Banach Journal of Mathematical Analysis

Translation theorems for the Fourier–Feynman transform on the product function space Ca,b2[0,T]

Seung Jun Chang, Jae Gil Choi, and David Skoug

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Abstract

In this article, we establish the Cameron–Martin translation theorems for the analytic Fourier–Feynman transform of functionals on the product function space Ca,b2[0,T]. The function space Ca,b[0,T] is induced by the generalized Brownian motion process associated with continuous functions a(t) and b(t) on the time interval [0,T]. The process used here is nonstationary in time and is subject to a drift a(t). To study our translation theorem, we introduce a Fresnel-type class FA1,A2a,b of functionals on Ca,b2[0,T], which is a generalization of the Kallianpur and Bromley–Fresnel class FA1,A2. We then proceed to establish the translation theorems for the functionals in FA1,A2a,b.

Article information

Source
Banach J. Math. Anal., Volume 13, Number 1 (2019), 192-216.

Dates
Received: 27 March 2018
Accepted: 27 June 2018
First available in Project Euclid: 6 December 2018

Permanent link to this document
https://projecteuclid.org/euclid.bjma/1544086815

Digital Object Identifier
doi:10.1215/17358787-2018-0022

Mathematical Reviews number (MathSciNet)
MR3892340

Zentralblatt MATH identifier
07002038

Subjects
Primary: 46G12: Measures and integration on abstract linear spaces [See also 28C20, 46T12]
Secondary: 28C20: Set functions and measures and integrals in infinite-dimensional spaces (Wiener measure, Gaussian measure, etc.) [See also 46G12, 58C35, 58D20, 60B11] 60J65: Brownian motion [See also 58J65] 42B10: Fourier and Fourier-Stieltjes transforms and other transforms of Fourier type

Keywords
generalized Brownian motion process generalized analytic Feynman integral generalized analytic Fourier–Feynman transform generalized Fresnel-type class translation theorem

Citation

Chang, Seung Jun; Choi, Jae Gil; Skoug, David. Translation theorems for the Fourier–Feynman transform on the product function space $C_{a,b}^{2}[0,T]$. Banach J. Math. Anal. 13 (2019), no. 1, 192--216. doi:10.1215/17358787-2018-0022. https://projecteuclid.org/euclid.bjma/1544086815


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References

  • [1] S. A. Albeverio and R. J. Høegh-Krohn, Mathematical Theory of Feynman Path Integrals, Lecture Notes in Math. 523, Springer, Berlin, 1976.
  • [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 D. A. Storvick, A translation theorem for analytic Feynman integrals, Trans. Math. Amer. Soc. 125 (1966), 1–6.
  • [5] R. H. Cameron and D. A. Storvick, “Some Banach algebras of analytic Feynman integrable functionals” in Analytic Functions (Kozubnik, 1979), Lecture Notes in Math. 798, Springer, Berlin, 1980, 18–67.
  • [6] R. H. Cameron and D. A. Storvick, A new translation theorem for the analytic Feynman integral, Rev. Roumaine Math. Pures Appl. 27 (1982), no. 9, 937–944.
  • [7] K. S. Chang, B. S. Kim, and I. Yoo, Analytic Fourier-Feynman transform and convolution of functionals on abstract Wiener space, Rocky Mountain J. Math. 30 (2000), no. 3, 823–842.
  • [8] S. J. Chang, J. G. Choi, and A. Y. Ko, Multiple generalized analytic Fourier-Feynman transform via rotation of Gaussian paths on function space, Banach J. Math. Anal. 9 (2015), no. 4, 58–80.
  • [9] S. J. Chang, J. G. Choi, and A. Y. Ko, A translation theorem for the generalised analytic Feynman integral associated with Gaussian paths, Bull. Aust. Math. Soc. 93 (2016), no. 1, 152–161.
  • [10] S. J. Chang, J. G. Choi, and A. Y. Ko, A translation theorem for the generalized Fourier–Feynman transform associated with Gaussian process on function space, J. Korean Math. Soc. 53 (2016), no. 5, 991–1017.
  • [11] S. J. Chang, J. G. Choi, and S. D. Lee, A Fresnel type class on function space, J. Korean Soc. Math. Educ. Ser. B Pure Appl. Math. 16 (2009), no. 1, 107–119.
  • [12] 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, no. 7 (2003), 2925–2948.
  • [13] S. J. Chang and D. M. Chung, Conditional function space integrals with applications, Rocky Mountain J. Math. 26 (1996), no. 1, 37–62.
  • [14] 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), no. 4, 441–462.
  • [15] S. J. Chang and D. Skoug, Generalized Fourier-Feynman transforms and a first variation on function space, Integral Transforms Spec. Funct. 14 (2003), no. 5, 375–393.
  • [16] J. G. Choi, D. Skoug, and S. J. Chang, Generalized analytic Fourier–Feynman transform of functionals in a Banach algebra $\mathcal{F}_{A_{1},A_{2}}^{a,b}$, J. Funct. Spaces Appl. 2013 (2013), no. 954098.
  • [17] D. M. Chung and S. J. Kang, Translation theorems for Feynman integrals on abstract Wiener and Hilbert spaces, Bull. Korean Math. Soc. 23 (1986), no. 2, 177–187.
  • [18] G. W. Johnson, The equivalence of two approaches to the Feynman integral, J. Math. Phys. 23 (1982), no. 11, 2090–2096.
  • [19] G. W. Johnson and M. L. Lapidus, The Feynman Integral and Feynman’s Operational Calculus, Oxford Math. Monogr., Oxford Univ. Press, New York, 2000.
  • [20] G. Kallianpur and C. Bromley, “Generalized Feynman integrals using analytic continuation in several complex variables” in Stochastic Analysis and Applications, Adv. Probab. Relat. Top. 7, Dekker, New York, 1984, 217–267.
  • [21] G. Kallianpur, D. Kannan, and R. L. Karandikar, Analytic and sequential Feynman integrals on abstract Wiener and Hilbert spaces, and a Cameron-Martin formula, Ann. Inst. Henri Poincaré Probab. Stat. 21 (1985), no. 4, 323–361.
  • [22] E. Nelson, Dynamical Theories of Brownian Motion, 2nd ed., Math. Notes, Princeton Univ. Press, Princeton, 1967.
  • [23] I. Pierce and D. Skoug, Integration formulas for functionals on the function space $C_{a,b}[0,T]$ involving the Paley-Wiener-Zygmund stochastic integrals, PanAmer. Math. J. 18 (2008), no. 4, 101–112.
  • [24] J. Yeh, Singularity of Gaussian measures on function spaces induced by Brownian motion processes with non-stationary increments, Illinois J. Math. 15 (1971), 37–46.
  • [25] J. Yeh, Stochastic Processes and the Wiener Integral, Pure Appl. Math. 13, Dekker, New York, 1973.