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Liouville theorem and coupling on negatively curved Riemannian manifolds

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  • Wang, Feng-Yu

Abstract

By using probabilistic approaches, Liouville theorems are proved for a class of Riemannian manifolds with Ricci curvatures bounded below by a negative function. Indeed, for these manifolds we prove that all harmonic functions (maps) with certain growth are constant. In particular, the well-known Liouville theorem due to Cheng for sublinear harmonic functions (maps) is generalized. Moreover, our results imply the Brownian coupling property for a class of negatively curved Riemannian manifolds. This leads to a negative answer to a question of Kendall concerning the Brownian coupling property.

Suggested Citation

  • Wang, Feng-Yu, 0. "Liouville theorem and coupling on negatively curved Riemannian manifolds," Stochastic Processes and their Applications, Elsevier, vol. 100(1-2), pages 27-39, July.
    • Handle: RePEc:eee:spapps:v:100:y::i:1-2:p:27-39
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    References listed on IDEAS

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    1. Cranston, Michael & Greven, Andreas, 1995. "Coupling and harmonic functions in the case of continuous time Markov processes," Stochastic Processes and their Applications, Elsevier, vol. 60(2), pages 261-286, December.
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