충청수학회지 (Journal of the Chungcheong Mathematical Society)

  • 제23권4호
  • /
  • Pages.711-720
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  • 2010
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  • 1226-3524(pISSN)
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  • 2383-6245(eISSN)
충청수학회 (The Chungcheong Mathematical Society)
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AN INTEGRATION FORMULA FOR ANALOGUE OF WIENER MEASURE AND ITS APPLICATIONS

  • Ryu, Kun Sik (Department of Mathematics Educations Hannam University)
  • 투고 : 2010.07.28
  • 심사 : 2010.11.09
  • 발행 : 2010.12.30
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초록

In this note, we will establish the integration formulae for functionals such as $F(x)=\prod_{j=1}^{n}\;x(s_j)^2$ and $G(x)=\exp\{{\lambda}{\int}_{0}^{t}\;x(s)^2dm_L(s)\}$ in the analogue of Wiener measure space and using our formulae, we will derive some formulae for series.

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참고문헌

  1. T. S. Chiang, Y. Chow and Y. J. Lee, A formula for EW exp$-2^{-1}a^{2}II{x+y}II^{2}_{2}$}, Proc. Amer. Math. Soc. 100 (1987), 721-724.
  2. C. George, Exercises in integration, Springer-Verlag New York, 1984.
  3. 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), 4921-4951. https://doi.org/10.1090/S0002-9947-02-03077-5
  4. K. S. Ryu and M. K. Im, An analogue of Wiener measure and its applications, J. Korean Math. Soc. 39 (2002), 801-819. https://doi.org/10.4134/JKMS.2002.39.5.801
  5. K. S. Ryu, The operational calculus for a measure-valued Dyson series, J. Korean Math. Soc. 43 (2006), 703-715. https://doi.org/10.4134/JKMS.2006.43.4.703
  6. K. S. Ryu, The Dobrakov integral over paths, J. Chungcheong Math. Soc. 19 (2006), 61-68.
  7. K. S. Ryu and M. K. Im, The measure-valued Dyson series and its stability theorem, J. Korean Math. Soc. 43 (2006), 461-489. https://doi.org/10.4134/JKMS.2006.43.3.461
  8. K. S. Ryu, The simple formula of conditional expectation on analogue of Wiener measure, Honam Math. J. 30 (2008), 723-732. https://doi.org/10.5831/HMJ.2008.30.4.723
  9. K. S. Ryu and S. H. Shim, The rotation theorem on analogue of Wiener mea- sure, Honam Math. J. 29 (2007), 577-588. https://doi.org/10.5831/HMJ.2007.29.4.577
  10. K. S. Ryu, The generalized Fernique's theorem for analogue of Wiener measure space, J. Chungcheong Math. Soc. 22 (2009), 743-748.
  11. N. Wiener, Differential space, J. Math. Phys. 2 (1923), 131-174. https://doi.org/10.1002/sapm192321131