A remark on subharmonicity for symmetric Dirichlet forms

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

💡 Research Summary

The paper “A remark on subharmonicity for symmetric Dirichlet forms” removes the local boundedness hypothesis that has been a standing assumption in the stochastic characterization of (\mathscr{E}_\alpha)-subharmonic functions. Working in the general framework of a regular symmetric Dirichlet form ((\mathscr{E},D(\mathscr{E}))) on a locally compact separable metric space (E) equipped with a full‑support Radon measure (m), the authors develop a theory that applies to Hunt processes with both diffusion and jump components, and even to processes with killing.

The first technical contribution is a careful construction of several function spaces that extend the classical domain (D(\mathscr{E})). The extended Dirichlet space (D(\mathscr{E})e) consists of (m)-a.e. limits of (\mathscr{E})-Cauchy sequences, and its local version (D(\mathscr{E}){e,\mathrm{loc}}) is defined by requiring such approximations on every relatively compact open subset. Within this framework the authors introduce three sub‑classes—denoted (D(\mathscr{E})^\dagger_{\mathrm{loc}}), (D(\mathscr{E})^\diamond_{\mathrm{loc}}) and (D(\mathscr{E}){\mathrm{loc}})—characterized by integrability conditions involving the jump measure (J). In particular, a function (u) belongs to (D(\mathscr{E})^\diamond{\mathrm{loc}}) if for every pair of relatively compact open sets (U\Subset V\Subset D) one has
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