DOI QR코드

DOI QR Code

A TIME-INDEPENDENT CONDITIONAL FOURIER-FEYNMAN TRANSFORM AND CONVOLUTION PRODUCT ON AN ANALOGUE OF WIENER SPACE

Cho, Dong Hyun

  • 투고 : 2013.03.18
  • 심사 : 2013.04.29
  • 발행 : 2013.06.25

초록

Let $C[0,t]$ denote the function space of all real-valued continuous paths on $[0,t]$. Define $X_n:C[0,t]{\rightarrow}\mathbb{R}^{n+1}$ by $Xn(x)=(x(t_0),x(t_1),{\cdots},x(t_n))$, where $0=t_0$ < $t_1$ < ${\cdots}$ < $t_n$ < $t$ is a partition of $[0,t]$. In the present paper, using a simple formula for the conditional expectation given the conditioning function $X_n$, we evaluate the $L_p(1{\leq}p{\leq}{\infty})$-analytic conditional Fourier-Feynman transform and the conditional convolution product of the cylinder functions which have the form $$f((v_1,x),{\cdots},(v_r,x))\;for\;x{\in}C[0,t]$$, where {$v_1,{\cdots},v_r$} is an orthonormal subset of $L_2[0,t]$ and $f{\in}L_p(\mathbb{R}^r)$. We then investigate several relationships between the conditional Fourier-Feynman transform and the conditional convolution product of the cylinder functions.

키워드

analogue of Wiener measure;analytic conditional Feynman integral;analytic conditional Fourier-Feynman transform;analytic conditional Wiener integral;conditional convolution product

참고문헌

  1. Cho D. H., Kim B. J., Yoo I., Analogues of conditional Wiener integrals and their change of scale transformations on a function space, J. Math. Anal. Appl. 359 (2009), 421-438. https://doi.org/10.1016/j.jmaa.2009.05.023
  2. Folland G. B., Real analysis, John Wiley & Sons, New York-Chichester-Brisbane, 1984.
  3. Huffman T., Park C., Skoug D., Analytic Fourier-Feynman transforms and convolution, Trans. Amer. Math. Soc. 347(2) (1995), 661-673. https://doi.org/10.1090/S0002-9947-1995-1242088-7
  4. Im M. K., Ryu K. S., An analogue of Wiener measure and its applications, J. Korean Math. Soc. 39(5) (2002), 801-819. https://doi.org/10.4134/JKMS.2002.39.5.801
  5. Johnson G. W., Skoug D. L., The Cameron-Storvick function space integral: an L($L_p,\;L_p'$)-theory, Nagoya Math. J. 60 (1976), 93-137. https://doi.org/10.1017/S0027763000017189
  6. Kim M. J., Conditional Fourier-Feynman transform and convolution product on a function space, Int. J. Math. Anal. 3(10) (2009), 457-471.
  7. Laha R. G., Rohatgi V. K., Probability theory, John Wiley & Sons, New York-Chichester-Brisbane, 1979.
  8. Ryu K. S., Im M. K., A measure-valued analogue of Wiener measure and the measure-valued Feynman-Kac formula, Trans. Amer. Math. Soc. 354(12) (2002), 4921-4951. https://doi.org/10.1090/S0002-9947-02-03077-5
  9. Ryu K. S., Im M. K., Choi K. S., Survey of the theories for analogue of Wiener measure space, Interdiscip. Inform. Sci. 15(3) (2009), 319-337.
  10. Stein E. M., Weiss G., Introduction to Fourier analysis on Euclidean spaces, Princeton Univ. Press, Princeton, 1971.
  11. Yeh J., Stochastic processes and the Wiener integral, Marcel Dekker, New York, 1973.
  12. Brue M. D., A functional transform for Feynman integrals similar to the Fourier transform, Thesis, Univ. of Minnesota, Minneapolis, 1972.
  13. Cameron R. H., Storvick D. A., Some Banach algebras of analytic Feynman integrable functionals, Lecture Notes in Mathematics 798, Springer, Berlin-New York, 1980.
  14. Chang K. S., Cho D. H., Kim B. S., Song T. S., Yoo I., Conditional Fourier-Feynman transform and convolution product over Wiener paths in abstract Wiener space, Integral Transforms Spec. Funct. 14(3) (2003), 217-235. https://doi.org/10.1080/1065246031000081652
  15. Chang S. J., Skoug D., The effect of drift on conditional Fourier-Feynman transforms and conditional convolution products, Int. J. Appl. Math. 2(4) (2000), 505-527.
  16. Cho D. H., A time-dependent conditional Fourier-Feynman transform and convolution product on an analogue of Wiener space, Houston J. Math. 2012, submitted.
  17. Cho D. H., Conditional integral transforms and conditional convolution products on a function space, Integral Transforms Spec. Funct. 23(6) (2012), 405-420. https://doi.org/10.1080/10652469.2011.596482
  18. Cho D. H., A simple formula for an analogue of conditional Wiener integrals and its applications II, Czechoslovak Math. J. 59(2) (2009), 431-452. https://doi.org/10.1007/s10587-009-0030-6
  19. Cho D. H., Conditional Fourier-Feynman transform and convolution product over Wiener paths in abstract Wiener space: an $L_p$ theory, J. Korean Math. Soc. 41(2) (2004), 265-294. https://doi.org/10.4134/JKMS.2004.41.2.265

피인용 문헌

  1. CONDITIONAL FOURIER-FEYNMAN TRANSFORMS AND CONVOLUTIONS OF UNBOUNDED FUNCTIONS ON A GENERALIZED WIENER SPACE vol.50, pp.5, 2013, https://doi.org/10.5831/HMJ.2013.35.2.179

과제정보

연구 과제 주관 기관 : Kyonggi University