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 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.
On the L1-Liouville property of stochastically incomplete manifolds
PIGOLA, STEFANO;SETTI, ALBERTO GIULIO
2013-01-01
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 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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G. Pacelli Bessa, S. Pigola, A.G. Setti, On the L 1-Liouville Property of Stochastically Incomplete Manifolds, PoTa 39 (2013).pdf
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