Tohoku Mathematical Journal
2020
June
SECOND SERIES VOL. 72, NO. 2

Tohoku Math. J.
72 (2020), 299-315

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

Author Kaneharu Tsuchida

(Received December 3, 2018, revised February 22, 2019)
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.

Mathematics Subject Classification. Primary 60J45; Secondary 60J75, 31C25.
Key words and phrases. Harmonic function, relativistic stable process, criticality.

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