On a paper of Berestycki-Hamel-Rossi and its relations to the weak maximum principle at innity, with applications

The aim of this paper is to study a new equivalent form of the weak maximum principle for a large class of dierential operators on Riemannian manifolds. This new form has been inspired by the work of Berestycki, Hamel and Rossi, [5], for trace operators and allows us to shed new light on it and to introduce a new sufficient bounded Khas'minskii type condition for its validity. We show its effectiveness by applying it to obtain some uniqueness results in a geometric setting.