Skip to main content
SpringerLink
Log in

Intertwinned Diffusions by Examples

  • Chapter
  • First Online:
Stochastic Analysis 2010

  • 1427 Accesses

Abstract

We discuss the geometry induced by pairs of diffusion operators on two states spaces related by a map from one space to the other. This geometry led us to an intrinsic point of view on filtering. This will be explained plainly by examples, in local coordinates and in the metric setting. This article draws largely from the books (“Elworthy et al., On the geometry of diffusion operators and stochastic flows, Lecture Notes in Mathematics 1720, Springer, 1999”, “Elworthy et al., The Geometry of Filtering. To appear in Frontiers in Mathematics Series, Birkhauser”) and aims to have a comprehensive account of the geometry for a general audience.

MSC (2010): 60Hxx, 60Dxx

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. Anderson, A., Camporesi, R.: Intertwining operators for solving differential equations, with applications to symmetric spaces. Commun. Math. Phys. 130, 61–82 (1990)

    Article  MATH  MathSciNet  Google Scholar 

  2. Arnaudon, M., Plank, H., Thalmaier, A.: A Bismut type formula for the Hessian of heat semigroups. C. R. Acad. Sci. Paris 336, 661–666 (2003)

    MATH  MathSciNet  Google Scholar 

  3. Bismut, J.-M.: Large deviations and the Malliavin calculus, vol. 45 of Progress in Mathematics. Birkhäuser Boston Inc., Boston, MA (1984)

    Google Scholar 

  4. Carmona, P., Petit, F., Yor, M.: Beta-gamma random variables and intertwining relations between certain Markov processes. Revista Matematica Iberoamericana 14(2) (1998)

    Google Scholar 

  5. Crisan, D., Kouritzin, M., Xiong, J.: Nonlinear filtering with signal dependent observation noise. Electron. J. Probab. 14(63), 1863–1883 (2009)

    MATH  MathSciNet  Google Scholar 

  6. Driver, B.K., Thalmaier, A.: Heat equation derivative formulas for vector bundles. J. Funct. Anal. 183(1), 42–108 (2001)

    Article  MATH  MathSciNet  Google Scholar 

  7. Driver, B.K., Melcher, T.: Hypoelliptic heat kernel inequalities on the Heisenberg group. J. Funct. Anal. 221(2), 340–365 (2005)

    Article  MATH  MathSciNet  Google Scholar 

  8. Elworthy, K.D.: Geometric aspects of diffusions on manifolds. In: Hennequin, P.L. (ed.) Ecole d’Eté de Probabilités de Saint-Flour XV-XVII, 1985–1987. Lecture Notes in Mathematics 1362, vol. 1362, pp. 276–425. Springer (1988)

    Google Scholar 

  9. Elworthy, K.D., Yor, M.: Conditional expectations for derivatives of certain stochastic flows. In: Azéma, J., Meyer, P.A., Yor, M. (eds.) Sem. de Prob. XXVII. Lecture Notes in Maths. 1557, pp. 159–172. Springer (1993)

    Google Scholar 

  10. Elworthy, K.D., LeJan, Y., Li, X.-M.: Concerning the geometry of stochastic differential equations and stochastic flows. In: Elworthy, K.D., Kusuoka, S., Shigekawa, I. (eds.) ‘New Trends in stochastic Analysis’, Proc. Taniguchi Symposium, Sept. 1994, Charingworth. World Scientific Press (1996)

    Google Scholar 

  11. Elworthy, K.D., LeJan, Y., Li, X.-M.: On the geometry of diffusion operators and stochastic flows, Lecture Notes in Mathematics 1720. Springer (1999)

    Google Scholar 

  12. Elworthy, K.D., Le Jan, Y., Li, X.-M.: Equivariant diffusions on principal bundles. In: Stochastic analysis and related topics in Kyoto, vol. 41 of Adv. Stud. Pure Math., pp. 31–47. Math. Soc. Japan, Tokyo (2004)

    Google Scholar 

  13. Elworthy, K.D., Le Jan, Y., Li, X.-M.: The Geometry of Filtering. To appear in Frontiers in Mathematics Series, Birkhauser

    Google Scholar 

  14. Kunita, H.: Cauchy problem for stochastic partial differential equations arising in nonlinear filtering theory. Syst. Control Lett. 1(1), 37–41 (1981/1982)

    Google Scholar 

  15. LeJan, Y., Watanabe, S.: Stochastic flows of diffeomorphisms. In: Stochastic analysis (Katata/Kyoto, 1982), North-Holland Math. Library, 32, pp. 307–332. North-Holland, Amsterdam (1984)

    Google Scholar 

  16. Lázaro-Camí, J., Ortega, J.: Reduction, reconstruction, and skew-product decomposition of symmetric stochastic differential equations. Stoch. Dyn. 9(1), 1–46 (2009)

    Article  MATH  MathSciNet  Google Scholar 

  17. Liao, M.: Factorization of diffusions on fibre bundles. Trans. Am. Math. Soc. 311(2), 813–827 (1989)

    Article  MATH  Google Scholar 

  18. Li, X.-M.: Stochastic flows on noncompact manifolds. Thesis, University Of Warwick (1992)

    Google Scholar 

  19. Li, X.-M.: Stochastic differential equations on non-compact manifolds: moment stability and its topological consequences. Probab. Theory Relat. Fields. 100(4), 417–428 (1994)

    Article  MATH  Google Scholar 

  20. Li, X.-M.: Strong p-completeness of stochastic differential equations and the existence of smooth flows on non-compact manifolds. Probab. Theory Relat. Fields 100(4), 485–511 (1994)

    Article  MATH  Google Scholar 

  21. Li, X.-M.: On extensions of Myers’ theorem. Bull. London Math. Soc. 27, 392–396 (1995)

    Article  MATH  MathSciNet  Google Scholar 

  22. Norris, J.R.: Path integral formulae for heat kernels and their derivatives. Probab. Theory Relat. Fields 94, 525–541 (1993)

    Article  MATH  MathSciNet  Google Scholar 

  23. Pitman, J.: One dimensional Brownian motion and the three dimensional Bessel process. Adv. Appl. Probab. 7, 511–526 (1975)

    Article  MATH  MathSciNet  Google Scholar 

  24. Thalmaier, A., Wang, F.: Gradient estimates for harmonic functions on regular domains in Riemannian manifolds. J. Funct. Anal. 155, 109–124 (1998)

    Article  MATH  MathSciNet  Google Scholar 

Download references

Acknowledgements

This article is based on the books [11, 13] and could and should be considered as joint work with K.D. Elworthy and Y. LeJan and I would like to thank them for helpful discussions. However, any short comings and errors are my sole responsibility. This research is supported by an EPSRC grant (EP/E058124/1).

Author information

Authors and Affiliations

  1. Mathematics Institute, University of Warwick, Coventry, CV4 7AL, United Kingdom

    Xue-Mei Li

Authors
  1. Xue-Mei Li
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Xue-Mei Li .

Editor information

Editors and Affiliations

  1. Department of Mathematics, Imperial College London, South Kensington Campus, London, SW7 2AZ, United Kingdom

    Dan Crisan

Rights and permissions

Reprints and permissions

Copyright information

© 2011 Springer-Verlag Berlin Heidelberg

About this chapter

Cite this chapter

Li, XM. (2011). Intertwinned Diffusions by Examples. In: Crisan, D. (eds) Stochastic Analysis 2010. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-15358-7_4

Download citation

Publish with us

Policies and ethics

Search

Navigation