A change of scale formula for Wiener integrals of cylinder functions on abstract Wiener space

Young Sik Kim

doi:10.1155/s0161171298000088

Young Sik Kim

Open Access

https://doi.org/10.1155/s0161171298000088

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Abstract

The purpose of this paper is to establish the existence of analytic Wiener and Feynman integrals for a class of certain cylinder functions which is of the form: urn:x-wiley:01611712:media:ijmm132848:ijmm132848-math-0001 on the abstract Wiener space, and to establish the relationship between the Wiener integral and the analytic Feynman integral for such cylinder functions on the abstract Wiener space. We then establish a change of scale formula for Wiener integrals of such cylinder functions on the abstract Wiener space.

Highlights

We show that for certain bounded cylinder functions of the form F (x) =
We prove a change of scale formula for Wiener integrals of F on the abstract Wiener space
We show that the analytic Feynman integral of F exists for certain bounded cylinder functions of the form F (x) = μ((h1, x)∼, . . . ,∼), x ∈ B, where μ : Rn → C is the Fourier-transform of the complex-valued Borel measure μ on Ꮾ(Rn), the Borel σ -algebra of Rn with μ < ∞

Summary

Introduction

F exists, Rn f (→u) the analytic exp{−(z/2)|→u|2} d→u, Feynman integral, limz→−iq Iaw(F ; z) = do not always exist for bounded cylinder functions F (x) = f ((h1, x)∼, . (hn, x)∼) is a bounded cylinder function, as we cannot apply the Lebesgue dominated convergence theorem to the limit whenever z → −iq through C+; limz→−iq Iaw(F ; z) = limz→−iq(z/2π )n/2 Rn f (→u) exp{−(z/2)|→u|2} d→u. We show that the analytic Feynman integral of F exists for certain bounded cylinder functions of the form F (x) = μ((h1, x)∼, .

Results

Conclusion