Skip to main content
SpringerLink
Log in

Brownian Motion and Negative Curvature

  • Conference paper
  • First Online:
Book cover Random Walks, Boundaries and Spectra

Part of the book series: Progress in Probability ((PRPR,volume 64))

Abstract

It is well known that on a Riemannian manifold, there is a deep interplay between geometry, harmonic function theory, and the long-term behaviour of Brownian motion. Negative curvature amplifies the tendency of Brownian motion to exit compact sets and, if topologically possible, to wander out to infinity. On the other hand, non-trivial asymptotic properties of Brownian paths for large time correspond with non-trivial bounded harmonic functions on the manifold. We describe parts of this interplay in the case of negatively curved simply connected Riemannian manifolds. Recent results are related to known properties and old conjectures.

Mathematics Subject Classification (2000). Primary 58J65; Secondary 60H30, 31C12, 31C35.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 84.99
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 109.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info
Hardcover Book
USD 109.99
Price excludes VAT (USA)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. A. Ancona, Negatively curved manifolds, elliptic operators, and the Martin boundary, Ann. of Math. (2) 125 (1987), no. 3, 495–536.

    Google Scholar 

  2. , Convexity at infinity and Brownian motion on manifolds with unbounded negative curvature, Rev. Mat. Iberoamericana 10 (1994), no. 1, 189–220.

    Google Scholar 

  3. M.T. Anderson, The Dirichlet problem at infinity for manifolds of negative curvature, J. Differential Geom. 18 (1983), no. 4, 701–721 (1984).

    Google Scholar 

  4. M.T. Anderson, R. Schoen, Positive harmonic functions on complete manifolds of negative curvature, Ann. of Math. (2) 121 (1985), no. 3, 429–461.

    Google Scholar 

  5. M. Arnaudon, A. Thalmaier, S. Ulsamer, Existence of non-trivial harmonic functions on Cartan-Hadamard manifolds of unbounded curvature, Math. Z. 263 (2009), 369– 409.

    Article  MathSciNet  MATH  Google Scholar 

  6. R.L. Bishop, B. O’Neill, Manifolds of Negative Curvature, Trans. Amer. Math. Soc. 145 (1968), 1–49.

    Article  MathSciNet  Google Scholar 

  7. A. Borb´ely, The nonsolvability of the Dirichlet problem on negatively curved manifolds, Differential Geom. Appl. 8 (1998), no. 3, 217–237.

    Google Scholar 

  8. H.I. Choi, Asymptotic Dirichlet problems for harmonic functions on Riemannian manifolds, Trans. Amer. Math. Soc. 281 (1984), no. 2, 691–716.

    Article  MathSciNet  MATH  Google Scholar 

  9. M. Cranston, On specifying invariant σ-fields. Seminar on Stochastic Processes 1991, Birkh¨auser, Progr. Probab., vol. 29 (1992), 15–37.

    MathSciNet  Google Scholar 

  10. M. Cranston, C. Mueller, A review of recent and older results on the absolute continuity of harmonic measure, Geometry of random motion (Ithaca, N.Y., 1987), Contemp. Math., vol. 73, Amer. Math. Soc., Providence, RI, 1988, pp. 9–19.

    Google Scholar 

  11. P. Eberlein, B. O’Neill, Visibility Manifolds. Pacific J. Math. 46 (1973), no. 1, 45– 109.

    MathSciNet  MATH  Google Scholar 

  12. M. ´Emery, Stochastic calculus in manifolds, Universitext, Springer-Verlag, Berlin, 1989, With an appendix by P.-A. Meyer.

    Google Scholar 

  13. R.E. Greene, H. Wu, Function theory on manifolds which possess a pole, Lecture Notes in Mathematics, vol. 699, Springer, Berlin, 1979.

    Google Scholar 

  14. W. Hackenbroch, A. Thalmaier, Stochastische Analysis. Eine Einf¨uhrung in die Theorie der stetigen Semimartingale, B.G. Teubner, Stuttgart, 1994.

    Google Scholar 

  15. I. Holopainen, A. V¨ah¨akangas, Asymptotic Dirichlet problem on negatively curved spaces. International Conference on Geometric Function Theory, Special Functions and Applications (R.W. Barnard and S. Ponnusamy, eds.), J. Analysis 15 (2007), 63–110.

    Google Scholar 

  16. E.P. Hsu, Brownian motion and Dirichlet problems at infinity, Ann. Probab. 31 (2003), no. 3, 1305–1319.

    Article  MathSciNet  MATH  Google Scholar 

  17. P. Hsu, W.S. Kendall, Limiting angle of Brownian motion in certain two-dimensional Cartan-Hadamard manifolds, Ann. Fac. Sci. Toulouse Math. (6) 1 (1992), no. 2, 169– 186.

    Google Scholar 

  18. P. Hsu, P. March, The limiting angle of certain Riemannian Brownian motions, Comm. Pure Appl. Math. 38 (1985), no. 6, 755–768.

