In recent years, the study of the interplay between (fully) non-linear potential theory and geometry received important new impulse. The purpose of our work is to move a step further in this direction by investigating appropriate versions of parabolicity and maximum principles at infinity for large classes of non-linear (sub)equations F on manifolds. The main goal is to show a unifying duality between such properties and the existence of suitable F-subharmonic exhaustions, called Khas'minskii potentials, which is new even for most of the ``standard" operators arising from geometry, and improves on partial results in the literature. Applications include new characterizations of the classical maximum principles at infinity (Ekeland, Omori-Yau and their weak versions by Pigola-Rigoli-Setti) and of conservation properties for stochastic processes (martingale completeness). Applications to the theory of submanifolds and Riemannian submersions are also discussed.

Duality between Ahlfors-Liouville and Khas'minskii properties for nonlinear equations

Mari, Luciano;
2020

Abstract

In recent years, the study of the interplay between (fully) non-linear potential theory and geometry received important new impulse. The purpose of our work is to move a step further in this direction by investigating appropriate versions of parabolicity and maximum principles at infinity for large classes of non-linear (sub)equations F on manifolds. The main goal is to show a unifying duality between such properties and the existence of suitable F-subharmonic exhaustions, called Khas'minskii potentials, which is new even for most of the ``standard" operators arising from geometry, and improves on partial results in the literature. Applications include new characterizations of the classical maximum principles at infinity (Ekeland, Omori-Yau and their weak versions by Pigola-Rigoli-Setti) and of conservation properties for stochastic processes (martingale completeness). Applications to the theory of submanifolds and Riemannian submersions are also discussed.
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Potential theory; Liouville theorem; Omori-Yau; maximum principles; stochastic completeness; martingale; completeness; Ekeland; Brownian motion
Mari, Luciano; Pessoa, Leandro de Freitas
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Utilizza questo identificativo per citare o creare un link a questo documento: http://hdl.handle.net/2318/1693164
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