Potential theory of subordinate Brownian motions and their perturbations

Subordinate Brownian motions;Perturbations;Green function;Boundary Harnack principle;Martin boundary;Minimal Martin boundary;Trace

Park, Hyunchul

关键词: Subordinate Brownian motions;    Perturbations;    Green function;    Boundary Harnack principle;    Martin boundary;    Minimal Martin boundary;    Trace;   

Others  :  https://www.ideals.illinois.edu/bitstream/handle/2142/44247/Hyunchul_Park.pdf?sequence=1&isAllowed=y

美国|英语

来源: The Illinois Digital Environment for Access to Learning and Scholarship

【 摘 要 】

In this thesis, we study potential theoretic properties of harmonic functions and spectral problemsof 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\beqG_{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})}.\eeqIn chapter \ref{chp:BHP} we use the Green function comparability result to obtain a version of the boundary Harnack principle for positive harmonicfunctions 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 establishthe 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 $t\rightarrow 0$.