On the range of ${\bf R}\sp 2$ or ${\bf R}\sp 3$-valued harmonic morphisms
Duheille, F.
Ann. Probab., Tome 26 (1998) no. 1, p. 308-315 / Harvested from Project Euclid
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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.
Publié le : 1998-01-14
Classification:  Harmonic morphism,  Picard's theorem,  Brownian motion,  probabilistic potential theory,  58E20,  31C05,  60J65,  60J45 
@article{1022855420,
     author = {Duheille, F.},
     title = {On the range of ${\bf R}\sp 2$ or ${\bf R}\sp 3$-valued harmonic
 morphisms},
     journal = {Ann. Probab.},
     volume = {26},
     number = {1},
     year = {1998},
     pages = { 308-315},
     language = {en},
     url = {http://dml.mathdoc.fr/item/1022855420}
}

Duheille, F. On the range of ${\bf R}\sp 2$ or ${\bf R}\sp 3$-valued harmonic
 morphisms. Ann. Probab., Tome 26 (1998) no. 1, pp.  308-315. http://gdmltest.u-ga.fr/item/1022855420/