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July  2020, 19(7): 3829-3842. doi: 10.3934/cpaa.2020169

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

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

Received  December 2019 Revised  January 2020 Published  April 2020

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.

Citation: Jae Gil Choi, David Skoug. Algebraic structure of the $ L_2 $ analytic Fourier–Feynman transform associated with Gaussian paths on Wiener space. Communications on Pure & Applied Analysis, 2020, 19 (7) : 3829-3842. doi: 10.3934/cpaa.2020169
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.  Google Scholar

[2]

M. D. Brue, A Functional Transform for Feynman Integrals Similar to the Fourier Transform, Ph.D Thesis, University of Minnesota, 1972.  Google Scholar

[3]

R. H. Cameron and D. A. Storvick, An $L_2$ analytic Fourier–Feynman transform, Michigan Math. J., 23 (1976), 1-30.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[6]

S. J. ChangH. 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[9]

D. M. ChungC. Park and D. Skoug, Generalized Feynman integrals via conditional Feynman integrals, Michigan Math. J., 40 (1993), 377-391.  doi: 10.1307/mmj/1029004758.  Google Scholar

[10]

T. HuffmanC. Park and D. Skoug, Analytic Fourier-Feynman transforms and convolution, Trans. Amer. Math. Soc., 347 (1995), 661-673.  doi: 10.2307/2154908.  Google Scholar

[11]

T. HuffmanC. Park and D. Skoug, Generalized transforms and convolutions, Int. J. Math. Math. Sci., 20 (1997), 19-32.  doi: 10.1155/S0161171297000045.  Google Scholar

[12]

T. HuffmanD. Skoug and D. Storvick, Integration formulas involving Fourier-Feynman transforms via a Fubini theorem, J. Korean Math., 38 (2001), 421-435.   Google Scholar

[13]

G. W. Johnson and D. L. Skoug, An $L_p$ analytic Fourier-Feynman transform, Michigan Math. J., 26 (1979), 103-127.   Google Scholar

[14]

G. W. Johnson and D. L. Skoug, Scale-invariant measurability in Wiener space, Pac. J. Math., 83 (1979), 157-176.   Google Scholar

[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.  Google Scholar

[16]

R. E. A. C. PaleyN. Wiener and A. Zygmund, Notes on random functions, Math. Z., 37 (1933), 647-668.  doi: 10.1007/BF01474606.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[22]

J. Yeh, Stochastic Processes and the Wiener Integral, Marcel Dekker, Inc., New York, 1973.  Google Scholar

show all references

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.  Google Scholar

[2]

M. D. Brue, A Functional Transform for Feynman Integrals Similar to the Fourier Transform, Ph.D Thesis, University of Minnesota, 1972.  Google Scholar

[3]

R. H. Cameron and D. A. Storvick, An $L_2$ analytic Fourier–Feynman transform, Michigan Math. J., 23 (1976), 1-30.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[6]

S. J. ChangH. 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[9]

D. M. ChungC. Park and D. Skoug, Generalized Feynman integrals via conditional Feynman integrals, Michigan Math. J., 40 (1993), 377-391.  doi: 10.1307/mmj/1029004758.  Google Scholar

[10]

T. HuffmanC. Park and D. Skoug, Analytic Fourier-Feynman transforms and convolution, Trans. Amer. Math. Soc., 347 (1995), 661-673.  doi: 10.2307/2154908.  Google Scholar

[11]

T. HuffmanC. Park and D. Skoug, Generalized transforms and convolutions, Int. J. Math. Math. Sci., 20 (1997), 19-32.  doi: 10.1155/S0161171297000045.  Google Scholar

[12]

T. HuffmanD. Skoug and D. Storvick, Integration formulas involving Fourier-Feynman transforms via a Fubini theorem, J. Korean Math., 38 (2001), 421-435.   Google Scholar

[13]

G. W. Johnson and D. L. Skoug, An $L_p$ analytic Fourier-Feynman transform, Michigan Math. J., 26 (1979), 103-127.   Google Scholar

[14]

G. W. Johnson and D. L. Skoug, Scale-invariant measurability in Wiener space, Pac. J. Math., 83 (1979), 157-176.   Google Scholar

[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.  Google Scholar

[16]

R. E. A. C. PaleyN. Wiener and A. Zygmund, Notes on random functions, Math. Z., 37 (1933), 647-668.  doi: 10.1007/BF01474606.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[22]

J. Yeh, Stochastic Processes and the Wiener Integral, Marcel Dekker, Inc., New York, 1973.  Google Scholar

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