PARABOLICITY, THE DIVERGENCE THEOREM FOR $\\delta$-SUBHARMONIC FUNCTIONS AND APPLICATIONS TO THE LIOUVILLE THEOREMS FOR HARMONIC MAPS

Abstract

We show that the parabolicity of a manifold is equivalent to the validity of the ‘divergence theorem’ for some class of $\delta$-subharmonic functions. From this property we can show a certain Liouville property of harmonic maps on parabolic manifolds. Elementary stochastic calculus is used as a main tool.