The author considers a regular Dirichlet space \(({\mathcal E, F})\) (where \({\mathcal E}\) is a symmetric bilinear form and \(F\) is a Hilbert space) with some properties ensuring that the corresponding Hunt process is a diffusion with no killing inside. If \(({\mathcal E, F})\) is irreducible and recurrent, the Liouville property for subharmonic functions is related to vanishing of the capacity at infinity. The existence of the exhaustion function for Liouville’s theorem, by using the Hellinger integral, is assured by the recurrence of the corresponding diffusion.
For the entire collection see [Zbl 0903.00037].