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Title:Potential theory of subordinate Brownian motions and their perturbations
Author(s):Park, Hyunchul
Director of Research:Song, Renming
Doctoral Committee Chair(s):Bauer, Robert
Doctoral Committee Member(s):Sowers, Richard B.; Song, Renming; Ruan, Zhong-Jin
Department / Program:Mathematics
Degree Granting Institution:University of Illinois at Urbana-Champaign
Subject(s):Subordinate Brownian motions
Green function
Boundary Harnack principle
Martin boundary
Minimal Martin boundary
Abstract:In this thesis, we study potential theoretic properties of harmonic functions and spectral problems of a large class of L\'evy processes using probabilistic techniques. In chapter \ref{chp:Green} we prove sharp two-sided Green function estimates in bounded $\kappa$-fat domains $D$ for a large class of L\'evy processes, which can be considered as perturbations of certain subordinate Brownian motions. In particular, we prove that in bounded $C^{1,1}$ domains $D$, the Green function $G_{D}^{Y}(x,y)$ of symmetric L\'evy processes $Y$ whose L\'evy densities are close to those of certain subordinate Brownian motions with characteristic exponent $\Psi(|\xi|)=\phi(|\xi|^{2})$ satisfies \beq G_{D}^{Y}(x,y) \asymp \left(1 \wedge \frac{\phi(|x-y|^{-2})}{\sqrt{\phi(\delta_D(x)^{-2}) \phi(\delta_D(y)^{-2})}}\right)\, \frac{1}{|x-y|^d\, \phi(|x-y|^{-2})}. \eeq In chapter \ref{chp:BHP} we use the Green function comparability result to obtain a version of the boundary Harnack principle for positive harmonic functions that vanish outside a part of the boundary of $D$ and some small ball with respect to perturbations of SBMs in bounded $\kappa$-fat domains $D$. In chapter \ref{chp:Martin} we use the boundary Harnack principle to prove that the Martin boundary and the minimal Martin boundary of $\kappa$-fat domains $D$ with respect to $Y$ can be identified with the Euclidean boundary of $D$. In chapter \ref{chp:trace} we turn our attention to some spectral problems about relativistic stable processes. We establish the asymptotic expansion of the trace (partition function) $Z_{D}^{m}(t)$ of relativistic stable processes on bounded $C^{1,1}$ open sets and Lipschitz open sets as $t\rightarrow 0$.
Issue Date:2013-05-24
Rights Information:Copyright 2013 Hyunchul Park
Date Available in IDEALS:2013-05-24
Date Deposited:2013-05

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