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Project Details
Funding Scheme : General Research Fund
Project Number : 401710
Project Title(English) : Boundary Theory and Analysis on Fractals 
Project Title(Chinese) : 分形上的邊界理論及分析 
Principal Investigator(English) : Prof Lau, Ka-sing 
Principal Investigator(Chinese) : 劉家成 
Department : Dept of Mathematics
Institution : The Chinese University of Hong Kong
E-mail Address : kslau@math.cuhk.edu.hk 
Tel : 2609 7966 
Co - Investigator(s) :
Prof Ngai, Sze-Man
Panel : Physical Sciences
Subject Area : Mathematics
Exercise Year : 2010 / 11
Fund Approved : 930,800
Project Status : Completed
Completion Date : 31-12-2013
Project Objectives :
Our investigation will focus mainly on connected self-similar $K$ that satisfy the open set condition. We want to find the admissible Markov chains on $K$, i.e., those chains whose Martin boundaries can be identified with $K$. The related Poisson boundary will also be considered.
We propose to use the harmonic structure of the Martin boundary and the Poisson boundary to study the existence problem of the Laplacian on a self-similar set. We will seek additional conditions on the self-similar set or the Markov chain for this purpose. We also propose studying the problem in another direction: Approximate the fractal set with open domains from outside, and use the limit of the classical Laplacians as the Laplacian on $K$.
We will apply the above results to study the intrinsic properties of the self-similar sets such as the spectral dimension, the walk dimension as well as the eigen-problem, the Green function, the heat kernel and the energy measure that arise from the Laplacian.
Abstract as per original application
(English/Chinese):

Realisation of objectives: Our goal was to identify the self-similar sets with the Martin boundaries of transient Markov chains, and to introduce the discrete potential theory on Markov chains to the self-similar sets. We used this to consider the existence of a Laplacian $Delta$, or more generally, a Dirichlet form on self-similar sets $K$. A self-similar set $K$ is defined by the limit set of an iterated function system, and the iteration is addressed by a symbolic space of finite words which form a tree. We introduced different Markov chains on such trees and study their limits (the Martin boundary). In particular, we considered two most basic situations, the Sierpinski gasket and the Hata tree [1,2], which revealed the essential features in this approach, and showed the compatibility with the known theory. We also studied another type of Markov chain on the symbolic space, which extended the Denker-Sato chain on the Sierpinski gasket to the most general setting [3]. These investigations set up a clear direction for the study of the class of post-critical finite self-similar sets. In another direction, we assume the existence of the Laplacian and Dirichlet forms on the fractal spaces, or more generally, on the metric measure space, and consider how the classical theory can be carried over. In [4] we studied the two-sided estimates of heat kernels with finite effective resistance. This setup is different from the above self-similar setting, and so is the associated Markov processes. It brings in another aspect of the analysis of fractals.
Summary of objectives addressed:
Objectives Addressed Percentage achieved
1.Our investigation will focus mainly on connected self-similar $K$ that satisfy the open set condition. We want to find the admissible Markov chains on $K$, i.e., those chains whose Martin boundaries can be identified with $K$. The related Poisson boundary will also be considered. Yes100%
2.We propose to use the harmonic structure of the Martin boundary and the Poisson boundary to study the existence problem of the Laplacian on a self-similar set. We will seek additional conditions on the self-similar set or the Markov chain for this purpose. We also propose studying the problem in another direction: Approximate the fractal set with open domains from outside, and use the limit of the classical Laplacians as the Laplacian on $K$. Yes95%
3.We will apply the above results to study the intrinsic properties of the self-similar sets such as the spectral dimension, the walk dimension as well as the eigen-problem, the Green function, the heat kernel and the energy measure that arise from the Laplacian. Yes95%
Research Outcome
Major findings and research outcome: In [1], we defined a new Markov chain on the symbolic space representing the Sierpinski gasket (SG), and showed that the corresponding Martin boundary is homeomorphic to the SG while the minimal Martin boundary is the three vertices of the SG. In addition, the $P$-harmonic structure induced by the Markov chain coincides with the canonical one on the SG. In [2], we ran the same type of Markov chain on the symbolic space of the Hata tree $K$, a non-symmetric case. In this case, the Martin boundary ${mathcal M}$ is not identify with $K$, it is homeomorphic to the "trunk" of $K$, and the minimal Martin boundary is the three vertices of the trunk. Still, the class of P-harmonic functions on ${mathcal M}$ coincides with Kigami's class of harmonic functions on K. The two special cases in [1], [2] suggested another approach to consider the existence of Laplacian on those self-similar sets for which the problem is still not settled. In [3], we considered a more general Denker-Sato type Markov chain associated with the self-similar sets $K$ with the open set condition. The chain is defined on the augmented tree of a symbolic space. Such tree was introduced by Kaimanovich, it is hyperbolic in the sense of Gromov. We showed that the Martin boundary, the hyperbolic boundary and the self-similar set $K$ are homeomorphic. The hitting distribution of the chain was also obtained. It brought together the probabilistic, geometric and fractal considerations. In [4], we carried out another direction of investigation. We studied certain class of non-local regular Dirichlet forms on metric measure spaces, including fractal spaces. We used a purely analytic method such as the parabolic maximum principle. We deduced an off-diagonal upper bound of the heat kernel from the on-diagonal one under the volume regularity hypothesis, restriction of the jump kernel and the survival hypothesis. As an application, we obtain two-sided estimates of heat kernels for non-local regular Dirichlet forms with finite effective resistance, including settings with the walk dimension greater than 2.
