Bessel Potentials, Hitting Distributions and Green Functions

Abstract

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&#39;s potential $q=-1$.