  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.
