## Tohoku Mathematical Journal 2005 September SECOND SERIES VOL. 57, NO. 3

 Tohoku Math. J. 57 (2005), 353-373

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

Author Atsushi Atsuji

(Received October 2, 2003, revised March 28, 2005)
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.

2000 Mathematics Subject Classification. Primary 31C05; Secondary 58J65.

Key words and phrases. Dirichlet form, Martingale, Harmonic map, Liouville theorem, $\delta$-subharmonic functions.