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Boundary behavior of harmonic functions for subordinate Brownian motion

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Authors
이윤주
김판기
Major
자연과학대학 수리과학부
Issue Date
2013-02
Publisher
서울대학교 대학원
Keywords
subordinate Brownian motionrelative Fatou type theoremMartin kernelMartin boundaryharmonic functionMartin representation
Description
학위논문 (박사)-- 서울대학교 대학원 : 수리과학부, 2013. 2. 김판기.
Abstract
In this thesis, we establish an oscillation estimate of nonnegative harmonic functions for a pure-jump subordinate Brownian motion. The infinitesimal generator of such subordinate Brownian motion $X$ is an integro-differential operator. As an application, we give a probabilistic proof of the following form of relative Fatou theorem for such subordinate Brownian motion $X$ in a bounded $\kappa$-fat open set
if $u$ is a positive harmonic function with respect to $X$ in a bounded $\kappa$-fat open set $D$ and $h$ is a positive harmonic function in $D$ vanishing on $D^c$, then the non-tangential limit of $u/h$ exists almost everywhere with respect to the Martin-representing measure of $h$. Under the gaugeability assumption, relative Fatou theorem is true for operators obtained from the generator of pure-jump subordinate Brownian motion in bounded $\kappa$-fat open set $D$ through non-local Feynman-Kac transforms.
Language
English
URI
https://hdl.handle.net/10371/121260
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College of Natural Sciences (자연과학대학)Dept. of Mathematical Sciences (수리과학부)Theses (Ph.D. / Sc.D._수리과학부)