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

TL;DR

This paper explores the equivalence of stochastic completeness, Liouville property, and Khas'minskii condition for certain operators in Riemannian geometry, extending previous results to nonlinear settings.

Contribution

It generalizes the equivalence between Liouville property and Khas'minskii condition to nonlinear operators like the p-Laplacian with potential in Riemannian environments.

Findings

Established equivalence between Liouville property and Khas'minskii condition for nonlinear operators

Extended previous linear results to nonlinear settings such as the p-Laplacian

Provided new insights into potential theory in Riemannian geometry

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.