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Teor. Veroyatnost. i Primenen., 1976, Volume 21, Issue 1, Pages 81–94 (Mi tvp3276)  

Brownian motion and harmonic functions on manifolds of negative curvature

Yu. I. Kifer

Moscow

Abstract: We investigate positive solutions of the equation $\Delta u=0$, where $\Delta$ is the Beltrami–Laplace operator on manifold $M$ of negative curvature $K$. In section 3 we prove the existence and uniqueness of the Dirichlet problem with a continuous boundary function defined on the absolute of the manifold $M$. If the curvature $K$ changes slowly at infinity (see condition 2), we prove that the structure of the space of minimal positive solutions of $\Delta u=0$ is the same as in the case of constant negative curvature, i. e. there is a one-to-one correspondence between points of the absolute and normalized minimal positive solutions of $\Delta u=0$.

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English version:
Theory of Probability and its Applications, 1976, 21:1, 81–95

Bibliographic databases:

Received: 08.07.1974

Citation: Yu. I. Kifer, “Brownian motion and harmonic functions on manifolds of negative curvature”, Teor. Veroyatnost. i Primenen., 21:1 (1976), 81–94; Theory Probab. Appl., 21:1 (1976), 81–95

Citation in format AMSBIB
\Bibitem{Kif76}
\by Yu.~I.~Kifer
\paper Brownian motion and harmonic functions on manifolds of negative curvature
\jour Teor. Veroyatnost. i Primenen.
\yr 1976
\vol 21
\issue 1
\pages 81--94
\mathnet{http://mi.mathnet.ru/tvp3276}
\mathscinet{http://www.ams.org/mathscinet-getitem?mr=420887}
\zmath{https://zbmath.org/?q=an:0361.60050}
\transl
\jour Theory Probab. Appl.
\yr 1976
\vol 21
\issue 1
\pages 81--95
\crossref{https://doi.org/10.1137/1121006}


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