Calculus of Variations and Geometric Measure Theory

L. Mari - D. Valtorta

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

created by mari1 on 04 Jan 2021



Inserted: 4 jan 2021
Last Updated: 4 jan 2021

Journal: Trans. Amer. Math. Soc.
Volume: 365
Number: 9
Pages: 4699-4727
Year: 2013
Doi: 10.1090/S0002-9947-2013-05765-0

ArXiv: 1106.1352 PDF


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.