Abstract
A classical result by Alexander Grigor’yan states that on a stochastically complete manifold non-negative superharmonic L 1-functions are necessarily constant. In this paper we construct explicit examples showing that, in the presence of an anisotropy of the space, the reverse implication does not hold. We also consider natural geometric situations where stochastically incomplete manifolds do not posses the above mentioned L 1-Liouville property for superharmonic functions.
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Bessa, G.P., Pigola, S. & Setti, A.G. On the L 1-Liouville Property of Stochastically Incomplete Manifolds. Potential Anal 39, 313–324 (2013). https://doi.org/10.1007/s11118-012-9331-8
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DOI: https://doi.org/10.1007/s11118-012-9331-8