S-Space College of Natural Sciences (자연과학대학) Dept. of Mathematical Sciences (수리과학부) Theses (Ph.D. / Sc.D._수리과학부)
Boundary behavior of harmonic functions for subordinate Brownian motion
- 자연과학대학 수리과학부
- Issue Date
- 서울대학교 대학원
- subordinate Brownian motion; relative Fatou type theorem; Martin kernel; Martin boundary; harmonic function; Martin representation
- 학위논문 (박사)-- 서울대학교 대학원 : 수리과학부, 2013. 2. 김판기.
- 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.