Wong, Ting Kam Leonard., Thesis (M.Phil.)--Chinese University of Hong Kong, 2011., Includes bibliographical references (leaves 91-97) and index., s in English and Chinese., Chapter 1 --- Introduction --- p.6, Chapter 1.1 --- Problems of fractal analysis --- p.6, Chapter 1.2 --- The boundary theory approach --- p.7, Chapter 1.3 --- Summary of the thesis --- p.9, Chapter 2 --- Martin boundary --- p.13, Chapter 2.1 --- Markov chains and discrete potential theory --- p.13, Chapter 2.2 --- Martin compactification --- p.18, Chapter 2.3 --- Convergence to boundary and integral representations --- p.20, Chapter 2.4 --- Dirichlet problem at infinity --- p.25, Chapter 3 --- Hyperbolic boundary --- p.27, Chapter 3.1 --- Random walks on infinite graphs --- p.27, Chapter 3.2 --- Hyperbolic compactification --- p.31, Chapter 3.3 --- Ancona's theorem --- p.33, Chapter 3.4 --- Self-similar sets as hyperbolic boundaries --- p.34, Chapter 3.5 --- Hyperbolic compactification of augmented rooted trees --- p.44, Chapter 4 --- Simple random walk on Sierpinski graphs --- p.47, Chapter 4.1 --- Hcuristic argument for d = 1 --- p.48, Chapter 4.2 --- Symmetries and group invariance --- p.51, Chapter 4.3 --- Reflection principle --- p.54, Chapter 4.4 --- Self-similar identity and hitting distribution --- p.60, Chapter 4.5 --- Remarks and Open Questions --- p.64, Chapter 5 --- Induced Dirichlet forms on self-similar sets --- p.66, Chapter 5.1 --- Basics of Dirichlet forms --- p.67, Chapter 5.2 --- Motivation: the classical Douglas integral --- p.68, Chapter 5.3 --- Graph energy and the induced forms --- p.69, Chapter 5.4 --- Induced Dirichlet forms on self-similar sets --- p.74, Chapter 5.5 --- A uniform tail estimate via coupling --- p.83, Chapter 5.6 --- Remarks and open questions --- p.89, Index of selected terms --- p.98, http://library.cuhk.edu.hk/record=b5894522, Use of this resource is governed by the terms and conditions of the Creative Commons “Attribution-NonCommercial-NoDerivatives 4.0 International” License (http://creativecommons.org/licenses/by-nc-nd/4.0/)