Acta Applicandae Mathematica

, Volume 35, Issue 1–2, pp 63–102 | Cite as

Nonlinear transformations of the canonical gauss measure on Hilbert space and absolute continuity

  • G. Kallianpur
  • R. L. Karandikar
Article
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Abstract

The papers of R. Ramer and S. Kusuoka investigate conditions under which the probability measure induced by a nonlinear transformation on abstract Wiener space(γ,H,B) is absolutely continuous with respect to the abstract Wiener measureμ. These conditions reveal the importance of the underlying Hilbert spaceH but involve the spaceB in an essential way. The present paper gives conditions solely based onH and takes as its starting point, a nonlinear transformationT=I+F onH. New sufficient conditions for absolute continuity are given which do not seem easily comparable with those of Kusuoka or Ramer but are more general than those of Buckdahn and Enchev. The Ramer-Itô integral occurring in the expression for the Radon-Nikodym derivative is studied in some detail and, in the general context of white noise theory it is shown to be an anticipative stochastic integral which, under a stronger condition on the weak Gateaux derivative of F is directly related to the Ogawa integral.

Mathematics subject classifications (1991)

Primary: 60B11 60H05 60H07 Secondary: 28C20 46G12 

Key words

Nonlinear transformation Wiener measure liftings Gohberg-Krein factorization absolute continuity Ramer-Ito integrals 

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References

  1. 1.
    Buckdahn, R.: Anticipative Girsanov transformations,Probab. Theory Related Fields 89 (1991), 211–238.Google Scholar
  2. 2.
    Cameron, R. H. and Martin, W. T.: The transformations of Wiener integral by nonlinear transformations,Trans. Amer. Math. Soc. 66 (1949), 253–283.Google Scholar
  3. 3.
    Dellacherie, C. and Meyer, P. A.:Probabilities and Potential, North-Holland, Amsterdam, 1975.Google Scholar
  4. 4.
    Enchev, O.: Anticipative Girsanov transformations, Preprint, 1991.Google Scholar
  5. 5.
    Feldman, J.: Equivalance and perpendicularity of Gaussian processes,Pacific J. Math. 8 (1958), 699–708.Google Scholar
  6. 6.
    Gohberg, I. C. and Krein, M. G.:Introduction to Linear Non-self Adjoint Operators (English trans.), Amer. Math. Soc. Providence, 1969.Google Scholar
  7. 7.
    Gohberg, I. C. and Krein, M. G.:Theory and Applications of Volterra Operators in Hilbert Space (English trans.) Amer. Math. Soc. Providence, 1970.Google Scholar
  8. 8.
    Gross, L.: Integration and nonlinear transformations in Hilbert space,Trans. Amer. Math. Soc. 94 (1960), 404–440.Google Scholar
  9. 9.
    Johnson, G. W. and Kallianpur, G.: Homogeneous chaos,p-forms, scaling and the Feynman integral,Trans. Amer. Math. Soc. 340 (1993), 503–548.Google Scholar
  10. 10.
    Kailath, T. and Duttweiler, D.: An RKHS approach to detection and estimation problems — Part III: Generalized innovations representations and a likelihood-ratio formula,IEEE Trans. Inf. Theory IT-18 (1972), 730–745.Google Scholar
  11. 11.
    Kallianpur, G. and Oodaira, H.: Non-anticipative representations of equivalent Gaussian processes,Ann. Probab. 1 (1973), 104–122.Google Scholar
  12. 12.
    Kallianpur, G.:Stochastic Filtering Theory, Springer-Verlag, New York, 1980.Google Scholar
  13. 13.
    Kallianpur, G. and Karandikar, R. L.:White Noise Theory of Prediction, Filtering and Smoothing, Gordon and Breach, New York, 1988.Google Scholar
  14. 14.
    Kusuoka, S.: The nonlinear transformation of Gaussian measure on Banach space and its absolute continuity (I),J. Fac. Sci. Univ. Tokyo Sec. 1A Math. 24 (1982), 567–597.Google Scholar
  15. 15.
    Nualart, D.: Nonlinear transformations of the Wiener measure and applications, in E. Meyer-Wolfet al. (eds),Stochastic Analysis, Academic Press, Boston, 1991.Google Scholar
  16. 16.
    Ramer, R.: On nonlinear transformations of Gaussian measures,J. Func. Anal. 15 (1974), 166–187.Google Scholar
  17. 17.
    Ustunel, A. S. and Zakai, M.: Transformation of Wiener measure under anticipative flows, Preprint, 1990.Google Scholar

Copyright information

© Kluwer Academic Publishers 1994

Authors and Affiliations

  • G. Kallianpur
    • 1
  • R. L. Karandikar
    • 2
    • 3
  1. 1.Center for Stochastic Processes, Department of StatisticsUniversity of North CarolinaChapel HillUSA
  2. 2.Indian Statistical InstituteNew DelhiIndia
  3. 3.Center for Stochastic ProcessesUniversity of North CarolinaChapel HillUSA

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