On the equivalence of stochastic completeness, Liouville and Khas'minskii condition in linear and nonlinear setting

Details On the equivalence of stochastic completeness, Liouville and Khas'minskii condition in linear and nonlinear setting On the equivalence of stochastic completeness, Liouville and Khas'minskii condition in linear and nonlinear setting

2011-06-07

arxiv.org/abs/1106.1352v5

Mathematics

Analysis of PDEs

34 pages. The pasting lemma has been improved to fix a technical problem in the main theorem. Final version, to appear on Tran

Scientific paper

Set in Riemannian enviroment, the aim of this paper is to present and discuss some equivalent characterizations of the Liouville property relative to special operators, in some sense modeled after the p-Laplacian with potential. In particular, we discuss the equivalence between the Lioville property and the Khas'minskii condition, i.e. the existence of an exhaustion functions which is also a supersolution for the operator outside a compact set. This generalizes a previous result obtained by one of the authors and answers to a question in "Aspects of potential theory, linear and nonlinear" by Pigola, Rigoli and Setti.

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