Maximum principles at infinity and the ahlfors-khas’minskii duality: An overview

Abstract

This note is meant to introduce the reader to a duality principle for nonlinear equations recently discovered in Valtorta (Reverse Khas’minskii condition. Math Z 270(1):65–177, 2011), Mari and Valtorta (Trans Am Math Soc 365(9):4699–4727, 2013), and Mari and Pessoa (Commun Anal Geom, to appear). Motivations come from the desire to give a unifying potential-theoretic framework for various maximum principles at infinity appearing in the literature (Ekeland, Omori-Yau, Pigola-Rigoli-Setti), as well as to describe their interplay with properties coming from stochastic analysis on manifolds. The duality involves an appropriate version of these principles formulated for viscosity subsolutions of fully nonlinear inequalities, called the Ahlfors property, and the existence of suitable exhaustion functions called Khas’minskii potentials. Applications, also involving the geometry of submanifolds, will be discussed in the last sections. We conclude by investigating the stability of these maximum principles when we remove polar sets.