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Potential theory for Lévy stable processes

Bogdan K., Stós A., Sztonyk P.

Bulletin of the Polish Academy of Sciences. Mathematics

2002

Vol. 50, no 3

361-372

We investigate properties of functions that are harmonic with respect to symmetric stable processes which are not necessarily rotation invariant. We prove the Harnack inequality and the boundary Harnack principle for a Lipschitz domain. To obtain this we use Poisson kernel estimates. We also give an estimate concerning the rate of decay of harmonic functions near the boundary of the domain.

On harmonic measure for Lévy processes

Sztonyk P.

Probability and Mathematical Statistics

2000

Vol. 20, Fasc. 2

383--390

Let {Xt} be a Lévy process in Rd, d ≥ 2, with infinite Lévy measure. If {Xt} has no Gaussian component, then the process does not hit the boundary of Lipschitz domain S ⊂ Rd at the first exit time of S under mild conditions on {Xt}. The conditions are met, e.g., if {Xt} is rotation invariant.