Abstract
In the theory of the analytic Feynman integral, the integrand is a functional of the standard Brownian motion process. In this note, we present an example of a bounded functional which is not Feynman integrable. The bounded functionals discussed in this note are defined in sample paths of the generalized Brownian motion process.
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References
R. H. Cameron, The Ilstow and Feynman integrals, J. Anal. Math. 10 (1962–1963), 287–361.
Cameron R. H., Storvick D. A.: An L 2 analytic Fourier–Feynman transform. Michigan Math. J. 23, 1–30 (1976)
R. H. Cameron and D. A. Storvick, Some Banach algebras of analytic Feynman integrable functionals, in: Analytic Functions, Kozubnik, 1979 (J. Ławrynowicz, ed.), pp. 18–67, Lecture Notes in Mathematics, 798, Springer, Berlin, 1980.
Chang S. J.: Conditional generalized Fourier–Feynman transform of functionals in a Fresnel type class. Commun. Korean Math. Soc. 26, 273–289 (2011)
Chang S. J., Choi J. G., Skoug D.: Integration by parts formulas involving generalized Fourier–Feynman transforms on function space. Trans. Amer. Math. Soc. 355, 2925–2948 (2003)
Chang S. J., Choi J. G., Ko A. Y.: A translation theorem for the generalized analytic Feynman integral associated with Gaussian paths. Bull. Aust. Math. Soc. 93, 152–161 (2016)
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, to appear in Journal of the Korean Mathematical Society.
Chang S. J., Choi J. G., Skoug D.: Evaluation formulas for conditional function space integrals I. Stoch. Anal. Appl. 25, 141–168 (2007)
Chang S. J., Choi J. G., Skoug D.: Generalized Fourier–Feynman transforms, convolution products, and first variations on function space. Rocky Mountain J. Math. 40, 761–788 (2010)
Chang S. J., Chung D. M.: Conditional function space integrals with applications. Rocky Mountain J. Math. 26, 37–62 (1996)
Chang S. J., Chung H. S., Skoug D.: Integral transforms of functionals in \({L^2(C_{a,b}[0,T])}\). J. Fourier Anal. Appl. 15, 441–462 (2009)
Chang S. J., Kang S. J., Skoug D.: Conditional generalized analytic Feynman integrals and a generalized integral equation. Internat. J. Math. Math. Sci. 23, 759–776 (2000)
Chang S.J., Lee W.G., Choi J.G.: L 2-sequential transforms on function space. J. Math. Anal. Appl. 421, 625–642 (2015)
Chang S. J., Skoug D.: Generalized Fourier–Feynman transforms and a first variation on function space. Integral Transforms Spec. Funct. 14, 375–393 (2003)
J. G. Choi and S. J. Chang, Generalized Fourier–Feynman transform and sequential transforms on function space, J. Korean Math. Soc. 49 (2012), 1065–1082. Addendums, J. Korean Math. Soc. 51 (2014), 1321–1322.
J. G. Choi, H. S. Chung, and S. J. Chang, Sequential generalized transforms on function space, Abstr. Appl. Anal. 20 (2013), Article ID 565832, 12 pp.
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), Article ID 954098, 12 pp.
Chung D. M., Lee J. H.: Generalized Brownian motions with application to finance. J. Korean Math. Soc. 43, 357–371 (2006)
Chung D. M., Park C., Skoug D.: Generalized Feynman integrals via conditional Feynman integrals. Michigan Math. J. 40, 377–391 (1993)
Chung H. S., Choi J. G., Chang S. J.: A Fubini theorem on a function space and its applications. Banach J. Math. Anal. 7, 173–185 (2013)
T. S. Chung and U. C. Ji, Gaussian white noise and applications to finance, Quantum Information and Complexity (T. Hida, K. Saitô, S. Si, Eds.), pp. 179–201, World Scientific, Singapore, 2004.
Huffman T., Park C., Skoug D.: Analytic Fourier–Feynman transforms and convolution. Trans. Amer. Math. Soc. 347, 661–673 (1995)
Huffman T., Park C., Skoug D.: Convolutions and Fourier–Feynman transforms of functionals involving multiple integrals. Michigan Math. J. 43, 247–261 (1996)
Huffman T., Park C., Skoug D.: Convolution and Fourier–Feynman transforms. Rocky Mountain J. Math. 27, 827–841 (1997)
Lee J. H.: The linear space of generalized Brownian motions with application. Proc. Amer. Math. Soc. 133, 2147–2155 (2005)
Feynman R. P.: Space-time approach to non-relativistic quantum mechanics. Rev. Mod. Phys. 20, 367–387 (1948)
Johnson G. W., Skoug D. L.: A Banach algebra of Feynman integrable functionals with application to an integral equation formally equivalent to Schroedinger’s equation. J. Math. Anal. Appl. 12, 129–152 (1973)
Johnson G. W., Skoug D. L.: An L p analytic Fourier–Feynman transform. Michigan Math. J. 26, 103–127 (1979)
Johnson G. W., Skoug D. L.: Notes on the Feynman integral, III: The Schroedinger equation. Pacific J. Math. 105, 321–358 (1983)
Yeh J.: Singularity of Gaussian measures on function spaces induced by Brownian motion processes with non-stationary increments. Illinois J. Math. 15, 37–46 (1971)
Yeh J.: Stochastic Processes and the Wiener Integral. Marcel Dekker, New York (1973)
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Chang, S.J., Choi, J.G. Effect of drift of the generalized Brownian motion process: an example for the analytic Feynman integral. Arch. Math. 106, 591–600 (2016). https://doi.org/10.1007/s00013-016-0899-x
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DOI: https://doi.org/10.1007/s00013-016-0899-x