A strong maximum principle for globally hypoelliptic operators

In this paper we study the strong maximum principle for globally hypoelliptic differential operators of second-order, and reveal the underlying analytical mechanism of propagation of maximums in terms of the Lie algebra generated by diffusion vector fields and the Fichera function. Our formulation of the strong maximum principle is coordinate-free. The results here may be applied to questions of uniqueness for degenerate elliptic boundary value problems on a manifold. Furthermore, the mechanism of propagation of maximums plays an important role in the interpretation and study of Markov processes from the viewpoint of functional analysis