Bessel Potentials, Hitting Distributions and Green Functions

Byczkowski, T.; Ryznar, M.; Malecki, J.

Bessel Potentials, Hitting Distributions and Green Functions

Authors

Byczkowski, T.

Ryznar, M.

Malecki, J.

Description

The purpose of this paper is to find explicit formulas for basic objects pertaining the local potential theory of the operator $(I-Δ)^{α/2}$, $0<α<2$. The potential theory of this operator is based on Bessel potentials $J_α=(I-Δ)^{-α/2}$. We compute the {\it harmonic measure} of the half-space and write a concise form of the corresponding {\it Green function} for the operator $(I-Δ)^{α/2}$. To achieve this we analyze the so-called {\it relativistic $α$-stable process} on $\R^d$ space, killed when exiting the half-space. In terms of this process we are dealing here with the 1-{\it potential theory} or, equivalently, potential theory of Schr{ö}dinger operator based on the generator of the process with Kato's potential $q=-1$.