TY - JOUR

T1 - On Potential Theory of Markov Processes with Jump Kernels Decaying at the Boundary

AU - Kim, Panki

AU - Song, Renming

AU - Vondraček, Zoran

N1 - Funding Information:
Renming Song: Research supported in part by a grant from the Simons Foundation (#429343, Renming Song)
Funding Information:
We thank the referee for helpful comments on the first version of this paper. Part of the research for this paper was done while the second-named author was visiting Jiangsu Normal University, where he was partially supported by a grant from the National Natural Science Foundation of China (11931004) and by the Priority Academic Program Development of Jiangsu Higher Education Institutions.
Funding Information:
Panki Kim: This work was supported by the National Research Foundation of Korea(NRF) grant funded by the Korea government(MSIT) (No. NRF-2021R1A4A1027378)
Funding Information:
Zoran Vondraček: Research supported in part by the Croatian Science Foundation under the project 4197. Acknowledgements
Publisher Copyright:
© 2021, The Author(s), under exclusive licence to Springer Nature B.V.

PY - 2021

Y1 - 2021

N2 - Motivated by some recent potential theoretic results on subordinate killed Lévy processes in open subsets of the Euclidean space, we study processes in an open set D⊂ ℝd defined via Dirichlet forms with jump kernels of the form JD(x, y) = j(| x− y|) B(x, y) and critical killing functions. Here j(|x − y|) is the Lévy density of an isotropic stable process (or more generally, a pure jump isotropic unimodal Lévy process) in ℝd. The main novelty is that the term B(x, y) tends to 0 when x or y approach the boundary of D. Under some general assumptions on B(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, β1, β2, β3, β4, , 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=(x~,xd):xd>0}, the α-stable kernel j(|x − y|) = |x − y|−d−α and the killing function κ(x)=cxd−α, α ∈ (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 > (α − 1)+, there are values of the parameters β1, β2, β3, β4, and the constant c such that non-negative harmonic functions of the process must decay at the rate xdp if they vanish near a portion of the boundary. We further show that there are values of the parameters β1, β2, β3, β4, for which the boundary Harnack principle fails despite the fact that Carleson’s estimate is valid.

AB - Motivated by some recent potential theoretic results on subordinate killed Lévy processes in open subsets of the Euclidean space, we study processes in an open set D⊂ ℝd defined via Dirichlet forms with jump kernels of the form JD(x, y) = j(| x− y|) B(x, y) and critical killing functions. Here j(|x − y|) is the Lévy density of an isotropic stable process (or more generally, a pure jump isotropic unimodal Lévy process) in ℝd. The main novelty is that the term B(x, y) tends to 0 when x or y approach the boundary of D. Under some general assumptions on B(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, β1, β2, β3, β4, , 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=(x~,xd):xd>0}, the α-stable kernel j(|x − y|) = |x − y|−d−α and the killing function κ(x)=cxd−α, α ∈ (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 > (α − 1)+, there are values of the parameters β1, β2, β3, β4, and the constant c such that non-negative harmonic functions of the process must decay at the rate xdp if they vanish near a portion of the boundary. We further show that there are values of the parameters β1, β2, β3, β4, for which the boundary Harnack principle fails despite the fact that Carleson’s estimate is valid.

KW - Boundary Harnack principle

KW - Carleson estimate

KW - Harnack inequality

KW - Jump processes

KW - Jumping kernel with boundary part

UR - http://www.scopus.com/inward/record.url?scp=85113754190&partnerID=8YFLogxK

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U2 - 10.1007/s11118-021-09947-8

DO - 10.1007/s11118-021-09947-8

M3 - Article

AN - SCOPUS:85113754190

SN - 0926-2601

VL - 58

SP - 465

EP - 528

JO - Potential Analysis

JF - Potential Analysis

IS - 3

ER -