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# Revista Matemática Iberoamericana

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**Volume 28, Issue 1, 2012, pp. 201–230**

**DOI: 10.4171/RMI/674**

Published online: 2012-01-20

On positive harmonic functions in cones and cylinders

Alano Ancona^{[1]}(1) Université Paris-Sud 11, Orsay, France

We first consider a question raised by Alexander Eremenko and show that if $\Omega $ is an arbitrary connected
open cone in $\mathbb{R}^d$, then any two positive harmonic functions in $\Omega $ that vanish on $\partial
\Omega $ must be proportional – an already known fact when $\Omega $ has a Lipschitz basis or more generally a
John basis. It is also shown however that when $d \geq 4$, there can be more than one Martin point at
infinity for the cone though non-tangential convergence to the canonical Martin point at infinity always
holds. In contrast, when $d \leq 3$, the Martin point at infinity is unique for every cone. These properties
connected with the dimension are related to well-known results of M. Cranston and T. R. McConnell about the
lifetime of conditioned Brownian motions in planar domains and also to subsequent results by R. Bañuelos and

*Keywords: *Cone, cylinder, Martin boundary, harmonic function, heat kernel, elliptic second order operator

Ancona Alano: On positive harmonic functions in cones and cylinders. *Rev. Mat. Iberoam.* 28 (2012), 201-230. doi: 10.4171/RMI/674