Algebraic structure of the $ L_2 $ analytic Fourier–Feynman transform associated with Gaussian paths on Wiener space

Jae Gil Choi; David Skoug

doi:10.3934/cpaa.2020169

Article Contents

2020, Volume 19, Issue 7: 3829-3842. Doi: 10.3934/cpaa.2020169

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Algebraic structure of the $ L_2 $ analytic Fourier–Feynman transform associated with Gaussian paths on Wiener space

Jae Gil Choi^1, , and
David Skoug^2,

1.
School of General Education, Dankook University, Cheonan 31116, Republic of Korea
2.
Department of Mathematics, University of Nebraska-Lincoln, Lincoln, NE 68588-0130, USA

^* Corresponding author
^* Corresponding author

Received: December 2019

Revised: January 2020

Published online: April 27, 2020

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Abstract

In this paper we study algebraic structures of the classes of the $ L_2 $ analytic Fourier–Feynman transforms on Wiener space. To do this we first develop several rotation properties of the generalized Wiener integral associated with Gaussian paths. We then proceed to analyze the $ L_2 $ analytic Fourier–Feynman transforms associated with Gaussian paths. Our results show that these $ L_2 $ analytic Fourier–Feynman transforms are actually linear operator isomorphisms from a Hilbert space into itself. We finally investigate the algebraic structures of these classes of the transforms on Wiener space, and show that they indeed are group isomorphic.

Keywords:

Paley-Wiener-Zygmund stochastic integral,
Gaussian process,
Fourier- Feynman transform associated with Gaussian paths,
monoid isomorphism,
linear operator isomorphism,
free group.

Mathematics Subject Classification: Primary: 28C20, 60G15, 60J65; Secondary: 46B09, 42B10, 46G12.

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References

[1]	J. E. Bearman, Rotations in the product of two Wiener spaces, Proc. Amer. Math. Soc., 3 (1952), 129-137. doi: 10.2307/2032469.
[2]	M. D. Brue, A Functional Transform for Feynman Integrals Similar to the Fourier Transform, Ph.D Thesis, University of Minnesota, 1972.
[3]	R. H. Cameron and D. A. Storvick, An $L_2$ analytic Fourier–Feynman transform, Michigan Math. J., 23 (1976), 1-30.
[4]	R. H. Cameron and D. A. Storvick, An operator valued Yeh–Wiener integral, and a Wiener integral equation, Indiana Univ. Math. J., 25 (1976), 235-258. doi: 10.1512/iumj.1976.25.25020.
[5]	S. J. Chang and J. G. Choi, Rotation of Gaussian paths on Wiener space with applications, Banach J. Math. Anal., 12 (2018), 651-672. doi: 10.1215/17358787-2017-0057.
[6]	S. J. Chang, H. S. Chung and J. G. Choi, Generalized Fourier–Feynman transforms and generalized convolution products on Wiener space, Indag. Math., 28 (2017), 566-579. doi: 10.1016/j.indag.2017.01.004.
[7]	J. G. Choi and S. J. Chang, Note on generalized Wiener integrals, Arch. Math., 101 (2013), 569-579. doi: 10.1007/s00013-013-0595-z.
[8]	J. G. Choi, D. Skoug and S. J. Chang, A multiple generalized Fourier–Feynman transform via a rotation on Wiener space, Int. J. Math., 23 (2012), Art. 1250068. doi: 10.1142/S0129167X12500681.
[9]	D. M. Chung, C. Park and D. Skoug, Generalized Feynman integrals via conditional Feynman integrals, Michigan Math. J., 40 (1993), 377-391. doi: 10.1307/mmj/1029004758.
[10]	T. Huffman, C. Park and D. Skoug, Analytic Fourier-Feynman transforms and convolution, Trans. Amer. Math. Soc., 347 (1995), 661-673. doi: 10.2307/2154908.
[11]	T. Huffman, C. Park and D. Skoug, Generalized transforms and convolutions, Int. J. Math. Math. Sci., 20 (1997), 19-32. doi: 10.1155/S0161171297000045.
[12]	T. Huffman, D. Skoug and D. Storvick, Integration formulas involving Fourier-Feynman transforms via a Fubini theorem, J. Korean Math., 38 (2001), 421-435.
[13]	G. W. Johnson and D. L. Skoug, An $L_p$ analytic Fourier-Feynman transform, Michigan Math. J., 26 (1979), 103-127.
[14]	G. W. Johnson and D. L. Skoug, Scale-invariant measurability in Wiener space, Pac. J. Math., 83 (1979), 157-176.
[15]	G. W. Johnson and D. L. Skoug, Notes on the Feynman integral, II, J. Funct. Anal., 41 (1981), 277-289. doi: 10.1016/0022-1236(81)90075-6.
[16]	R. E. A. C. Paley, N. Wiener and A. Zygmund, Notes on random functions, Math. Z., 37 (1933), 647-668. doi: 10.1007/BF01474606.
[17]	C. Park and D. Skoug, A note on Paley–Wiener–Zygmund stochastic integrals, Proc. Amer. Math. Soc., 103 (1988), 591-601. doi: 10.2307/2047184.
[18]	C. Park and D. Skoug, A Kac-Feynman integral equation for conditional Wiener integrals, J. Integral Equ. Appl., 3 (1991), 411-427. doi: 10.1216/jiea/1181075633.
[19]	C. Park and D. Skoug, Generalized Feynman integrals: The $\mathcal L(L_2, L_2)$ theory, Rocky Mountain J. Math., 25 (1995), 739-756. doi: 10.1216/rmjm/1181072247.
[20]	D. Robinson, A course in the Theory of Groups, 2$^{nd}$ edition, Graduate texts in mathematics, Vol. 80, Springer, New York, 1996. doi: 10.1007/978-1-4419-8594-1.
[21]	D. Skoug and D. Storvick, A survey of results involving transforms and convolutions in function space, Rocky Mountain J. Math., 34 (2004), 1147-1175. doi: 10.1216/rmjm/1181069848.
[22]	J. Yeh, Stochastic Processes and the Wiener Integral, Marcel Dekker, Inc., New York, 1973.