November 2018, 17(6): 2225-2238. doi: 10.3934/cpaa.2018106

A Cameron-Storvick theorem for the analytic Feynman integral associated with Gaussian paths on a Wiener space and applications

Department of Mathematics, Dankook University, Cheonan 330-714, Republic of Korea

* Corresponding author

Received  March 2017 Revised  February 2018 Published  June 2018

The purpose of this paper is to establish a Cameron-Storvick theorem for the analytic Feynman integral of functionals in non-stationary Gaussian processes on Wiener space. As interesting applications, we apply this theorem to evaluate the generalized analytic Feynman integral of certain polynomials in terms of Paley-Wiener-Zygmund stochastic integrals.

Citation: Seung Jun Chang, Jae Gil Choi. A Cameron-Storvick theorem for the analytic Feynman integral associated with Gaussian paths on a Wiener space and applications. Communications on Pure & Applied Analysis, 2018, 17 (6) : 2225-2238. doi: 10.3934/cpaa.2018106
References:
[1]

R. H. Cameron, The first variation of an indefinite Wiener integral, Proc. Amer. Math. Soc., 2 (1951), 914-924.

[2]

R. H. Cameron and R. E. Graves, Additive functionals on a space of continuous functions. Ⅰ, Trans. Amer. Math. Soc., 70 (1951), 160-176.

[3]

R. H. Cameron and W. T. Martin, Transformations of Wiener integrals under translations, Ann. of Math. (2), 45 (1944), 386-396.

[4]

R. H. Cameron and W. T. Martin, The orthogonal development of non-linear functionals in series of Fourier-Hermite functionals, Annal. of Math. (2), 48 (1947), 385-392.

[5]

R. H. Cameron and D. A. Storvick, A translation theorem for analytic Feynman integrals, Trans. Math. Amer. Soc., 125 (1966), 1-6.

[6]

R. H. Cameron and D. A. Storvick, A new translation theorem for the analytic Feynman integral, Rev. Roumaine Math. Pures Appl., 27 (1982), 937-944.

[7]

R. H. Cameron and D. A. Storvick, Feynman integral of variations of functionals, in Gaussian Random Fields (eds. K. Itô and T. Hida, 1990), Series on Probability and Statistics, vol. 1, World Scientific, Singapore, (1991), 144-157.

[8]

K. S. ChangD. H. ChoB. S. KimT. S. Song and I. Yoo, Relationships involving generalized Fourier-Feynman transform, convolution and first variation, Integral Transforms Spec. Funct., 16 (2005), 391-405.

[9]

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), Article ID: 1250068.

[10]

D. M. ChungC. Park and D. Skoug, Generalized Feynman integrals via conditional Feynman integrals, Michigan Math. J., 40 (1993), 377-391.

[11]

M. D. Donsker, On function space integrals, in Analysis in Function Space (eds. W. T. Martin and I. Segal), MIT Press, Cambridge, Massachusetts, (1964), 17-30.

[12]

G. B. Folland, Real Analysis, 2nd edition, John Wiley & Sons, New York, 1999.

[13]

S. Janson, Gaussian Hilbert Spaces, Cambridge Tracts in Mathematics (129), Cambridge University Press, 1997.

[14]

G. W. Johnson and D. L. Skoug, Notes on the Feynman integral, Ⅱ, J. Funct. Anal., 41 (1981), 277-289.

[15]

J. Kuelbs, Abstract Wiener spaces and applications to analysis, Pacific J. Math., 31 (1969), 433-450.

[16]

H.-H. Kuo, Integration by parts for abstract Wiener measures, Duke Math. J., 41 (1974), 373-379.

[17]

H.-H. Kuo and Y.-J. Lee, Integration by parts formula and the Stein lemma on abstract Wiener space, Commun. Stoch. Anal., 5 (2011), 405-418.

[18]

R. E. A. C. PaleyN. Wiener and A. Zygmund, Notes on random functions, Math. Z., 37 (1933), 647-668.

[19]

C. Park and D. Skoug, A note on Paley-Wiener-Zygmund stochastic integrals, Proc. Amer. Math. Soc., 103 (1988), 591-601.

[20]

C. Park and D. Skoug, A Kac-Feynman integral equation for conditional Wiener integrals, J. Integral Equations Appl., 3 (1991), 411-427.

