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Extended Stochastic Integral and Wick Calculus on Spaces of Regular Generalized Functions Connected with Gamma Measure

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Ukrainian Mathematical Journal Aims and scope

Abstract

We introduce and study an extended stochastic integral, a Wick product, and Wick versions of holomorphic functions on Kondrat'ev-type spaces of regular generalized functions. These spaces are connected with the Gamma measure on a certain generalization of the Schwartz distribution space \(S'\). As examples, we consider stochastic equations with Wick-type nonlinearity.

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REFERENCES

  1. Yu. G. Kondratiev, J. Luis da Silva, L. Streit, and G. F. Us, “Analysis on Poisson and Gamma spaces,” Quantum. Probab. Relat. Top., 1, 91–117 (1998).

    MATH  Google Scholar 

  2. A. Dermoune, “Distributions sur l'espace de P. Levy et calcul stochastique,” Ann. IHP. Prob. Stat., 26, No.1, 101–120 (1990).

    MathSciNet  MATH  Google Scholar 

  3. N. A. Kachanovsky, “On the extended stochastic integral connected with the Gamma measure on an infinite-dimensional space,” Meth. Funct. Anal. Top., 8, No.2, 10–32 (2002).

    MathSciNet  MATH  Google Scholar 

  4. Yu. M. Berezansky and D. A. Merzejewski, “The structure of extended symmetric Fock space,” Meth. Funct. Anal. Top., 6, No.4, 1–13 (2000).

    MATH  Google Scholar 

  5. Yu. G. Kondratiev and E. W. Lytwynov, “Operators of Gamma white noise calculus,” Infin. Dimen. Anal., Quantum Probab. Relat. Top., 3, No.3, 303–335 (2000).

    MathSciNet  MATH  Google Scholar 

  6. J. Meixner, “Orthogonale polynom systeme mit einer besonderen gestalt der erzeugenden funktion,” J. London Math. Soc., 9, No.1, 6–13 (1934).

    MATH  Google Scholar 

  7. S. Dineen, Complex Analysis in Locally Convex Spaces, North Holland, Amsterdam (1981).

    MATH  Google Scholar 

  8. Yu. M. Berezans'kyi and Yu. G. Kondrat'ev, Spectral Methods in Infinite-Dimensional Analysis [in Russian], Naukova Dumka, Kiev (1988). English translation: Kluwer, Dordrecht (1995).

    Google Scholar 

  9. N. A. Kachanovsky, “A generalized Malliavin derivative connected with the Poisson-and Gamma-measures,” Meth. Funct. Anal. Top., 9, No.3, 213–240 (2003).

    MathSciNet  MATH  Google Scholar 

  10. A. V. Skorokhod, Integration in a Hilbert Space, Springer, Berlin (1974).

    Google Scholar 

  11. M. Grothaus, Yu. G. Kondratiev, and L. Streit, “Regular generalized functions in Gaussian analysis,” Infin. Dimen. Anal., Quantum Probab. Relat. Top., 2, No.3, 359–380 (1999).

    MathSciNet  MATH  Google Scholar 

  12. N. A. Kachanovsky, “Biorthogonal Appell-like systems in a Hilbert space,” Meth. Funct. Anal. Top., 2, No.3–4, 36–52 (1996).

    MathSciNet  Google Scholar 

  13. Yu. G. Kondratiev, J. Luis da Silva, and L. Streit, “Generalized Appell systems,” Meth. Funct. Anal. Top., 3, No.3, 28–61 (1997).

    MATH  Google Scholar 

  14. N. A. Kachanovskii, “Dual Appell system and Kondrat'ev spaces in analysis on Schwartz spaces,” Ukr. Mat. Zh., 49, No.4, 527–534 (1997).

    Article  MathSciNet  MATH  Google Scholar 

  15. K. Ito, “Multiple Wiener integral,” J. Math. Soc. Jpn., 3, 157–169 (1951).

    Article  MATH  Google Scholar 

  16. T. Hida, H. H. Kuo, J. Potthoff, and L. Streit, White Noise. An Infinite Dimensional Calculus, Kluwer, Dordrecht (1993).

    MATH  Google Scholar 

  17. Yu. M. Kabanov and A. V. Skorokhod, “On extended stochastic integrals,” in: Proceedings of the School-Seminar on the Theory of Stochastic Processes [in Russian], Part 1, Institute of Physics and Mathematics, Vilnius (1975), pp. 123–167.

    Google Scholar 

  18. Yu. G. Kondratiev, L. Streit, W. Westerkamp, and J. Yan, “Generalized functions in infinite-dimensional analysis,” Hiroshima Math. Zh., 28, 213–260 (1998).

    MathSciNet  MATH  Google Scholar 

  19. N. A. Kachanovsky, “On an analog of stochastic integral and Wick calculus in non-Gaussian infinite-dimensional analysis,” Meth. Funct. Anal. Top., 3, No.3, 1–12 (1997).

    MathSciNet  Google Scholar 

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Authors and Affiliations

  1. Institute of Mathematics, Ukrainian Academy of Sciences, Kiev, Ukraine

    N. A. Kachanovskii

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  1. N. A. Kachanovskii
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This paper is dedicated to Professor Yu. M. Berezansky, who is one of my mentors

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Published in Ukrains'kyi Matematychnyi Zhurnal, Vol. 57, No. 8, pp. 1030–1057, August, 2005.

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Kachanovskii, N.A. Extended Stochastic Integral and Wick Calculus on Spaces of Regular Generalized Functions Connected with Gamma Measure. Ukr Math J 57, 1214–1248 (2005). https://doi.org/10.1007/s11253-005-0258-8

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  • DOI: https://doi.org/10.1007/s11253-005-0258-8

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