Naslov
On Potential Theory of Markov Processes with Jump Kernels Decaying at the Boundary

Autori
Kim, Panki ; Song, Renming ; Vondraček, Zoran

Izvornik
Potential analysis (0926-2601) 58 (2023); 465-528

Vrsta, podvrsta i kategorija rada
Radovi u časopisima, članak, znanstveni

Ključne riječi
Jump processes ; Jumping kernel with boundary part ; Harnack inequality ; Carleson estimate ; Boundary Harnack principle

Sažetak
Motivated by some recent potential theoretic results on subordinate killed L\'evy processes in open subsets of the Euclidean space, we study processes in an open set $D\subset \R^d$ defined via Dirichlet forms with jump kernels of the form $J^D(x, y)=j(|x-y|)\sB(x, y)$ and critical killing functions. Here $j(|x-y|)$ is the L\'evy density of an isotropic stable process (or more generally, a pure jump isotropic unimodal L\'evy process) in $\R^d$. The main novelty is that the term $\sB(x, y)$ tends to 0 when $x$ or $y$ approach the boundary of $D$. Under some general assumptions on $\sB(x, y)$, we construct the corresponding process and prove that non-negative harmonic functions of the process satisfy the Harnack inequality and Carleson's estimate. We give several examples of boundary terms satisfying those assumptions. The examples depend on four parameters, $\beta_1, \beta_2, \beta_3$, $\beta_4$, roughly governing the decay of the boundary term near the boundary of $D$. In the second part of this paper, we specialize to the case of the half- space $D=\R_+^d=\{; ; ; ; ; x=(\wt{; ; ; ; ; x}; ; ; ; ; , x_d):\, x_d>0\}; ; ; ; ; $, the $\alpha$-stable kernel $j(|x-y|)=|x-y|^{; ; ; ; ; -d-\alpha}; ; ; ; ; $ and the killing function $\kappa(x)=c x_d^{; ; ; ; ; -\alpha}; ; ; ; ; $, $\alpha\in (0, 2)$, where $c$ is a positive constant. Our main result in this part is a boundary Harnack principle which says that, for any $p>(\alpha- 1)_+$, there are values of the parameters $\beta_1, \beta_2, \beta_3$, $\beta_4$, and the constant $c$ such that non-negative harmonic functions of the process must decay at the rate $x_d^p$ if they vanish near a portion of the boundary. We further show that there are values of the parameters $\beta_1, \beta_2, \beta_3$, $\beta_4$, for which the boundary Harnack principle fails despite the fact that Carleson's estimate is valid.

Izvorni jezik
Engleski

Znanstvena područja
Matematika

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