Analytic Feynman Integral and a Change of Scale Formula for Wiener Integrals of an Unbounded Cylinder Function

Kim Young Sik,Dimitri Volchenkov

doi:10.1155/2020/2671474

Abstract

We investigate the behavior of the unbounded cylinder function F x = ∫ 0 T α 1 t d x t 2 k ⋅ ∫ 0 T α 2 t d x t 2 k ⋅ ⋯ ⋅ ∫ 0 T α n t d x t 2 k , k = 1,2 , … whose analytic Wiener integral and analytic Feynman integral exist, we prove some relationships among the analytic Wiener integral, the analytic Feynman integral, and the Wiener integral, and we prove a change of scale formula for the Wiener integral about the unbounded function on the Wiener space C 0 0 , T .

Highlights

In [1], Cameron and Martin initially worked about the behavior of measure and measurability under change of scale in the Wiener space in 1947
In [5], Kim proved a change of scale formula for Wiener integrals about a function F(x) f((h1, x) ∼, . . . , ∼ ) with f ∈ Lp(Rn), 1 ≤ p ≤ ∞: the analytic Wiener integral exists for f ∈ Lp(Rn), 1 ≤ p ≤ ∞ and the analytic Feynman integral exists for f ∈ L1(Rn)
We prove some relationships among the analytic Wiener integral, the analytic Feynman integral, and the Wiener integral and prove a change of scale formula for the Wiener integral of the unbounded function F(x) on the Wiener space C0[0, T]

Summary

Introduction

In [1], Cameron and Martin initially worked about the behavior of measure and measurability under change of scale in the Wiener space in 1947. Whose analytic Wiener cylinder function integral and analytic α1(t)dx(t))2k· In [3, 4], Cameron and Storvick proved a change of scale formula for bounded functions on the Wiener space in 1987.

Results

Conclusion