Geodesic
Stochastically complete manifolds and summable harmonic functions
Izvestiya. Mathematics , Tome 33 (1989) no. 2, pp. 425-432

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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. 
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     author = {A. A. Grigor'yan},
     title = {Stochastically complete manifolds and summable harmonic functions},
     journal = {Izvestiya. Mathematics },
     pages = {425--432},
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     volume = {33},
     number = {2},
     year = {1989},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/IM2_1989_33_2_a11/}
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A. A. Grigor'yan. Stochastically complete manifolds and summable harmonic functions. Izvestiya. Mathematics , Tome 33 (1989) no. 2, pp. 425-432. http://geodesic.mathdoc.fr/item/IM2_1989_33_2_a11/