We study the appropriate versions of parabolicity stochastic completeness and related Liouville properties for a general class of operators which include the $p$-Laplace operator, and the non linear singular operators in non-diagonal form considered by J. Serrin and collaborators.
"Some non-linear function theoretic properties of Riemannian manifolds." Rev. Mat. Iberoamericana 22 (3) 801 - 831, December, 2006.