A remark on subharmonicity for symmetric Dirichlet forms

We remove the local boundedness for ${\mathcal{E}}_{\alpha }$-subharmonicity in the framework of (not necessarily strongly local) regular symmetric Dirichlet form $(\mathcal{E},D(\mathcal{E}))$ with $\alpha \ge 0$ and establish the stochastic characterization for $\mathcal{E}$-subharmonic functions without assuming the local boundedness.