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

Abstract

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.

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

References in corpus (2)

• Stokes' theorem, volume growth and parabolicity
• Reverse Khas'minskii condition

Cited by in corpus (6)

• Yamabe type equations with a sign-changing nonlinearity, and the prescribed curvature problem
• Duality between Ahlfors-Liouville and Khas'minskii properties for nonlinear equations
• Bernstein and half-space properties for minimal graphs under Ricci lower bounds
• Detecting the completeness of a Finsler manifold via potential theory for its infinity Laplacian
• Recent rigidity results for graphs with prescribed mean curvature
• A splitting theorem for capillary graphs under Ricci lower bounds