Abstract

The purpose of this paper is to investigate the behavior of a scale factor for Wiener integrals about the unbounded function , where {a1,a2,...,an} is an orthonormal set of elements in L2[0,T] on the Wiener space C0[0,T].

Highlights

  • Storvick proved relationships between Wiener integrals and analytic Feynman integrals to prove the change of scale formula for Wiener integral on the Wiener space in 1987

  • We investigate the behavior of a scale factor ρ > 0 for the Wiener integral ∫C0[0,T]F ( ρ x) dm ( x) which is defined on the Wiener space

  • Property. We investigate the interesting behavior of the scale factor for the Wiener integral by analyzing the Wiener integral as followings: For real ρ > 0 and for a finite real number a > 0

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Summary

Introduction

H. Cameron wrote the paper about the translation pathology of a Wiener spac (1972). T. Martin proved some theorems on the transformation and the translation and used the expression of the change of scale for Wiener integrals (1944, 1947). A. Storvick proved relationships between Wiener integrals and analytic Feynman integrals to prove the change of scale formula for Wiener integral on the Wiener space in 1987. S. Kim proved a change of scale formula for Wiener ( ) integrals about cylinder functions f (h1x) , ,(hn , x) with ( ) f ∈ Lp Rn ,1 ≤ p ≤ ∞ on the abstract Wiener space: the analytic Wiener ( ) integral exists for f ∈ Lp Rn ,1 ≤ p ≤ ∞ , and the analytic Feynman integral ( ) exists for f ∈ L1 Rn (1998) and (2001). About the unbounded function dx with a > 0 , where {α1,α2 , ,αn} is an orthonormal set of elements in L2 [0,T ] on the Wiener space C0 [0,T ]

Definitions and Preliminaries
Main Results
Conclusions
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