On the $L^{1}$-Liouville property of stochastically incomplete manifolds

TL;DR

This paper investigates whether the property that non-negative superharmonic $L^1$-functions are constant on stochastically complete manifolds extends to stochastically incomplete manifolds, exploring the $L^{1}$-Liouville property in this context.

Contribution

The paper examines the extent to which the $L^{1}$-Liouville property characterizes stochastically incomplete manifolds, challenging the classical equivalence known for complete manifolds.

Findings

Analysis of conditions under which the $L^{1}$-Liouville property holds for incomplete manifolds

Identification of cases where the property fails or holds in stochastically incomplete settings

Insights into the relationship between stochastic completeness and harmonic function properties

Abstract

A classical result by Alexander Grigor'yan states that on a stochastically complete manifold the non-negative superharmonic $L^1$-functions are necessarily constant. In this paper we address the question of whether and to what extent the reverse implication holds.