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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, ZhongJin 
Department / Program:  Mathematics 
Discipline:  Mathematics 
Degree Granting Institution:  University of Illinois at UrbanaChampaign 
Degree:  Ph.D. 
Genre:  Dissertation 
Subject(s):  Subordinate Brownian motions
Perturbations Green function Boundary Harnack principle Martin boundary Minimal Martin boundary Trace 
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 twosided 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(xy^{2})}{\sqrt{\phi(\delta_D(x)^{2}) \phi(\delta_D(y)^{2})}}\right)\, \frac{1}{xy^d\, \phi(xy^{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:  20130524 
URI:  http://hdl.handle.net/2142/44247 
Rights Information:  Copyright 2013 Hyunchul Park 
Date Available in IDEALS:  20130524 
Date Deposited:  201305 
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Dissertations and Theses  Mathematics

Graduate Dissertations and Theses at Illinois
Graduate Theses and Dissertations at Illinois