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Boundary potential theory for stable Lévy processes

100%
Sztonyk P.
Colloquium Mathematicum
|
2003
|
tom 95
|
nr 2
191-206
EN
We investigate properties of harmonic functions of the symmetric stable Lévy process on $ℝ^{d}$ without the assumption that the process is rotation invariant. Our main goal is to prove the boundary Harnack principle for Lipschitz domains. To this end we improve the estimates for the Poisson kernel obtained in a previous work. We also investigate properties of harmonic functions of Feynman-Kac semigroups based on the stable process. In particular, we prove the continuity and the Harnack inequality for such functions.
2
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Estimates of the potential kernel and Harnack's inequality for the anisotropic fractional Laplacian

63%
Bogdan K., Sztonyk P.
Studia Mathematica
|
2007
|
tom 181
|
nr 2
101-123
EN
We characterize those homogeneous translation invariant symmetric non-local operators with positive maximum principle whose harmonic functions satisfy Harnack's inequality. We also estimate the corresponding semigroup and the potential kernel.
3
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Harnack inequality for stable processes on d-sets

51%
Bogdan K., Stós A., Sztonyk P.
Studia Mathematica
|
2003
|
tom 158
|
nr 2
163-198
EN
We investigate properties of functions which are harmonic with respect to α-stable processes on d-sets such as the Sierpiński gasket or carpet. We prove the Harnack inequality for such functions. For every process we estimate its transition density and harmonic measure of the ball. We prove continuity of the density of the harmonic measure. We also give some results on the decay rate of harmonic functions on regular subsets of the d-set. In the case of the Sierpiński gasket we even obtain the Boundary Harnack Principle.
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