On $L^p$ Liouville theorems for Dirichlet forms

TL;DR

This paper investigates harmonic functions associated with general Dirichlet forms, establishing Liouville-type theorems under $L^p$ growth conditions and providing criteria for recurrence.

Contribution

It extends Liouville theorems to weakly harmonic functions for general Dirichlet forms with $L^p$ conditions, and links these results to recurrence criteria.

Findings

Weakly harmonic functions satisfying certain $L^p$ growth are constant.

Liouville theorems analogous to Yau's and Karp's are proved.

An integral criterion for recurrence is established.

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.