Ukrainian Mathematical Journal

, Volume 52, Issue 9, pp 1403–1431 | Cite as

Nonlinear Transformations of Smooth Measures on Infinite-Dimensional Spaces

  • A. M. Kulik
  • A. Yu. Pilipenko


We investigate the properties of the image of a differentiable measure on an infinitely-dimensional Banach space under nonlinear transformations of the space. We prove a general result concerning the absolute continuity of this image with respect to the initial measure and obtain a formula for density similar to the Ramer–Kusuoka formula for the transformations of the Gaussian measure. We prove the absolute continuity of the image for classes of transformations that possess additional structural properties, namely, for adapted and monotone transformations, as well as for transformations generated by a differential flow. The latter are used for the realization of the method of characteristics for the solution of infinite-dimensional first-order partial differential equations and linear equations with an extended stochastic integral with respect to the given measure.


Differential Equation Banach Space Partial Differential Equation Linear Equation Structural Property 
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.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    I. I. Gikhman and A. V. Skorokhod, “On densities of probability measures in functional spaces,” Usp. Mat. Nauk, 21, No.6, 83–152 (1966).Google Scholar
  2. 2.
    A. V. Skorokhod, Integration in Hilbert Spaces[in Russian], Nauka, Moscow (1975).Google Scholar
  3. 3.
    R. Ramer, “On nonlinear transformations of Gaussian measures,” J. Funct. Anal., 15, 166–187 (1974).Google Scholar
  4. 4.
    S. Kusuoka, “The nonlinear transformation of Gaussian measure on Banach space and its absolute continuity. I, II,” J. Fac. Sci. Univ. Tokyo Sec. 1A, 29, No.3, 567–598 (1982); 30, No. 1, 199–220 (1983).Google Scholar
  5. 5.
    V. I. Bogachev, “Differentiable measures and Malliavin calculus,” J. Math. Sci., 87, No.5, 3577–3731 (1997).Google Scholar
  6. 6.
    Yu. L. Daletskii and G. A. Sokhadze, “Absolute continuity of smooth measures,” Funkts. Anal. Prilozh., 22, No.2, 77–78 (1988).Google Scholar
  7. 7.
    O. G. Smolyanov and H. V. Weizsacer, “Differentiable families of measures,” J. Funct. Anal., 118, 454–476 (1993).Google Scholar
  8. 8.
    V. I. Averbukh, O. G. Smolyanov, and S. V. Fomin, “Generalized functions and differential equations in linear spaces. 1. Differentiable measures,” Tr. Mosk. Mat. Obshch., 24, 133–174 (1971).Google Scholar
  9. 9.
    H. Federer, Geometric Measure Theory[Russian translation], Nauka, Moscow (1987).Google Scholar
  10. 10.
    A. V. Skorokhod, Random Linear Operators[in Russian], Naukova Dumka, Kiev (1978).Google Scholar
  11. 11.
    O. Enchev and D. W. Strook, “Rademacher's theorem for Wiener functionals,” Ann. Probab., 21, No.1, 25–33 (1993).Google Scholar
  12. 12.
    A. Yu. Pilipenko, “Anticipate analogues of diffusion processes,” Theory Stochast. Process., 3(19), Nos. 3–4, 363–372 (1997).Google Scholar
  13. 13.
    A. M. Kulik, “Large deviations for smooth measures,” Theory Stochast. Process., 4(20), Nos. 1–2, 180–188 (1998).Google Scholar
  14. 14.
    I. Ts. Gokhberg and M. G. Krein, Introduction to the Theory of Linear Nonself-Adjoint Operators in Hilbert Spaces[in Russian], Nauka, Moscow (1965).Google Scholar
  15. 15.
    A. A. Dorogovtsev, Stochastic Equations with Anticipation[in Russian], Institute of Mathematics, Ukrainian Academy of Sciences, Kiev (1996).Google Scholar
  16. 16.
    V. I. Bogachev, Gaussian Measures[in Russian], Nauka, Moscow (1997).Google Scholar
  17. 17.
    S. Smale, “An infinite dimensional version of Sard's theorem,” Amer. J. Math., 87, 861–866 (1965).Google Scholar
  18. 18.
    A. M. Kulik, “Integral representation for functionals on a space with a smooth measure,” Theory Stochast. Process., 3, Nos.1–2, 235–244 (1997).Google Scholar
  19. 19.
    A. B. Cruseiro, “Equations differentielles sur l'espace de Wiener et formules de Cameron – Martin non-lineaires,” J. Funct. Anal., 54, 206–227 (1983).Google Scholar
  20. 20.
    V. I. Bogachev and E. Mayer-Wolf, Absolutely Continuous Flows Generated by Sobolev-Class Vector Fields in Finite and Infinite Dimensions, Preprint No. SFB343, University of Bielefeld, Bielefeld (1994).Google Scholar
  21. 21.
    E. A. Coddington and N. Levinson, Theory of Ordinary Differential Equations[Russian translation], Inostrannaya Literatura, Moscow (1958).Google Scholar
  22. 22.
    Yu. L. Daletskii and Ya. I. Belopol'skaya, Stochastic Differential Geometry[in Russian], Vyshcha Shkola, Kiev (1989).Google Scholar
  23. 23.
    A. A. Dorogovtsev, “Linear equations with generalized stochastic integrals,” Ukr. Mat. Zh., 41, No.12, 1714–1716 (1989).Google Scholar
  24. 24.
    A. A. Dorogovtsev, Stochastic Analysis and Random Maps in Hilbert Space, VSP, Utrecht (1994).Google Scholar
  25. 25.
    R. Courant, Partial Differential Equations[Russian translation], Mir, Moscow (1964).Google Scholar
  26. 26.
    R. Buckdahn, “Anticipate Girsanov transformations and Skorokhod stochastic differential equations,” Mem. Amer. Math. Soc., 111, No.533, 1–88 (1994).Google Scholar
  27. 27.
    O. Enchev and D. W. Stroock, “Anticipate diffusion and related changes of measure,” J. Funct. Anal., 116, 449–473 (1993).Google Scholar
  28. 28.
    A. S. Ustunel and M. Zakai, “Transformation of Wiener measure under anticipative flows,” Probab. Theory Related Fields, 93, 91–136 (1994).Google Scholar
  29. 29.
    A. N. Kolmogorov and S. V. Fomin, Elements of the Theory of Functions and Functional Analysis[in Russian], Nauka, Moscow (1989).Google Scholar

Copyright information

© Plenum Publishing Corporation 2000

Authors and Affiliations

  • A. M. Kulik
    • 1
  • A. Yu. Pilipenko
    • 1
  1. 1.Institute of MathematicsUkrainian Academy of SciencesKiev

Personalised recommendations