Transactions of the American Mathematical Society

Integration by parts formulas involving generalized Fourier-Feynman transforms on function space

Author(s): Seung Jun Chang; Jae Gil Choi; David Skoug
Journal: Trans. Amer. Math. Soc. 355 (2003), 2925-2948.
MSC (2000): Primary 60J65, 28C20
Posted: February 25, 2003
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Abstract: In an upcoming paper, Chang and Skoug used a generalized Brownian motion process to define a generalized analytic Feynman integral and a generalized analytic Fourier-Feynman transform. In this paper we establish several integration by parts formulas involving generalized Feynman integrals, generalized Fourier-Feynman transforms, and the first variation of functionals of the form $F(x)=f(\langle {\alpha _{1} , x}\rangle, \dots , \langle {\alpha _{n} , x}\rangle )$ where $\langle {\alpha ,x}\rangle$ denotes the Paley-Wiener-Zygmund stochastic integral $\int _{0}^{T} \alpha (t) d x(t)$ .

[1]: R. H. Cameron and D. A. Storvick, An $L_{2}$ analytic Fourier-Feynman transform, Michigan Math. J. 23 (1976), 1-30. MR 53:8371
[2]: -, Feynman integral of variations of functions, in Gaussian random fields, Ser. Prob. Statist. 1 (1991), 144-157. MR 93b:28035
[3]: K. S. Chang, B. S. Kim, and I. Yoo, Fourier-Feynman transform, convolution and first variation of functionals on abstract Wiener space, Integral transforms and Special Functions 10 (2000), 179-200. MR 2001m:28023
[4]: S. J. Chang and D. L. Skoug, The effect of drift on the Fourier-Feynman transform, the convolution product and the first variation, Panamerican Math. J. 10 (2000), 25-38.
[5]: -, Generalized Fourier-Feynman transforms and a first variation on function space, to appear in Integral Transforms and Special Functions.
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[7]: -, Generalized transforms and convolutions, Internat. J. Math. and Math. Sci. 20 (1997), 19-32. MR 97k:46047
[8]: G. W. Johnson and D. L. Skoug, An $L_{p}$ Analytic Fourier-Feynman transform, Michigan Math. J. 26 (1979), 103-127. MR 81a:46050
[9]: -, Scale-invariant measurability in Wiener space, Pacific J. Math 83 (1979), 157-176. MR 81b:28016
[10]: E. Nelson, Dynamical theories of Brownian motion (2nd edition), Math. Notes, Princeton University Press, Princeton (1967). MR 35:5001
[11]: C. Park, and D. Skoug, Integration by parts formulas involving analytic Feynman integrals, Panamerican Math. J. 8 (1998), 1-11. MR 99i:46031
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[13]: J. Yeh, Stochastic Processes and the Wiener Integral, Marcel Dekker, Inc., New York (1973). MR 57:14166

Retrieve articles in Transactions of the American Mathematical Society with MSC (2000): 60J65, 28C20

Seung Jun Chang
Affiliation: Department of Mathematics, Dankook University, Cheonan 330-714, Korea
Email: sejchang@dankook.ac.kr

Jae Gil Choi
Affiliation: Department of Mathematics, Dankook University, Cheonan 330-714, Korea
Email: jgchoi@dankook.ac.kr

David Skoug
Affiliation: Department of Mathematics and Statistics, University of Nebraska, Lincoln, Nebraska, 68588-0323
Email: dskoug@math.unl.edu

DOI: 10.1090/S0002-9947-03-03256-2
PII: S 0002-9947(03)03256-2
Keywords: Generalized Brownian motion process, generalized analytic Feynman integral, generalized analytic Fourier-Feynman transform, first variation, Cameron-Storvick type theorem
Received by editor(s): September 6, 2002
Received by editor(s) in revised form: November 15, 2002
Posted: February 25, 2003
Additional Notes: The present research was conducted by the research fund of Dankook University in 2000
Copyright of article: Copyright 2003, American Mathematical Society

R. H. Cameron and D. A. Storvick, An $L_2$ analytic Fourier-Feynman transforms , Michigan Math. J. 23 (1976), 1-30. MR 53:8371

G. W. Johnson and D. L. Skoug, An $L_p$ Analytic Fourier-Feynman transform, Michigan Math. J 26 (1979), 103-127. MR 81a:46050

S. J. Chang and D. L. Skoug, Scale-invariant measurability in Wiener space, Pacific J. Math 83 (1979), 157-176. MR 81b:28016

R. H. Cameron and D. A. Storvick, Feynman integral of variations, in Gaussian random fields, Ser.Prob. Statist. 1 (1991), 144-157. MR 93b:28035

T. Huffman, C. Park and D. Skoug, Analytic Fourier-Feynman transforms and convolution, Trans. Amer. Math. Soc. 347 (1995), 661-673. MR 95d:28017

S. J. Chang and D. L. Skoug, Generalized transforms and convolutions, Internat. J. Math. and Math. Sci 20 (1997), 19-32. MR 97k:46047

C. Park, and D. Skoug, Integration by parts formulas involving analytic Feynman integrals, Panamerican Math. J 8 (1998), 1-11. MR 99i:46031

K. S. Chang, B. S. Kim, and I. Yoo, Fourier-Feynman transform, convolution and first variation of functionals on abstract Wiener space, Integral transforms and Special Functions 10 (2000), 179-200. MR 2001m:28023

S. J. Chang and D. Skoug, The effect of drift on the Fourier-Feynman transforms, the convolution product and the first variation, Panamerican Math. J. 10 (2000), 25-38.

E. Nelson, Dynamical theories of Brownian motion (2nd edition), Math Notes, Princeton University Press, Princeton, 1967. MR 35:5001

J. Yeh, Stochastic Processes and the Wiener Integral, Marcel Dekker, Inc., New York, 1973. MR 57:14166

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