On the $L^{1}$-Liouville property of stochastically incomplete manifolds
  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. 
G. Pacelli Bessa, Stefano Pigola, Alberto G. Setti
math.DG