Harmonic functions for recurrent symmetric $\hat{I}\pm$-stable processeses with non-local perturbations

Abstract

Let $\mathbf{M}$ be the recurrent symmetric (relativistic) $\hat{I}\pm$-stable process on ${\mathbb{R}}^d$. Let ${\mathcal{H}}^{\hat{I}¼+F}(:=\mathcal{H}+\hat{I}¼+F)$ be a Schrödinger type operator with local and non-local perturbations $\hat{I}¼$ and $F$. If $\hat{I}¼$ and $F$ satisfy suitable conditions associated with Kato class, we prove the existence of ground state for a Schrödinger type operator relating to ${\mathcal{H}}^{\hat{I}¼+F}$. Furthermore, we prove the ground state becomes a probabilistically harmonic function of the Schrödinger operator generated by ${\mathcal{H}}^{\hat{I}¼+F}$.

16 pages