Brownian motion, negative curvature, and harmonic maps

  • W. S. Kendall
Papers Based On Splinter-group Talks
Part of the Lecture Notes in Mathematics book series (LNM, volume 851)

Abstract

As mentioned above, it has been the purpose of this article to show that Brownian motion and probabilistic techniques can be applied to prove results in geometric function theory. The main result obtained, at 4.2, is weaker than the corresponding result at 3.1 proved by geometric methods. However it is possible to extend 4.2, for example by relaxing the curvature conditions to hold only off a compact set.

Keywords

Brownian Motion Riemannian Manifold Sectional Curvature Universal Covering Negative Curvature 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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7. References

  1. Bismut, J-M. Formulation geometrique du calcul de Ito, relevement de connexions et calcul des variations. C.R.A.S.(A) 290 427–429 (1980).MathSciNetMATHGoogle Scholar
  2. Cheeger, J. and Ebin, D. Comparison theorems in Riemannian Geometry. North-Holland, Amsterdam (1975).MATHGoogle Scholar
  3. Davis, Burgess. Picard's Theorem and Brownian motion. Trans. Amer. Math. Soc. 213 353–362 (1975).MathSciNetMATHGoogle Scholar
  4. Ducourtioux unpublished.Google Scholar
  5. Eells, J. and Lemaire, L. A report on harmonic maps. Bull. London Math. Soc. 10 1–68 (1978).MathSciNetCrossRefMATHGoogle Scholar
  6. Fuglede, B. Harmonic morphisms between Riemannian manifolds. Ann. l'inst. Fourier 28 107–144 (1978).MathSciNetCrossRefMATHGoogle Scholar
  7. Goldberg, S.I., Ishihara, T. and Petridis, N.C. Mappings of bounded dilatation of Riemannian manifolds. J. diff. Geom. 10 619–630 (1975).MathSciNetMATHGoogle Scholar
  8. Kendall,W.S. Brownian motion and a generalised Picard's Theorem. in preparation.Google Scholar
  9. McKean, H.P. Stochastic Integrals. Academic Press, New York (1969).MATHGoogle Scholar
  10. McKean,H.P. and Lyons,T. On the winding of Brownian motion about two points in R2. in preparation.Google Scholar
  11. Prat, J-J. Etude asymptotique et convergence angulaire du mouvement brownien sur une variete a courbure negative. C.R.A.S.(A) 280 1539–1542 (1975).MathSciNetMATHGoogle Scholar

Copyright information

© Springer-Verlag 1981

Authors and Affiliations

  • W. S. Kendall
    • 1
  1. 1.Department of Mathematical StatisticsThe UniversityHull

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