, Laplacian and energy of the system. We can therefore transfer this theory onto selfsimilar sets. In this regard, our first major achievement in this project [1] is to investigate a class of random walk on the augmented tree that have return ratio lambda, we call it lambdanatural random walk (NRW). The corresponding energy on the graph induces a nonlocal regular Dirichlet form on the selfsimilar set. Through the hyperbolic structure, we obtain an explicit expression of the kernel of the Dirichlet form, which is corresponding to a fractional Laplacian. The nonlocal Dirichlet forms arisen has the class of Besov spaces as its domain. The parameter beta in the expression of the Dirichlet form is interesting; it depends on the return ratio of the random walk, and the domain can be trivial when beta is large. The critical lambda is the largest value that the Besov space is nontrivial, it carries a lot of important information of the underlying set; in particular in the cases of the Sierpinski gasket and some other standard fractals, the critical exponent is where the local regular Dirichlet form is located. A local regular Dirichlet form associates with a Laplacian, and is of central importance in the study of analysis on fractal. We have considerable progress in in that aspect [2,4]. Recently, we discover that on a compact space of homogenous type, there is a refining net of partitions of the space, which gives a hyperbolic augmented tree, and our consideration of the boundary theories can be applied to those sets. The class of space of homogenous type is very broad, it covers the classical domains, Reimannian manifolds, and the fractal sets called dsets, it is known for a long time, and has been studied extensively in harmonic analysis and Hpspace theory. We can extend the previous results to this much broader setting [3]. In a recent paper [5], we study the convergence properties of Besov norms on homogeneous p.c.f. selfsimilar sets near critical exponent sigmastar, and that of the associated Dirichlet forms. Our convergence result extends a celebrate theorem of Bourgain, Brezis and Mironescu on Euclidean space.  
Summary of objectives addressed: 


Research Outcome  
Major findings and research outcome:  In [1] we investigate a class of random walk on the augmented tree that have return ratio lambda, we call it lambdanatural random walk (NRW). The corresponding energy on the graph induces a nonlocal regular Dirichlet form on the selfsimilar set. Through the hyperbolic structure, we obtain an explicit expression of the kernel of the Dirichlet form, which is corresponding to a fractional Laplacian. In [2], we study the partition systems and the tree structures for compact spaces of homogeneous type and establish the hyperbolicity of the augmented trees. We investigate the λNRW on the augmented trees; we identify the underlying set with the hyperbolic boundary and the Martin boundary. The estimations of the Martin kernel and the Naim kernel are stated. Finally, we study the induced Dirichlet forms and give a criterion of the critical exponents. In [4], we study Dirichlet forms and critical exponents of fractals and explore new situations that the underlying fractal sets admit inhomogeneous resistance scalings, which yield two types of critical exponents. We restrict our consideration on the p.c.f. (post critically finite) sets. We first develop a technique of quotient networks to study the general theory of these critical exponents. We then construct two asymmetric p.c.f. sets, and use them to illustrate the theory and examine the function properties of the associated Besov spaces at the critical exponents; the various Dirichlet forms on these fractals will also be studied.  
Potential for further development of the research and the proposed course of action: 
To develop the theory of analysis (Dirichlet forms and Laplacians) for some nonp.c.f. fractals.  
Layman's Summary of Completion Report:  In this proposal, we study Dirichlet forms (equivalently, energy forms) on fractals. The theory includes the important Laplace operator. As is known, in classical analysis, the theories of heat diffusion, wave propagation and all considerations in partial differential equations depend on the Laplacian. The operator is differential in nature while fractals are highly nonsmooth, and hence many classical perceptions need to be adjusted. Therefore defining the Laplacian on fractals and studying its properties are some central problems in the subject. In this project we obtained some new and interesting properties of the Dirichlet forms and Laplace operators on selfsimilar sets and compact spaces of homogeneous type. These works lead to better understanding of analysis on fractals.  
Research Output  
Peerreviewed journal publication(s) arising directly from this research project : (* denotes the corresponding author) 


Recognized international conference(s) in which paper(s) related to this research project was/were delivered : 


Other impact (e.g. award of patents or prizes, collaboration with other research institutions, technology transfer, etc.): 
SCREEN ID: SCRRM00542 