Translation theorem for the analytic Feynman integral associated with bounded linear operators on abstract Wiener spaces and an application
1
조회
0
다운로드
0
인용
0
참고문헌
키워드
초록
The Cameron--Martin translation theorem describes how \linebreak Wiener measure changes under translation by elements of the Cameron--Martin space in an abstract Wiener space (AWS). Translation theorems for the analytic Feynman integrals also have been established in the literature. In this article, we derive a more general translation theorem for the analytic Feynman integral associated with bounded linear operators (B.L.OP.) on AWSs. To do this, we use a certain behavior which exists between the analytic Fourier--Feynman transform (FFT) and the convolution product (CP) of functionals on AWS. As an interesting application, we apply this translation theorem to evaluate the analytic Feynman integral of the functional \[ F(x)=\exp\bigg(-iq \int_0^Tx(t)y(t)d t \bigg),\quad y\in C_0[0,T], \, q\in\mathbb R\setminus\{0\} \] defined on the classical Wiener space $C_0[0,T]$.
Abstract
The Cameron--Martin translation theorem describes how \linebreak Wiener measure changes under translation by elements of the Cameron--Martin space in an abstract Wiener space (AWS). Translation theorems for the analytic Feynman integrals also have been established in the literature. In this article, we derive a more general translation theorem for the analytic Feynman integral associated with bounded linear operators (B.L.OP.) on AWSs. To do this, we use a certain behavior which exists between the analytic Fourier--Feynman transform (FFT) and the convolution product (CP) of functionals on AWS. As an interesting application, we apply this translation theorem to evaluate the analytic Feynman integral of the functional \[ F(x)=\exp\bigg(-iq \int_0^Tx(t)y(t)d t \bigg),\quad y\in C_0[0,T], \, q\in\mathbb R\setminus\{0\} \] defined on the classical Wiener space $C_0[0,T]$.
참고문헌 (24)
- J. M. Ahn, "Fourier-Feynman transform, convolution and first variation", Acta Math. Hungar, 100(3), pp.215~235 (2003)[DOI]
- R. H. Cameron, "Additive functionals on a space of continuous functions I", Trans. Amer. Math. Soc, 70, pp.160~176 (1951)[DOI]
- R. H. Cameron, "Transformations of Wiener integrals under translations", Ann. of Math, 45(2), pp.386~396 (1944)[DOI]
- R. H. Cameron, "Transformations of Wiener integrals under a general class of linear transformations", Trans. Amer. Math. Soc, 58, pp.184~219 (1945)[DOI]
- R. H. Cameron, "A translation theorem for analytic Feynman integrals", Trans. Amer. Math. Soc, 125, pp.1~6 (1966)[DOI]
- R. H. Cameron, "A new translation theorem for the analytic Feynman integral", Rev. Roumaine Math. Pures Appl, 27(9), pp.937~944 (1982)
- S. J. Chang, "Evaluation formula for Wiener integral of polynomials in terms of natural dual pairings on abstract Wiener spaces", Bull. Korean Math. Soc, 59(5), pp.1093~1103 (2022)[DOI]
- K. S. Chang, "convolution and first variation of functionals on abstract Wiener space", Integral Transform. Spec. Funct, 10(3-4), pp.179~200 (2000)[DOI]
- K. S. Chang, "Analytic Fourier-Feynman transform and first variation on abstract Wiener space", J. Korean Math. Soc, 38(2), pp.485~501 (2001)
- J. G. Choi, "Analytic Fourier-Feynman transforms associated with bounded linear operators on abstract Wiener spaces", J. Math. Anal. Appl, 521(1), pp.16~ (2023)[DOI]
- D. M. Chung, "Scale-invariant measurability in abstract Wiener spaces", Pacific J. Math, 130(1), pp.27~40 (1987)
- D. M. Chung, "Evaluation formulas for conditional abstract Wiener integrals", Stochastic Anal. Appl, 7(2), pp.125~144 (1989)[DOI]
- X. Fernique, "Int´egrabilit´e des vecteurs gaussiens", C. R. Acad. Sci. Paris S´er. A-B, 270, pp.A1698~A1699 (1970)
- L. Gross, "Abstract Wiener spaces", Proc. Fifth Berkeley Sympos. Math. Statist. and Probability, pp.31~42 (1965), Univ. California Press
- M.-K. Im, "Relations among the first variation, the convolutions and the generalized Fourier-Gauss transforms", Bull. Korean Math. Soc, 48(2), pp.291~302 (2011)[DOI]
- G. Kallianpur, "Generalized Feynman integrals using analytic continuation in several complex variables, Stochastic Analysis and Applications", Adv. Probab. Related Topics, 7, pp.217~267 (1984), Dekker
- G. Kallianpur, "Analytic and sequential Feynman integrals on abstract Wiener and Hilbert spaces, and a Cameron Martin formula", Ann. Inst. H. Poincar´e Probab. Statist, 21(4), pp.323~361 (1985)
- J. Kuelbs, "Abstract Wiener spaces and applications to analysis", Pacific J. Math, 31, pp.433~450 (1969)
- H.-H. Kuo, "Gaussian Measures in Banach Spaces, Lecture Notes in Mathematics Vol. 463" (1975), Springer
- U. G. Lee, "An extension of the Cameron-Martin translation theorem via Fourier-Hermite functionals", Arch. Math. (Basel), 115(6), pp.679~689 (2020)[DOI]
- R. E. A. C. Paley, "Notes on random functions", Math. Z, 37(1), pp.647~668 (1933)[DOI]
- C. Park, "A note on Paley-Wiener-Zygmund stochastic integrals", Proc. Amer. Math. Soc, 103(2), pp.591~601 (1988)[DOI]
- D. Skoug, "A survey of results involving transforms and convolutions in function space", Rocky Mountain J. Math, 34(3), pp.1147~1175 (2004)[DOI]
- J. Yeh, "Inequalities and a limit theorem for certain Wiener integrals", Proc. Amer. Math. Soc, 16, pp.915~918 (1965)[DOI]