    Article  MathSciNet  MATH  Google Scholar 

  19. A. Katok, Four applications of conformal equivalence to geometry and dynamics, Ergodic Theory Dynam. Systems 8 ∗ (1988), no. Charles Conley Memorial Issue, 139–152.

    Google Scholar 

  20. W.S. Kendall, Brownian motion on a surface of negative curvature, Seminar on probability, XVIII, Lecture Notes in Math., vol. 1059, Springer, Berlin, 1984, pp. 70– 76.

    Google Scholar 

  21. Ju.I. Kifer, Brownian motion and harmonic functions on manifolds of negative curvature, Theor. Probability Appl. 21 (1976), no. 1, 81–95.

    Article  MathSciNet  MATH  Google Scholar 

  22. , Brownian motion and positive harmonic functions on complete manifolds of nonpositive curvature, From local times to global geometry, control and physics (Coventry, 1984/85), Pitman Res. Notes Math. Ser., vol. 150, Longman Sci. Tech., Harlow, 1986, pp. 187–232.

    Google Scholar 

  23. H. Le, Limiting angle of Brownian motion on certain manifolds, Probab. Theory Related Fields 106 (1996), no. 1, 137–149.

    Article  MathSciNet  MATH  Google Scholar 

  24. , Limiting angles of Γ-martingales, Probab. Theory Related Fields 114(1999), no. 1, 85–96.

    Google Scholar 

  25. F. Ledrappier, Propri´et´e de Poisson et courbure n´egative, C. R. Acad. Sci. Paris S´er. I Math. 305 (1987), no. 5, 191–194.

    Google Scholar 

  26. P. March, Brownian motion and harmonic functions on rotationally symmetric manifolds, Ann. Probab. 11 (1986) 793–801.

    Article  MathSciNet  Google Scholar 

  27. F. Mouton, Comportement asymptotique des fonctions harmoniques en courbure n´egative, Comment. ath. Helv. 70 (1995), 475–505.

    Article  MathSciNet  MATH  Google Scholar 

  28. ´E. Pardoux, Grossissement d’une filtration et retournement du temps d’une diffusion, S´eminaire de Probabilit´es, XX, 1984/85, Lecture Notes in Math., vol. 1204, Springer, Berlin, 1986, pp. 48–55.

    Google Scholar 

  29. J.-J. Prat, ´ Etude asymptotique du mouvement brownien sur une vari´et´e riemannienne `a courbure n´egative, C. R. Acad. Sci. Paris S´er. A-B 272 (1971), A1586–A1589.

    Google Scholar 

  30. , ´ Etude asymptotique et convergence angulaire du mouvement brownien surune vari´et´e `a courbure n´egative, C. R. Acad. Sci. Paris S´er. A-B 280 (1975), A1539– A1542.

    Google Scholar 

  31. D. Sullivan, The Dirichlet problem at infinity for a negatively curved manifold, J. Differential Geom. 18 (1983), no. 4, 723–732 (1984).

    Google Scholar 

  32. A. V¨ah¨akangas, Dirichlet problem at infinity for A-harmonic functions, Potential Anal. 27 (2007), no. 1, 27–44.

    Google Scholar 

  33. H.H. Wu, Function theory on noncompact K¨ahler manifolds, Complex differential geometry, DMV Sem., vol. 3, Birkh¨auser, Basel, 1983, pp. 67–155.

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Laboratoire de Mathématiques et Applications, CNRS: UMR6086, Université de Poitiers, Téléport 2 – BP 30179, F-86962, Futuroscope Chasseneuil Cedex, France

    Marc Arnaudon

  2. Unité de Recherche en Mathématiques, FSTC, Université du Luxembourg, 6, rue Richard Coudenhove-Kalergi, L-1359, Luxembourg, Grand-Duchy of Luxembourg

    Anton Thalmaier

Authors
  1. Marc Arnaudon
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Anton Thalmaier
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Marc Arnaudon .

Editor information

Editors and Affiliations

  1. Mathematisches Inst., Universität Jena, Ernst-Abbe-Platz 2, Jena, 07737, Germany

    Daniel Lenz

  2. Inst. Mathematik C, TU Graz, Steyrergasse 30, Graz, 8010, Austria

    Florian Sobieczky

  3. Inst. Mathematik C, TU Graz, Steyrergasse 30, Graz, 8010, Austria

    Wolfgang Woess

Rights and permissions

Reprints and permissions

Copyright information

© 2011 Springer Basel AG

About this paper

Cite this paper

Arnaudon, M., Thalmaier, A. (2011). Brownian Motion and Negative Curvature. In: Lenz, D., Sobieczky, F., Woess, W. (eds) Random Walks, Boundaries and Spectra. Progress in Probability, vol 64. Springer, Basel. https://doi.org/10.1007/978-3-0346-0244-0_8

Download citation

Publish with us

Policies and ethics

Search

Navigation