Remote Access Transactions of the American Mathematical Society
Green Open Access

Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



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
MathSciNet review: 682729
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

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.

References [Enhancements On Off] (What's this?)

  • [1] J. Cheeger and D. G. Ebin, Comparison theorems in Riemannian geometry, North-Holland, Amsterdam, 1975. MR 0458335 (56:16538)
  • [2] B. Davis, Picard's theorem and Brownian motion, Trans. Amer. Math. Soc. 213 (1975), 353-362. MR 0397900 (53:1756)
  • [3] J. Eells and L. Lemaire, A report on harmonic maps, Bull. London Math. Soc. 10 (1978), 1-68. MR 495450 (82b:58033)
  • [4] B. Fuglede, Harmonic morphisms between Riemannian manifolds, Ann. Inst. Fourier (Grenoble) 28 (1978), 107-144. MR 499588 (80h:58023)
  • [5] S. I. Goldberg, T. Ishihara and N. C. Petridis, Mappings of bounded dilatation of Riemannian manifolds, J. Differential Geom. 10 (1975), 619-630. MR 0390964 (52:11787)
  • [6] R. E. Greene and H. Wu, Function theory on manifolds which possess a pole, Lecture Notes in Math., vol. 699, Springer-Verlag, Berlin and New York, 1979. MR 521983 (81a:53002)
  • [7] W. S. Kendall, Brownian motion, negative curvature, and harmonic maps, Stochastic Integrals: Proc. L.M.S. Durham Sympos. 1980, (D. Williams, Editor), Lecture Notes in Math., vol. 851, Springer-Verlag, Berlin and New York, 1981. MR 621002 (82k:58099)
  • [8] M. Pinsky, Stochastic Riemannian geometry, Probabilistic Analysis and Related Topics. Vol. 1, (A. T. Bharucha-Reid, Editor), Academic Press, New York, 1978. MR 0501385 (58:18752)
  • [9] J.-J. Prat, Étude asymptotique et convergence angulaire du mouvement brownien sur une variété à courbure négative, C. R. Acad. Sci. Paris Sér. A 280 (1975), 1539-1542. MR 0388557 (52:9393)
  • [10] D. W. Stroock and S. R. S. Varadhan, On the support of diffusion processes with applications to the strong maximum principle, Proc. Sixth Berkeley Sympos. on Math. Stat. and Prob. III (1970/71), L. M. LeCam et al., Editors, Univ. of California Press, Berkeley and Los Angeles, 1972, pp. 333-360. MR 0400425 (53:4259)
  • [11] S. T. Yau, Harmonic functions on complete Riemannian manifolds, Comm. Pure Appl. Math. 28 (1975), 201-228. MR 0431040 (55:4042)

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC: 58G32, 32H30, 53C20

Retrieve articles in all journals with MSC: 58G32, 32H30, 53C20

Additional Information

Keywords: Brownian motion, zero-one laws, asymptotic behaviour of Brownian motion, harmonic maps, bounded dilatation, negative curvature
Article copyright: © Copyright 1983 American Mathematical Society

American Mathematical Society