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

We investigate the partial derivative approach to the change of scale formula for the functon space integral and we investigate the vector calculus approach to the directional derivative on the function space and prove relationships among the Wiener integral and the Feynman integral about the directional derivative of a Fourier transform.


Motivation and Introduction
The solution of the heat (or diffusion)equation: is of the form: where ψ ∈ L 2 (R d ) and ξ ∈ R d and x(·) is a R d −valued continuous function defined on [0, t] such that x(0) = 0. E denotes the expectation with respect to the Wiener path starting at time t = 0 (E is the Wiener integral). H = − + V is the energy operator (or, Hamiltonian) and is a Laplacian and V : R d → R is a potential. (1) is called the Feynman-Kac formula.
Applications of the Feynman-Kac formula (in various settings) have been given in the area of diffusion equations, the spectral theory of the Schrödinger operator, quantum mechanics, statistical physics. (For more details about the application, see [1]). In [2][3][4][5][6][7][8], formulas for linear transformations of Wiener integrals have been given and the behavior of measure and measurability and the change of scale were investigated and a change of scale formula and a scale invariant measurability were proven.
In [9][10][11], the author proved the change of scale formula on the abstract Wiener space and on the Wiener space and established those relationships in [12] and proved relationships among Fourier Feynman transforms and Wiener integrals for the Fourier transform on the abstract Wiener space in [13]. In [14], 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 investigate the partial derivative approach and the vector calculus approach to the change of scale formula for the Wiener integral of a Fourier transform and prove relationships among the Wiener integral and the Feynman integral.

Definitions and Notations
Let C 0 [0, T] be the one parameter Wiener space. That is the class of real-valued continuous functions x on [0, T] with x(0) = 0. Let M denote the class of all Wiener measurable subsets of C 0 [0, T] and let m denote the Wiener measure. (C 0 [0, T], M, m) is a complete measure space and we denote the Wiener integral of a functional F by A subset E of C 0 [0, T] is said to be a 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. For more details about the scale-invarant measurability on the Wiener space, see [15].
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.
Definition 2 (Ref. [16]). The first variation of a Wiener measurable functional F in the direction w ∈ C 0 [0, T] is defined by the partial derivative: Remark 1. We will denote the Formula (5) by (D w F)(x) whose notation is motivated from in the Calculus and we call (D w F)(x) by the directional derivative on the function space C 0 [0, T].

Remark 2.
We will use several times the following formula to prove the main result: For a ∈ C + and b ∈ R,

Main Results
Define where {α 1 , α 2 , · · · , α n } is an orthonormal class of L 2 [0, T] and is the Fourier transform of the measure µ on R n and u = (u 1 , · · · , u n ) and Proof. By Definition 2, 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, and by a Hölder inequality in L 2 [0, T]. Therefore (D w F)(x) exists.
In the next Theorem, we obtain the analytic Wiener integral of (D w F)(x) on the function space as a vector calculus form: Proof. By (12), we have that for z ∈ C + , By (13), we have To prove the relationship between the function space integral and the directional derivative on the functions space, we have to prove the following theorem: Proof. By Equation (6), and Therefore, the function in (17) is a Wiener integrable function of x ∈ C 0 [0, T]. Now, we prove that the analytic Wiener integral of the directional derivative on the function space is expressed as the sequence of Wiener integrals and we express the formula as a vector calculus form: Proof. By Theorems 3 and 4, Now, we prove that the directional derivative on the function space satisfies the change of scale formula for the function space integral and we express the formula as a vector calculus form: Theorem 6 (Change of scale formula). For real ρ > 0, Proof. By Theorem 5, we have that for real z > 0, Taking z = ρ −2 , we have (23). Now, we prove that the analytic Feynman integral of the directional derivative on the function space exists and we express it as a vector calculus form: Proof. By Theorem 3, whenever z → −i q through C + . By (16) and by (25), we have Finally, we prove that the analytic Feynman integral of the directional derivative on the function space is expressed as the sequence of Wiener integrals of the directional derivative on the function space and we express the formula as a vector calculus form:  = lim k→∞ z k n 2 E x exp 1−z k 2 || < x, α > || 2 (D w F)(x) whenever {z k } → − iq through C + .

Conclusions
In this paper, we find a new expression of the vector calculus approach to the change of scale formula for the Wiener integral (which is motivated from the Heat Equaton in Quantum Mechanics) about the directional derivative on the function space of a Fourier transform.

Remark 3.
Notations and Theorems of this paper are upgraded from the reviewer's comment. The author is very grateful to reviewers.
Funding: Fund of this paper was supported by NRF-2017R1A6A3A11030667.

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