Integration by parts formulas for analytic Feynman integrals of unbounded functionals

Abstract

Chang, Choi and Skoug established several integration by parts formulas involving generalized Feynman integrals, generalized Fourier–Feynman transforms, and the first variation of cylinder-type functionals. We establish integration by parts formulas for analytic Feynman integrals of unbounded functionals on abstract Wiener space having the form

where G belongs to the Fresnel class ℱ(B) and Ψ=ψ+φ with ψ∈L ₁(ℝ ⁿ ) and φ is the Fourier transform of a complex Borel measure of bounded variation on ℝ ⁿ . We also study integration by parts formulas for the analytic Feynman integral involving the Fourier–Feynman transform of those functionals in ℱ(B).

Keywords:

• abstract Wiener space
• Fresnel class
• analytic Feynman integral
• integration by parts
• Fourier–Feynman transform

1991 Mathematics Subject Classification :

• 28C20
• 60J65

Acknowledgements

This work was supported by the Daejin University Research Grants in 2008.