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Brownian motion and a generalised little Picard's theorem


Author: Wilfrid S. Kendall
Journal: Trans. Amer. Math. Soc. 275 (1983), 751-760
MSC: Primary 58G32; Secondary 32H30, 53C20
DOI: https://doi.org/10.1090/S0002-9947-1983-0682729-8
MathSciNet review: 682729
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Abstract: Goldberg, Ishihara, and Petridis have proved a generalised little Picard's theorem for harmonic maps; if a harmonic map of bounded dilatation maps euclidean space, for example, into a space of negative sectional curvatures bounded away from zero then that map is constant. In this paper a probabilistic proof is given of a variation on this result, requiring in addition that the image space has curvatures bounded below. The method involves comparing asymptotic properties of Brownian motion with the asymptotic behaviour of its image under such a map.


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Additional Information

DOI: https://doi.org/10.1090/S0002-9947-1983-0682729-8
Keywords: Brownian motion, zero-one laws, asymptotic behaviour of Brownian motion, harmonic maps, bounded dilatation, negative curvature
Article copyright: © Copyright 1983 American Mathematical Society

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