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Feynman Integral and a Change of Scale Formula about the First Variation and a Fourier–Stieltjes Transform

Author

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  • Young Sik Kim

    (Department of Mathematics, College of Natural Sciences, Industry-University Cooperation Foundation, Hanyang University, 222 Wangshmni-ro, Seongdong-gu, Seoul 04763, Korea)

Abstract

We prove that the Wiener integral, the analytic Wiener integral and the analytic Feynman integral of the first variation of F ( x ) = exp { ∫ 0 T θ ( t , x ( t ) ) d t } successfully exist under the certain condition, where θ ( t , u ) = ∫ R exp { i u v } d σ t ( v ) is a Fourier–Stieltjes transform of a complex Borel measure σ t ∈ M ( R ) and M ( R ) is a set of complex Borel measures defined on R. We will find this condition. Moreover, we prove that the change of scale formula for Wiener integrals about the first variation of F ( x ) sucessfully holds on the Wiener space.

Suggested Citation

  • Young Sik Kim, 2020. "Feynman Integral and a Change of Scale Formula about the First Variation and a Fourier–Stieltjes Transform," Mathematics, MDPI, vol. 8(10), pages 1-14, September.
    • Handle: RePEc:gam:jmathe:v:8:y:2020:i:10:p:1666-:d:420706
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