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Potential theory of Schrödinger operator based on fractional Laplacian

Bogdan K., Byczkowski T.

Probability and Mathematical Statistics

2000

Vol. 20, Fasc. 2

293--335

We develop potential theory of Schrödinger operators based on fractional Laplacian on Euclidean spaces of arbitrary dimension. We focus on questions related to gaugeability and existence of q-harmonic functions. Results are obtained by analyzing properties of a symmetric α-stable Lévy process on Rd, including the recurrent case. We provide some relevant techniques and apply them to give explicit examples of gauge functions for a general class of domains.

Intrinsic ultracontractivity for symmetric stable processes

Kulczycki T.

Bulletin of the Polish Academy of Sciences. Mathematics

1998

Vol. 46, no 3

325-334

We prove the intrinsic ultracontractivity for the semigroup P[...] generated by the symmetric [alpha]-stable process killed on exiting a bounded open set D. The same property is valid for the alpha-stable Feynman-Kac semigroup T[...] provided D is a gaugeable bounded Lipschitz domain.