Bulletin of the Korean Mathematical Society (대한수학회보)

Korean Mathematical Society (대한수학회)
권호 표지
DOI QR코드

DOI QR Code

OPERATOR-VALUED FUNCTION SPACE INTEGRALS VIA CONDITIONAL INTEGRALS ON AN ANALOGUE WIENER SPACE II

  • Received : 2015.06.01
  • Published : 2016.05.31
Download PDF
⟨ Previous Next ⟩

Abstract

In the present paper, using a simple formula for the conditional expectations given a generalized conditioning function over an analogue of vector-valued Wiener space, we prove that the analytic operator-valued Feynman integrals of certain classes of functions over the space can be expressed by the conditional analytic Feynman integrals of the functions. We then provide the conditional analytic Feynman integrals of several functions which are the kernels of the analytic operator-valued Feynman integrals.

Keywords

Acknowledgement

Supported by : Kyonggi University

References

  1. R. H. Cameron and D. A. Storvick, An operator-valued function space integral and a related integral equation, J. Math. Mech. 18 (1968), 517-552.
  2. R. H. Cameron and D. A. Storvick, An operator-valued function space integral applied to integrals of functions of class $L_{1}$, Proc. London Math. Soc. (3) 27 (1973), 345-360.
  3. R. H. Cameron and D. A. Storvick, Some Banach algebras of analytic Feynman integrable functionals, Analytic functions, Kozubnik 1979 (Proc. Seventh Conf., Kozubnik, 1979), pp. 18-67, Lecture Notes in Math., 798, Springer, Berlin-New York, 1980.
  4. D. H. Cho, A simple formula for an analogue of conditional Wiener integrals and its applications, Trans. Amer. Math. Soc. 360 (2008), no. 7, 3795-3811. https://doi.org/10.1090/S0002-9947-08-04380-8
  5. D. H. Cho, Operator-valued Feynman integral via conditional Feynman integrals on a function space, Cent. Eur. J. Math. 8 (2010), no. 5, 908-927. https://doi.org/10.2478/s11533-010-0059-7
  6. D. H. Cho, B. J. Kim, and I. Yoo, Analogues of conditional Wiener integrals and their change of scale transformations on a function space, J. Math. Anal. Appl. 359 (2009), no. 2, 421-438. https://doi.org/10.1016/j.jmaa.2009.05.023
  7. D. M. Chung, C. Park, and D. Skoug, Operator-valued Feynman integrals via conditional Feynman integrals, Pacific J. Math. 146 (1990), no. 1, 21-42. https://doi.org/10.2140/pjm.1990.146.21
  8. G. B. Folland, Real Analysis, John Wiley & Sons, New York, 1984.
  9. M. K. Im and K. S. Ryu, An analogue of Wiener measure and its applications, J. Korean Math. Soc. 39 (2002), no. 5, 801-819. https://doi.org/10.4134/JKMS.2002.39.5.801
  10. G. W. Johnson and D. L. Skoug, The Cameron-Storvick function space integral: the $L_{1}$ theory, J. Math. Anal. Appl. 50 (1975), 647-667. https://doi.org/10.1016/0022-247X(75)90017-7
  11. K. S. Ryu and M. K. Im, A measure-valued analogue of Wiener measure and the measure-valued Feynman-Kac formula, Trans. Amer. Math. Soc. 354 (2002), no. 12, 4921-4951. https://doi.org/10.1090/S0002-9947-02-03077-5