Liouville Property of Harmonic Functions of Finite Energy for Dirichlet Forms

Abstract

A quasi-regular Dirichlet form is said to have a Liouvill e property if any associated harmonic function of finite energy is constant. We first examine this property for the energy form \({\mathscr {E}}^\rho \) on \(\mathbb {R}^n\) generated by a positive function \(\rho .\) We next make a general consideration on a regular, strongly local and transient Dirichlet form \({\mathscr {E}}\) and an associated time changed symmetric diffusion process \(\check{X}\) with finite lifetime. We show that \(\check{X}\) always admits its one-point reflection \(\check{X}^*\) at infinity by constructing the corresponding regular Dirichlet form. We then prove that, if \({\mathscr {E}}\) satisfies the Liouville property, a symmetric conservative diffusion extension Y of \(\check{X}\) is unique up to a quasi-homeomorphism, and in fact, a quasi-homeomorphic image of Y equals the one-point reflection \(\check{X}^*\) of \(\check{X}\) at infinity.