Boundary theory of random walk and fractal analysis.

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