[21]

C. ParkD. Skoug and D. Storvick, Fourier-Feynman transforms and the first variation, Rend. Circ. Mat. Palermo (2), 47 (1998), 277-292.

[22]

Y. Yamasaki, Measures on Infinite Dimensional Spaces, World Sci. Ser. Pure Math. 5, World Sci. Publishing, Singapore, 1985.

[23]

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

show all references

References:
[1]

R. H. Cameron, The first variation of an indefinite Wiener integral, Proc. Amer. Math. Soc., 2 (1951), 914-924.

[2]

R. H. Cameron and R. E. Graves, Additive functionals on a space of continuous functions. Ⅰ, Trans. Amer. Math. Soc., 70 (1951), 160-176.

[3]

R. H. Cameron and W. T. Martin, Transformations of Wiener integrals under translations, Ann. of Math. (2), 45 (1944), 386-396.

[4]

R. H. Cameron and W. T. Martin, The orthogonal development of non-linear functionals in series of Fourier-Hermite functionals, Annal. of Math. (2), 48 (1947), 385-392.

[5]

R. H. Cameron and D. A. Storvick, A translation theorem for analytic Feynman integrals, Trans. Math. Amer. Soc., 125 (1966), 1-6.

[6]

R. H. Cameron and D. A. Storvick, A new translation theorem for the analytic Feynman integral, Rev. Roumaine Math. Pures Appl., 27 (1982), 937-944.

[7]

R. H. Cameron and D. A. Storvick, Feynman integral of variations of functionals, in Gaussian Random Fields (eds. K. Itô and T. Hida, 1990), Series on Probability and Statistics, vol. 1, World Scientific, Singapore, (1991), 144-157.

[8]

K. S. ChangD. H. ChoB. S. KimT. S. Song and I. Yoo, Relationships involving generalized Fourier-Feynman transform, convolution and first variation, Integral Transforms Spec. Funct., 16 (2005), 391-405.

[9]

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), Article ID: 1250068.

[10]

D. M. ChungC. Park and D. Skoug, Generalized Feynman integrals via conditional Feynman integrals, Michigan Math. J., 40 (1993), 377-391.

[11]

M. D. Donsker, On function space integrals, in Analysis in Function Space (eds. W. T. Martin and I. Segal), MIT Press, Cambridge, Massachusetts, (1964), 17-30.

[12]

G. B. Folland, Real Analysis, 2nd edition, John Wiley & Sons, New York, 1999.

[13]

S. Janson, Gaussian Hilbert Spaces, Cambridge Tracts in Mathematics (129), Cambridge University Press, 1997.

[14]

G. W. Johnson and D. L. Skoug, Notes on the Feynman integral, Ⅱ, J. Funct. Anal., 41 (1981), 277-289.

[15]

J. Kuelbs, Abstract Wiener spaces and applications to analysis, Pacific J. Math., 31 (1969), 433-450.

[16]

H.-H. Kuo, Integration by parts for abstract Wiener measures, Duke Math. J., 41 (1974), 373-379.

[17]

H.-H. Kuo and Y.-J. Lee, Integration by parts formula and the Stein lemma on abstract Wiener space, Commun. Stoch. Anal., 5 (2011), 405-418.

[18]

R. E. A. C. PaleyN. Wiener and A. Zygmund, Notes on random functions, Math. Z., 37 (1933), 647-668.

[19]

C. Park and D. Skoug, A note on Paley-Wiener-Zygmund stochastic integrals, Proc. Amer. Math. Soc., 103 (1988), 591-601.

[20]

C. Park and D. Skoug, A Kac-Feynman integral equation for conditional Wiener integrals, J. Integral Equations Appl., 3 (1991), 411-427.

[21]

C. ParkD. Skoug and D. Storvick, Fourier-Feynman transforms and the first variation, Rend. Circ. Mat. Palermo (2), 47 (1998), 277-292.

[22]

Y. Yamasaki, Measures on Infinite Dimensional Spaces, World Sci. Ser. Pure Math. 5, World Sci. Publishing, Singapore, 1985.

[23]

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

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