Stochastically complete manifolds and summable harmonic functions

Main result: if on a geodesically complete Riemannian manifold $M$ the volume $V_R$ of a geodesic ball of radius $R$ with fixed center satisfies the condition $\displaystyle\int^\infty\frac{R\,dR}{\ln V_R}=\infty$ then every nonnegative integrable superharmonic function on $M$ is equal to a constant.
Bibliography: 18 titles.