On $L^p$ Liouville theorems for Dirichlet forms

Abstract

We study harmonic functions for general Dirichlet forms. First we review consequences of Fukushima's ergodic theorem for the harmonic functions in the domain of the $L^p$ generator. Secondly we prove analogues of Yau's and Karp's Liouville theorems for weakly harmonic functions. Both say that weakly harmonic functions which satisfy certain $L^p$ growth criteria must be constant. As consequence we give an integral criterion for recurrence.