The Annals of Probability

On the range of ${\bf R}\sp 2$ or ${\bf R}\sp 3$-valued harmonic morphisms

F. Duheille

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Abstract

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.

Article information

Source
Ann. Probab., Volume 26, Number 1 (1998), 308-315.

Dates
First available in Project Euclid: 31 May 2002

Permanent link to this document
https://projecteuclid.org/euclid.aop/1022855420

Digital Object Identifier
doi:10.1214/aop/1022855420

Mathematical Reviews number (MathSciNet)
MR1617050

Zentralblatt MATH identifier
0933.31006

Subjects
Primary: 58E20: Harmonic maps [See also 53C43], etc. 31C05: Harmonic, subharmonic, superharmonic functions 60J65: Brownian motion [See also 58J65] 60J45: Probabilistic potential theory [See also 31Cxx, 31D05]

Keywords
Harmonic morphism Picard's theorem Brownian motion probabilistic potential theory

Citation

Duheille, F. On the range of ${\bf R}\sp 2$ or ${\bf R}\sp 3$-valued harmonic morphisms. Ann. Probab. 26 (1998), no. 1, 308--315. doi:10.1214/aop/1022855420. https://projecteuclid.org/euclid.aop/1022855420


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References

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  • UNIVERSITE CLAUDE BERNARD, LYON 1 ´ 43, BOULEVARD DU 11 NOVEMBRE 1918 69622 VILLEURBANNE CEDEX FRANCE E-MAIL: duheille@jonas.univ-lyon1.fr