Maximum Principles at Infinity and the Ahlfors-Khas’minskii Duality: An Overview
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.
KeywordsPotential theory Liouville theorem Omori-Yau Maximum principles Stochastic completeness Martingale Completeness Ekeland Brownian motion
2010 Mathematics Subject ClassificationPrimary 31C12 35B50; Secondary 35B53 58J65 58J05 53C42
This work was completed when the second author was visiting the Abdus Salam International Center for Theoretical Physics (ICTP), Italy. He is grateful for the warm hospitality and for financial support. The authors would also like to thank the organizing and local committees of the INdAM workshop “Contemporary Research in elliptic PDEs and related topics” (Bari, May 30/31, 2017) for the friendly and pleasant environment.
The first author “Luciano Mari” is supported by the grants SNS17_B_MARI and SNS_RB_MARI of the Scuola Normale Superiore.
The second author “Leandro F. Pessoa” was partially supported by CNPq-Brazil.
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