Brownian motion and harmonic functions on manifolds of negative curvature
Teoriâ veroâtnostej i ee primeneniâ, Tome 21 (1976) no. 1, pp. 81-94
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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 $\Delta u=0$.
@article{TVP_1976_21_1_a5,
author = {Yu. I. Kifer},
title = {Brownian motion and harmonic functions on manifolds of negative curvature},
journal = {Teori\^a vero\^atnostej i ee primeneni\^a},
pages = {81--94},
publisher = {mathdoc},
volume = {21},
number = {1},
year = {1976},
language = {ru},
url = {http://geodesic.mathdoc.fr/item/TVP_1976_21_1_a5/}
}
Yu. I. Kifer. Brownian motion and harmonic functions on manifolds of negative curvature. Teoriâ veroâtnostej i ee primeneniâ, Tome 21 (1976) no. 1, pp. 81-94. http://geodesic.mathdoc.fr/item/TVP_1976_21_1_a5/