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

Young Sik Kim
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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

Mathematics, 2020, vol. 8, issue 10, 1-14

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.

Keywords: Wiener space; Wiener integral; Feynman integral; Fourier–Stieltjes transform; first variation (search for similar items in EconPapers)
JEL-codes: C (search for similar items in EconPapers)
Date: 2020
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