Potential for further development of the research
and the proposed course of action:
Up to now, there are two basic approaches to construct the Laplacian on self-similar fractals: the discrete approximation on the p.c.f. fractals, as well as the probabilistic method on the finitely ramified fractals and the Sierpinski carpet. The approach in [1-3] using the Markov change offers a new way to look for the Laplacian and Dirichlet forms on a fractal set $K$. This is one of our future lines of investigation. On the other hand, by assuming the existence of the Laplacian, Dirichlet forms and harmonic structure, there is a wide scope of topics to be considered. Our investigation in [4] is related to diffusion and heat conduction in the general metric measure space. Some shaper estimate for the more specific cases are to be studied.
Layman's Summary of
Completion Report:
There is a lot interest to extend and assimilate the classical theories and ideas on analysis and probability theory to the fractal setting. One of the central questions is to establish the existence of the {it Laplace operator} on fractal sets. Another problem is to study the consequence and properties of the operator, for example, diffusion and heat conduction. In this project we identified certain classes of the self-similar sets and the boundaries of Markov chains. As the boundary theory is well connected with harmonic analysis, potential theory, as well as random walks on groups and graphs, we successfully integrated this theory into the study of fractals. In particular, by using this, we brought in a new method to study the Laplacian on fractals different from the existing approaches, which are still limited to small classes of fractal sets. In regard to the application of such operator, we studied the heat kernel on the general metric measure spaces, which include the classical Euclidean spaces and the fractal spaces.
Research Output
Peer-reviewed journal publication(s)
arising directly from this research project :
(* denotes the corresponding author)
Year of
Publication
Author(s) Title and Journal/Book Accessible from Institution Repository
2012 Lau Ka-Sing & Ngai Sze-Man*  Martin boundary and exit space on the Sierpinski gasket, Sc. China, Math, 55 (2012) 475-494  No 
2014 Ka-Sing Lau & Sze-Man Ngai*  Boundary theory on the Hata tree, Nonlinear Analysis, vol.95, 292-307 (2014)  No 
Ka-Sing Lau and Xiang-Yang Wang*  Denker-Sato type Markov Chains on Self-similar sets, Math Zeitschrift (to appear)  No 
2014 Alexander Grigor'yan, Jiaxin Hu and Ka-Sing Lau  Estimates of heat kernels for non-local regular Dirichlet forms, Transaction of the American Mathematical Society, vol.366 (12), 6391-6441 (2014)  No 
Recognized international conference(s)
in which paper(s) related to this research
project was/were delivered :
Month/Year/City Title Conference Name
March 2011, Georgia Self-similar sets and Martin boundaries  AMS Regional Meeting, Statesboro 
May 2011, Zhangjiajie Boundary theory on self-similar sets  2011 National conference on Fractals and Dynamical Systems 
June 2011, Porquerolles Self-similar sets and Martin bouondary  Fractals and Related Topics II 
September 2011, Ithaca Random walk and induced Dirichlet form on self-similar set  Conference on Analysis on Fractals 
Other impact
(e.g. award of patents or prizes,
collaboration with other research institutions,
technology transfer, etc.):

  SCREEN ID: SCRRM00542