Duality between Ahlfors–Liouville and Khas’minskii properties for non-linear equations

In recent years, the study of the interplay between (fully) nonlinear potential theory and geometry received important new impulse. The purpose of this 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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The first author was partially supported by CNPq-Brazil and by research funds of the Scuola Normale Superiore. The second author was partially supported by Capes and CNPq-Brazil.

Received 31 July 2016

Accepted 25 November 2017

Published 6 May 2020