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.