Intertwinned Diffusions by Examples



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.


Linear and equi-variant connections stochastic differential equations geometry of diffusion operators 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.



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).


  1. 1.
    Anderson, A., Camporesi, R.: Intertwining operators for solving differential equations, with applications to symmetric spaces. Commun. Math. Phys. 130, 61–82 (1990)MATHCrossRefMathSciNetGoogle Scholar
  2. 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)MATHMathSciNetGoogle Scholar
  3. 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. 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. 5.
    Crisan, D., Kouritzin, M., Xiong, J.: Nonlinear filtering with signal dependent observation noise. Electron. J. Probab. 14(63), 1863–1883 (2009)MATHMathSciNetGoogle Scholar
  6. 6.
    Driver, B.K., Thalmaier, A.: Heat equation derivative formulas for vector bundles. J. Funct. Anal. 183(1), 42–108 (2001)MATHCrossRefMathSciNetGoogle Scholar
  7. 7.
    Driver, B.K., Melcher, T.: Hypoelliptic heat kernel inequalities on the Heisenberg group. J. Funct. Anal. 221(2), 340–365 (2005)MATHCrossRefMathSciNetGoogle Scholar
  8. 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. 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. 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. 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. 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. 13.
    Elworthy, K.D., Le Jan, Y., Li, X.-M.: The Geometry of Filtering. To appear in Frontiers in Mathematics Series, BirkhauserGoogle Scholar
  14. 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. 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. 16.
    Lázaro-Camí, J., Ortega, J.: Reduction, reconstruction, and skew-product decomposition of symmetric stochastic differential equations. Stoch. Dyn. 9(1), 1–46 (2009)MATHCrossRefMathSciNetGoogle Scholar
  17. 17.
    Liao, M.: Factorization of diffusions on fibre bundles. Trans. Am. Math. Soc. 311(2), 813–827 (1989)MATHCrossRefGoogle Scholar
  18. 18.
    Li, X.-M.: Stochastic flows on noncompact manifolds. Thesis, University Of Warwick (1992)Google Scholar
  19. 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)MATHCrossRefGoogle Scholar
  20. 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)MATHCrossRefGoogle Scholar
  21. 21.
    Li, X.-M.: On extensions of Myers’ theorem. Bull. London Math. Soc. 27, 392–396 (1995)MATHCrossRefMathSciNetGoogle Scholar
  22. 22.
    Norris, J.R.: Path integral formulae for heat kernels and their derivatives. Probab. Theory Relat. Fields 94, 525–541 (1993)MATHCrossRefMathSciNetGoogle Scholar
  23. 23.
    Pitman, J.: One dimensional Brownian motion and the three dimensional Bessel process. Adv. Appl. Probab. 7, 511–526 (1975)MATHCrossRefMathSciNetGoogle Scholar
  24. 24.
    Thalmaier, A., Wang, F.: Gradient estimates for harmonic functions on regular domains in Riemannian manifolds. J. Funct. Anal. 155, 109–124 (1998)MATHCrossRefMathSciNetGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2011

Authors and Affiliations

  1. 1.Mathematics InstituteUniversity of WarwickCoventryUnited Kingdom

Personalised recommendations