On a construction of harmonic function for recurrent relativistic $\alpha$-stable processes

Abstract

In this paper, we study a $\lambda(\mu)$-ground state of the Schrödinger operator $\mathcal{H}^{\mu}$ 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 $\mathcal{H}^{\mu}$. Finally, we show that if $\alpha > d = 1$, the relativistic $\alpha$-stable process is point recurrent.