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

TL;DR

This paper establishes a new duality between parabolicity and maximum principles at infinity for nonlinear equations on manifolds, linking geometric potential theory with stochastic and submanifold properties.

Contribution

It introduces a unifying duality between parabolicity and Khas'minskii potentials for nonlinear equations, advancing the understanding of potential theory in geometric contexts.

Findings

New duality between parabolicity and Khas'minskii potentials.

Characterizations of maximum principles at infinity.

Applications to stochastic processes and submanifold theory.

Abstract

In recent years, the study of the interplay between (fully) non-linear 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.