Random Walks and Induced Dirichlet Forms on Self-similar Sets

• Other Authors: Kong, Shilei (author.)
• Format: Others
• Language: English; Chinese
• Published: 2017
• Online Access: http://repository.lib.cuhk.edu.hk/en/item/cuhk-1292202

眾所週知，一個自相似集K 可等同於它關聯的擴充樹(augmented tree) (X,E) 的雙曲邊界(hyperbolic boundary) ∂HX。在本文裡，我們將上述思路推 廣到緊的α 正則度量測度空間(K,ρ,µ)，併研究一類在(X,E) 上返比(return ratio) 為λ ∈ (0,1) 的可逆(reversible) 隨機游動。我們證明Martin 邊界可以與 ∂HX 及K 等同，並且游動的首中分佈(hitting distribution) 即為測度µ。 通過這一設定結合Silverstein 的方法，我們能夠得到對Martin 核(Martin kernel) 和Naim 核(Naim kernel) 的包含Gromov 積(Gromov product) 項的雙 向估計；同時，X上的離散能量EX 誘導出了一個K 上的能量型(energy form) EK，Naim 核則作為它的跳核(jump kernel) Θ(ξ,η) ρ(ξ,η)−(α+β)，這裡β 依 賴於λ，由此推出EK 的定義域是一類Besov 空間Λα,β/2 2,2 。 為使EK 成為一個非局部正則狄氏型(non-local regular Dirichlet form)， 我們進一步考察了兩個能量型EX 和EK 之間的函數關係，併通過EX 的 等效電阻(effective resistances) 給出了對Besov 空間Λα,β/2 2,2 鄰界指數(critical exponents) 的判定法則。在本文最後，我們運用網絡簡化(network reduction)的方法計算了某些自相似集上的指數。 === It is known that a self-similar set K can be identified with the hyperbolic boundary ∂HX of its associated augmented tree (X,E). In this thesis, we extend the above consideration to compact α-regular metric measure spaces (K,ρ,µ), and study certain reversible random walks with return ratio λ ∈ (0,1) on (X,E). We show that the Martin boundary M can be identified with ∂HX and K, and the hitting distribution of the walk is exactly the measure µ. With this setup and a device of Silverstein, we are able to obtain a two-sided estimates of the Martin kernel and the Naim kernel in terms of the Gromov product; meanwhile, the discrete energy EX on X induces an energy form EK on K. The Naim kernel turns out to be a jump kernel Θ(ξ,η) ρ(ξ,η)−(α+β) where β depends on λ, which implies that the domain ofEK is a Besov space Λα,β/2 2,2 . In order for thisEK to be a non-local regular Dirichlet form, we further investigate the functional relationship of two energy forms EX and EK, and provide some criteria to determine the critical exponents of Besov spaces Λα,β/2 2,2 through the effective resistances of EX. By means of network reduction, we calculate the exponents for some concrete examples on self-similar sets at the end of the thesis. === Kong, Shilei. === Thesis Ph.D. Chinese University of Hong Kong 2017. === Includes bibliographical references (leaves ). === Abstracts also in Chinese. === Title from PDF title page (viewed on …). === Detailed summary in vernacular field only.