On the equivalence of stochastic completeness and Liouville and Khas’minskii conditions in linear and nonlinear settings

Abstract

Set in the Riemannian enviroment, the aim of this paper is to present and discuss some equivalent characterizations of the Liouville property relative to special operators, which in some sense are modeled after the p p -Laplacian with potential. In particular, we discuss the equivalence between the Liouville property and the Khas’minskii condition, i.e. the existence of an exhaustion function which is also a supersolution for the operator outside a compact set. This generalizes a previous result obtained by one of the authors.