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January 1998 On the range of ${\bf R}\sp 2$ or ${\bf R}\sp 3$-valued harmonic morphisms
F. Duheille
Ann. Probab. 26(1): 308-315 (January 1998). DOI: 10.1214/aop/1022855420


We prove that, under some general assumptions, the range of any nonconstant harmonic morphism from a simply connected open set $U$ in $\mathbf{R}^n$ to $\mathbf{R}^3$, $n > 3$, cannot avoid three concurrent half-lines, which is an extension to Picard’s little theorem. To this end, we will prove two results concerning the windings of Brownian motion around three concurrent half-lines in $\mathbf{R}^3$ and the recurrence of some domains linked with the harmonic morphism.


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F. Duheille. "On the range of ${\bf R}\sp 2$ or ${\bf R}\sp 3$-valued harmonic morphisms." Ann. Probab. 26 (1) 308 - 315, January 1998.


Published: January 1998
First available in Project Euclid: 31 May 2002

zbMATH: 0933.31006
MathSciNet: MR1617050
Digital Object Identifier: 10.1214/aop/1022855420

Primary: 31C05 , 58E20 , 60J45 , 60J65

Keywords: Brownian motion , harmonic morphism , Picard's theorem , Probabilistic potential theory

Rights: Copyright © 1998 Institute of Mathematical Statistics

Vol.26 • No. 1 • January 1998
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