Advertisement

An Analytic Operator-Valued Generalized Feynman Integral on Function Space

  • Seung Jun Chang
  • Jae Gil Choi
  • Il Yong LeeEmail author
Article
  • 34 Downloads

Abstract

In this paper, we use a generalized Brownian motion process to define an analytic operator-valued Feynman integral. We then establish the existence of the analytic operator-valued generalized Feynman integral. We next investigate a stability theorem for the analytic operator-valued generalized Feynman integral.

Keywords

Analytic operator-valued function space integral Analytic operator-valued generalized Feynman integral Stability theorem 

Mathematics Subject Classification

Primary 60J25 28C20 

Notes

Acknowledgements

The authors thank the referees for their helpful suggestions which led to the present version of this paper.

References

  1. 1.
    Cameron, R.H., Storvick, D.A.: An operator valued function space integral and a related integral equation. J. Math. Mech. 18, 517–552 (1968)MathSciNetzbMATHGoogle Scholar
  2. 2.
    Chang, K.S., Ko, J.W., Ryu, K.S.: Stability theorems for the operator-valued Feynman integral: the \(\cal{L}(L_1(\mathbb{R}), C_0(\mathbb{R}))\) theory. J. Korean Math. Soc. 35, 999–1018 (1998)MathSciNetzbMATHGoogle Scholar
  3. 3.
    Chang, S.J., Choi, J.G., Skoug, D.: Integration by parts formulas involving generalized Fourier–Feynman transforms on function space. Trans. Am. Math. Soc. 355, 2925–2948 (2003)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Chang, S.J., Chung, D.M.: Conditional function space integrals with applications. Rocky Mt. J. Math. 26, 37–62 (1996)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Chang, S.J., Lee, I.Y.: Analytic operator-valued generalized Feynman integrals on function space. J. Chungcheong Math. Soc. 23, 37–48 (2010)Google Scholar
  6. 6.
    Chang, S.J., Skoug, D.: Generalized Fourier–Feynman transforms and a first variation on function space. Integral Transforms Special Funct. 14, 375–393 (2003)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Johnson, G.W.: A bounded convergence theorem for the Feynman integral. J. Math. Phys. 25, 1323–1326 (1984)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Johnson, G.W., Lapidus, M.L.: Generalized Dyson series, generalized Feynman diagrams, the Feynman integral and Feynman’s operational calculus. Mem. Am. Math. Soc. 62, 1–78 (1986)MathSciNetzbMATHGoogle Scholar
  9. 9.
    Reed, M., Simon, B.: Methods of Modern Mathematical Physics, vol. I, Rev. and enlarged edn. Academic Press, New York (1980)zbMATHGoogle Scholar
  10. 10.
    Yeh, J.: Singularity of Gaussian measures on function spaces induced by Brownian motion processes with non-stationary increments. Ill. J. Math. 15, 37–46 (1971)MathSciNetzbMATHGoogle Scholar
  11. 11.
    Yeh, J.: Stochastic Processes and the Wiener Integral. Marcel Dekker Inc., New York (1973)zbMATHGoogle Scholar

Copyright information

© Malaysian Mathematical Sciences Society and Penerbit Universiti Sains Malaysia 2017

Authors and Affiliations

  1. 1.Department of MathematicsDankook UniversityCheonanKorea

Personalised recommendations