On Potential Theory of Markov Processes with Jump Kernels Decaying at the Boundary
نویسندگان
چکیده
Motivated by some recent potential theoretic results on subordinate killed L\'evy processes in open subsets of the Euclidean space, we study an set $D\subset {\mathbb R}^d$ defined via Dirichlet forms with jump kernels form $J^D(x,y)=j(|x-y|)\mathcal{B}(x,y)$ and critical killing functions. Here $j(|x-y|)$ is density isotropic stable process (or more generally, a pure unimodal process) $\mathbb{R}^d$. The main novelty that term $\mathcal{B}(x,y)$ tends to 0 when $x$ or $y$ approach boundary $D$. Under general assumptions $\mathcal{B}(x,y)$, construct corresponding prove non-negative harmonic functions satisfy Harnack inequality Carleson's estimate. We give several examples terms satisfying those assumptions. depend four parameters, $\beta_1, \beta_2, \beta_3$, $\beta_4$, roughly governing decay near In second part this paper, specialise case half-space $D=\mathbb{R}_+^d=\{x=(\widetilde{x},x_d):\, x_d>0\}$, $\alpha$-stable kernel $j(|x-y|)=|x-y|^{-d-\alpha}$ function$\kappa(x)=c x_d^{-\alpha}$, $\alpha\in (0,2)$, where $c$ positive constant. Our result principle which says that, for any $p>(\alpha-1)_+$, there are values parameters constant such must at rate $x_d^p$ if they vanish portion boundary. further show fails despite fact estimate valid.
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ژورنال
عنوان ژورنال: Potential Analysis
سال: 2021
ISSN: ['1572-929X', '0926-2601']
DOI: https://doi.org/10.1007/s11118-021-09947-8