Liouville theorem and coupling on negatively curved Riemannian manifolds

Feng-Yu Wang

Stochastic Processes and their Applications, vol. 100, issue 1-2, 27-39

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.

Keywords: Liouville; theorem; Harmonic; function; Diffusion; process; Semigroup; Coupling (search for similar items in EconPapers)
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