Annals of Functional Analysis

Applications to the Cameron--Storvick type theorem with respect to the Gaussian process

Abstract

In this paper, we establish a Cameron--Storvick type theorem with respect to the Gaussian process. We then use this theorem to obtain various integration formulas involving the transform, the $\diamond$-product and the first variation.

Article information

Source
Ann. Funct. Anal., Volume 6, Number 2 (2015), 11-25.

Dates
First available in Project Euclid: 19 December 2014

https://projecteuclid.org/euclid.afa/1418997762

Digital Object Identifier
doi:10.15352/afa/06-2-2

Mathematical Reviews number (MathSciNet)
MR3292511

Zentralblatt MATH identifier
1321.60073

Citation

Lee, Il Yong; Chung, Hyun Soo; Chang, Seung Jun. Applications to the Cameron--Storvick type theorem with respect to the Gaussian process. Ann. Funct. Anal. 6 (2015), no. 2, 11--25. doi:10.15352/afa/06-2-2. https://projecteuclid.org/euclid.afa/1418997762

References

• R.H. Cameron and W.T. Martin, Fourier-Wiener transforms of functionals belonging to $L_2$ over the space $C$, Duke Math. J. 14 (1947), 99–107.
• R.H. Cameron and D.A. Storvick, Feynman integral of variations of functionals, in: Gaussian Random Fields, World Scientific, Singapore (1991), 144-157.
• S.J. Chang, J.G. Choi and D. Skoug, Integration by parts formulas involving generalized Fourier-Feynman transforms on function space, Trans. Amer. Math. Soc. 355 (2003), 2925–2948.
• S.J. Chang and H.S. Chung, Generalized Fourier-Wiener function space transforms, J. Korean Math. Soc. 46 (2009), 327-345.
• S.J. Chang, H.S. Chung and D. Skoug, Integral transforms of functionals in $L^2(C_{a,b}[0,T])$, J. Fourier Anal. Appl. 15 (2009), 441-462.
• S.J. Chang, H.S. Chung and D. Skoug, Some basic relationships among transforms, convolution products, first variations and inverse transforms, Cent. Eur. J. Math. 11 (2013), 538-551.
• S.J. Chang and D. Skoug, Generalized Fourier-Feynman transforms and a first variation on function space, Integral Transforms Spec. Funct. 14 (2003), 375-393.
• D.M. Chung, Scale-invariant measurability in abstract Wiener spaces, Pacific J. Math. 130 (1987), 27–40.
• D.M. Chung, C. Park and D. Skoug, Generalized Feynman integrals via conditional Feynamn integrals, Michigan Math. J. 40 (1993), 337-391.
• H.S. Chung, D. Skoug and S.J. Chang, Relationships involving transform and convolutions via the translation theorem, Stoch. Anal. Appl. 32 (2014), 348–363.
• T. Huffman, D. Skoug and D. Storvick, Integration formulas involving Fourier-Feynman transforms via a Fubini theorem, J. Korean Math. Soc. 38 (2001), 421–435.
• G.W. Johnson and D.L. Skoug, Scale-invariant measurability in Wiener space, Pacific J. Math. 83 (1979), 157–176.
• B.S. Kim and D. Skoug, Integral transforms of functionals in $L_2(C_0[0,T])$, Rocky Mountain J. Math. 33 (2003), 1379-1393.
• B.S. Kim and B.S. Kim, Parts formulas involving integral transforms on function space, Commun. Korean Math. Soc. 22 (2007), 553–564.
• I.Y. Lee, H.S. Chung and S.J. Chang, Relationships among the transform with respect to the Gaussian process, the $\diamond$-product and the first variation, of functionals on function space, preprint.
• I.Y. Lee, H.S. Chung and S.J. Chang, Integration formulas involving the transform with respect to the Gaussian process on Wiener space, preprint.
• C. Park, D. Skoug and D.A. Storvick, Relationships among the first variation, the convolution product and the Fourier-Feynman transform, Rocky Mountain J. Math. 27 (1997), no. 3, 827–841.
• J. Yeh, Stochastic Processes and the Wiener Integral, Marcel Dekker, Inc., New York, 1973.