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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

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


Authors: Luciano Mari and Daniele Valtorta
Journal: Trans. Amer. Math. Soc. 365 (2013), 4699-4727
MSC (2010): Primary 31C12; Secondary 35B53, 58J65, 58J05
Published electronically: February 28, 2013
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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$-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.


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Additional Information

Luciano Mari
Affiliation: Dipartimento di Matematica, Università degli studi di Milano, via Saldini 50, 20133 Milano, Italy
Email: luciano.mari@unimi.it, lucio.mari@libero.it

Daniele Valtorta
Affiliation: Dipartimento di Matematica, Università degli studi di Milano, via Saldini 50, 20133 Milano, Italy
Email: danielevaltorta@gmail.com

DOI: http://dx.doi.org/10.1090/S0002-9947-2013-05765-0
PII: S 0002-9947(2013)05765-0
Keywords: Khas’minskii condition, stochastic completeness, parabolicity
Received by editor(s): July 21, 2011
Published electronically: February 28, 2013
Dedicated: Sui quisque laplaciani faber
Article copyright: © Copyright 2013 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.