A Modified Analytic Function Space Feynman Integral of Functionals on Function Space

Abstract

In this paper, the authors introduce a class of functionals. This class forms a Banach algebra for the special cases. The main purpose of this paper is to investigate some properties of the modified analytic function space Feynman integral of functionals in the class. Those properties contain various results and formulas which were not obtained in previous papers. Also, the authors establish some relationships involving the first variation via the translation theorem on function space. In particular, the authors establish the Fubini theorem for the modified analytic function space Feynman integral which was not obtained in previous researches yet.

Keywords

Generalized Brownian motion process / Modified analytic Feynman integral / First variation / Cameron-Storvick type theorem / Fubini theorem

References

[1] Albeverio S, Mazzucchi S. Feynman path integrals for polynomially growing potentials. J. Fund. Anal., 2005, 221: 83-121

[2] Albeverio S, Mazzucchi S. The time-dependent quadratic oscillator—a Feynman path integral approach. J. Fund. Anal., 2006, 238: 471-488

[3] Cameron R H, Storvick D A. Feynman integral of variation of functionals, Gaussian Random Fields, 1980, Singapore: World Scientific 144-157

[4] Cameron R H, Storvick D A. Analytic Feynman integral solutions of an integral equation related to the Schrödinger equation. J. Anal Math., 1980, 38: 34-66

[5] Cameron R H, Storvick D A. Some Banach algebras of analytic Feynman integrable functionals, Analytic Functions, Kozubnik, 1979, 1980, Berlin, New York: Springer-Verlag 18-67

[6] Chang J G, Lee I Y. A Fubini theorem for generalized analytic Feynman integrals and Fourier-Feynman transforms on function space. Bull Korean Math. Soc., 2003, 40: 437-456

[7] Chang K S, Johnson G W, Skoug G W. The Feynman integral of quadratic potentials depending on two time variables. Pacific J. Math., 1986, 122: 11-33

[8] Chang S J, Choi J G, Chung H S. Generalized analytic Feynman integral via function space integral of bounded cylinder functionals. Bulletin of the Korean Mathematical Society, 2011, 48: 475-489

[9] Chang Seung Jun, Choi Jae Gil, Chung Hyun Soo. A Modified Analytic Function Space Feynman Integral and Its Applications. Journal of Function Spaces, 2014, 2014: 1-9

[10] Chang S J, Chung D M. Conditional function space integrals with applications. Rocky Mountain J. Math., 1996, 26: 37-62

[11] Chang S J, Skoug D. Generalized Fourier-Feynman transforms and a first variation on function space. Integral Transforms and Special Functions, 2003, 14: 375-393

[12] Chung H S, Choi J G, Chang S J. A Fubini theorem on a function space and its applications. Banach J. Math. Anal., 2013, 7: 172-185

[13] Chung H S, Choi J G, Chang S J. The approach to solution of the Schrödinger equation using Fourier-type functionals. J. Korean Math. Soc., 2013, 50: 259-274

[14] Chung H S, Tuan V K, Chang S J. Analytic Feynman integral of functionals of functionals in a Banach algebra involving the first variation. Chinese Annals of Mathematics, Series B, 2016, 37: 281-290

[15] Johnson G W, Skoug D L. Scale-invariant measurability in Wiener space. Pacific J. Math., 1979, 83: 157-176

[16] Kongioumtzoglou I A, Spanos P D. An analytical Wiener path integral technique for non-stationary response determination of nonlinear oscillator. Prob. Engineering Mech., 2012, 28: 125-131

[17] Mazzucchi S. Mathematical Feynman Path Integrals, 2009, Hackensack: World Scientific

[18] Nelson E. Dynamical Theories of Brownian Motion, 1967 2nd ed. Princeton: Math. Notes, Princeton University Press

[19] Yeh J. Singularity of Gaussian measures on function spaces induced by Brownian motion processes with non-stationary increments. Illinois J. Math., 1971, 15: 37-46

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