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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

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

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.

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.
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 and 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.

[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.

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.

[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.

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