A remark on subharmonicity for symmetric Dirichlet forms

TL;DR

This paper extends the theory of subharmonic functions associated with symmetric Dirichlet forms by removing the local boundedness condition and providing a stochastic characterization, broadening the applicability of the theory.

Contribution

It introduces a new approach to characterize -subharmonic functions without the local boundedness assumption in the framework of regular symmetric Dirichlet forms.

Findings

Removed the local boundedness requirement for -subharmonicity.

Established a stochastic characterization for -subharmonic functions.

Applicable to non-strongly local Dirichlet forms.

Abstract

We remove the local boundedness for $\mathscr{E}_{\alpha}$-subharmonicity in the framework of (not necessarily strongly local) regular symmetric Dirichlet form $(\mathscr{E},D(\mathscr{E}))$ with $\alpha\geq0$ and establish the stochastic characterization for $\mathscr{E}$-subharmonic functions without assuming the local boundedness.