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.
"Parabolicity, the divergence theorem for $\delta$-subharmonic functions and applications to the Liouville theorems for harmonic maps." Tohoku Math. J. (2) 57 (3) 353 - 373, September 2005. https://doi.org/10.2748/tmj/1128703002