Dirichlet parabolicity and $L^1$-Liouville property under localized geometric conditions

TL;DR

This paper explores how localized geometric conditions influence the $L^1$-Liouville property for superharmonic functions on manifolds, showing it is weaker than stochastic completeness and emphasizing potential theory tools.

Contribution

It demonstrates the dependence of the $L^1$-Liouville property on localized geometry and clarifies its relation to stochastic completeness using potential theory and boundary conditions.

Findings

$L^1$-Liouville property depends on localized geometric conditions.

The property is strictly weaker than stochastic completeness.

Potential theory with Dirichlet boundary conditions is central to the analysis.

Abstract

We shed a new light on the $L^1$-Liouville property for positive, superharmonic functions by providing many evidences that its validity relies on geometric conditions localized on large enough portions of the space. We also present examples in any dimension showing that the $L^1$-Liouville property is strictly weaker than the stochastic completeness of the manifold. The main tool in our investigations is represented by the potential theory of a manifold with boundary subject to Dirichlet boundary conditions. The paper incorporates, under a unifying viewpoint, some old and new aspects of the theory, with a special emphasis on global maximum principles and on the role of the Dirichlet Green's kernel.