Partial Derivative Approach to the Change of Scale Formula for the Function Space Integral

We investigate the behavior of the partial derivative approach to the change of scale formula and prove relationships among the analytic Wiener integral and the analytic Feynman integral of the partial derivative for the function space integral.


Introduction
In [2,3,4], various formulas for linear transformations of Wiener integrals have been given and the behavior of measure and measurability under the change of scale was investigated in [5] on the Wiener space. The scale invariant measurability was proved on the Wiener space in [9]. The relationship between the Wiener integral and the analytic Feynman integral was proved in [6] and [8]. Using these results, they found a change of scale formula for Wiener integrals on C 0 [0, T ] in [7].
In [10], the author proved the change of scale formula for F (x) = f ((h 1 , x) ∼ , · · · , (h n , x) ∼ ) where f ∈ L p (R n ) with 1 ≤ p ≤ ∞ on abstract Wiener spaces. In [11], the author established those relationships for the Fourier transform of a measureμ((h 1 , x) ∼ , · · · , (h n , x) ∼ ) on abstract Wiener spaces. In [13], the author investigated the partial derivative approach to the integral transform for the function space in some Banach algebra on the Wiener space.
In this paper, we prove the change of scale formula for the function space integral about the partial derivative of a Fourier transform on the Wiener space and prove some relationships among the analytic Wiener integral and the analytic Feynman integral and the Wiener integral of the partial derivative for the function space integral.

Definitions and Preliminaries
A subset E of C 0 [0, T ] is said to be scale-invariant measurable provided ρE ∈ M for all ρ > 0, and scale invariant measurable set N is said to be scale-invariant null provided m(ρN ) = 0 for each ρ > 0. A property that holds except on a scale-invariant null set is said to hold scale-invariant almost everywhere (s-a.e.). If two functionals F and G are equal s-a.e., we write F ≈ G.
exists for all real λ > 0. If there exists an analytic function J * F (z) analytic on C + such that J * F (λ) = J F (λ) for all real λ > 0, then we define J * F (z) to be the analytic Wiener integral of F over C 0 [0, T ] with parameter z and for each z ∈ C + , we write Let q be a non-zero real number and let F be a function whose analytic Wiener integral exists for each z in C + . If the following limit exists, then we call it the analytic Feynman integral of F over C 0 [0, T ] with parameter q, and we write where z approaches −iq through C + and i 2 = −1.
The first variation of a Wiener measurable functional F in the direction w ∈ C 0 [0, T ] is defined by the partial derivative as We will denote it by [D, F, x, w].(See [13]).
The following is a well-known Wiener integration formula for Wiener integrals on the Wiener Then for 1 ≤ j ≤ n and " = " mean that if either side exists, then both sides exists and they are equal and {α 1 , α 2 , · · · , α n } is an orthonormal class of Remark. For a ∈ C + and b ∈ R, where {α 1 , α 2 , · · · , α n } is an orthonormal class of L 2 [0, T ] and wherê is the Fourier transform of the measure µ on R n and u = (u 1 , · · · , u n ) and The Paley-Wiener-Zygmund integral equals to the Riemann Stieltzes integral as w is an absolutely continuous function in [0, T ] with w (t) ∈ L 2 [0, T ]. Therefore, is a Wiener integrable function of x ∈ C 0 [0, T ].
Proof. By the Wiener integration theorem, we have that As we have that we have the desired result.
E anfq Therefore the analytic Feynman integral for the partial derivative of the Fourier transformμ of the measure µ on R n exists. 2 Proof. By the theorem 3.4,

Coclusions
We prove the change of scale formula for the function space integral about the partial derivative of a function defined on the function space which is constructed by a Fourier transform of a measure. And we prove some relationships among the analytic Wiener integral and the analytic Feynman integral and the Wiener integral of the partial deriavtive for the function space integral.
Remark 1. In this paper, we prove new results by extending those results [11] to the first variation theory in [1] and to the partial derivative approach developed in [13] and to the change of scale formula in [7] using the Fourier transform of a measure.
Remark 2. In [12], the author extended those results in [11] to the first variation theory in [1] and to the Fourier Feynman transform theory which is an integral transform on the function space.
Funding: Fund of this paper was supported by NRF-2017R1A6A3A11030667.

Conflicts of Interest:
The author declares no conflict of interest.