ON A CONSTRUCTION OF HARMONIC FUNCTION FOR RECURRENT RELATIVISTIC alpha-STABLE PROCESSES

摘要

In this paper, we study a lambda(mu)-ground state of the Schrodinger operator H-mu based on recurrent relativistic alpha-stable processes. For a signed measure mu = mu(+)-mu(-) being in a suitable Kato class, we construct the lambda(mu)-ground state which is bounded and continuous. Moreover, we prove that the lambda(mu)-ground state has the mean-value property. In particular, if lambda(mu) = 1, the mean-value property means the probabilistical harmonicity of H-mu. Finally, we show that if alpha > d = 1, the relativistic a-stable process is point recurrent.