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

Authors: Kim Young Sik
Journal: Complexity
 (2020)

---

## Abstract

We investigate the behavior of the unbounded cylinder functionFx=∫0Tα1tdxt2k⋅∫0Tα2tdxt2k⋅⋯⋅∫0Tαntdxt2k,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 C00,T.


---

## Body

## 1. 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. In [2], Johnson and Skoug proved the scale-invariant measurability on the Wiener space in 1979. In [3, 4], Cameron and Storvick proved a change of scale formula for bounded functions on the Wiener space in 1987. In [5], Kim proved a change of scale formula for Wiener integrals about a function Fx=fh1,x∼,…,hn,x∼ with f∈LpRn,1≤p≤∞: the analytic Wiener integral exists for f∈LpRn,1≤p≤∞ and the analytic Feynman integral exists for f∈L1Rn. In general, the analytic Feynman integral does not always exist for f∈LpRn with 1<p. In [6], Brue worked about the transform for Feynman integrals in 1972. In [7], Huffman et al. expanded the Fourier Feynman transform theory of f∫0Tα1dx,…,∫0Tαndx for f∈LpRn with 1≤p≤2. In [8, 9], Kim extended these results to the function μ^h1x∼,…,hn,x∼, where μ^ is a Fourier transform of a complex-valued Borel measure μ in ℳRn, which is a space of complex-valued Borel measures. In [10], Kim investigated the behavior of a scale factor for Wiener integrals on the Wiener space.In [11]‐[13], Cameron and Martin expanded the theory about the translation and transformation theory for the Wiener integral. In [14], Chung expanded the generalized integral transforms for Wiener integrals. In [15], Gaysinsky and Goldstein expanded the self-adjointness of Schrodinger operator and Wiener integrals. In [16], Johnson and Lapidus wrote the book about the Feynman integral and the Feynman's operational calculus. In [20], Kim proved the change of scale formula for Wiener integrals of cylinder functions of a Fourier transform of a measure.In this paper, we investigate the behavior of a Wiener integral for the unbounded functionFx=∫0Tα1tdxt2k⋅∫0Tα2tdxt2k⋅⋯⋅∫0Tαntdxt2k,k=1,2,… and we prove that Fx is Wiener integrable and the analytic Wiener integral and the analytic Feynman integral of Fx exist. We also 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 Fx on the Wiener space C00,T.


## 2. Definitions and Preliminaries

Throughout this paper, letRn denote the n-dimensional Euclidean space and let C,C+, and C+∼ denote the set of complex numbers, the set of complex numbers with positive real part, and the set of nonzero complex numbers with nonnegative real part, respectively.LetC00,T denote the space of real-valued continuous functions x on 0,T such that x0=0. Let ℳ denote the class of all Wiener measurable subsets of  C00,T, let m denote a Wiener measure, and let C00,T,ℳ,m be a Wiener measure space; we denote the Wiener integral of a function F:C00,T⟶C by ∫C00,TFxdmx.A subsetE of C00,T is said to be scale-invariant measurable if ρE∈ℳ for each ρ>0, and a scale-invariant measurable set N is said to be scale-invariant null if mρN=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.Definition 1.
LetF be a complex-valued measurable function on C00,T such that the integral(1)JF;λ=∫C00,TFλ−1/2xdmxexists for all real λ>0. If there exists a function J∗F;z analytic on C+ such that J∗F;λ=JF;λ for all real λ>0, then we define J∗F;z to be the analytic Wiener integral of F over C00,T with parameter z, and for each z∈C+, we write(2)IawF;z=J∗F;z≡∫C00,TanwzFxdmx.
Letq be a nonzero real number and let F be a function defined on C00,T 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 C00,T with parameter q, and we write(3)IafF;q=limz⟶−iqIawF;z≡∫C00,TanwqFxdmx,where z approaches −iq through C+ and i2=−1.Theorem 1.
Wiener integration formula:(4)∫C00,Tf∫0Tα1tdxt,…,∫0Tαntdxtdmx=12πn/2∫Rnfu⟶exp−12∑j=1nuj2du⟶,where α1t,α2t,…,αnt is an orthonormal set of elements in L20,T, f:Rn⟶C is a Lebesgue measurable function, u⟶=u1,u2,…,un, and du⟶=du1,du2,…,dun.Remark 1.
We will use several times the following well-known integration formula:(5)∫Rexp−au2+ibudu=πaexp−b24a,where a is a complex number with Rea>0 and b is a real number.


## 3. Main Results

Define the unbounded functionFx=f∫0Tα1tdxt,…,∫0Tαntdxt:(6)Fx=∫0Tα1tdxt2k⋅∫0Tα2tdxt2k⋅⋯⋅∫0Tαntdxt2k,k=1,2,…,where α1t,α2t,…,αnt is an orthonormal set in L20,T and fu⟶=u12ku22k,…,un2k∉LpRn, 1≤p≤∞ and fu⟶ is unbounded.To expand main results of this paper, we prove some lemmas.Lemma 1.
Fork=1,2,… and for z∈C+, we have that(7)∫−∞+∞u2kexp−12zu2du=2πz⋅2k!k!⋅12zk.Proof.
First we know that forz∈C+ and for v∈R,(8)∫−∞+∞exp−z2x2+vxdx=2πz⋅expv22z.
Using the series expansion of the exponential functionez=∑n=0∞1/n!zn  with z∈C, we have that for z∈C+ and for v∈R,(9)∫−∞+∞exp−z2x2+vxdx=∫−∞+∞exp−z2x2⋅1+vx+12!vx2+⋯+1k!vxk+⋯dx=∫−∞+∞exp−z2x2+vx·exp−z2x2+12!vx2⋅exp−z2x2+⋯⋯+12k!vxk·exp−z2x2+⋯dx=∫−∞+∞exp−z2x2+12!vx2⋅exp−z2x2+⋯⋯+12k!vx2k⋅exp−z2x2+⋯dx=∫−∞+∞∑k=0∞12k!v2k⋅u2k⋅e−z/2u2du=∑k=0∞12k!v2k⋅∫−∞+∞u2k⋅e−z/2u2du.
And forz∈C+,(10)2πzexpv22z=2πz∑k=0∞1k!v22zk.
By (8)–(10), we have the desired result.Lemma 2.
LetF:C00,T⟶C be the unbounded function defined by (6). Then, for k=1,2,…, Fx is a Wiener integrable function of x∈C00,T and the Wiener integral is(11)∫C00,TFxdmx=2k!k!2kn.Proof.
By (4) and by Lemma 1, we have that(12)∫C00,TFxdmx=12πn/2·∫Rnu12k⋅u22k⋅⋯⋅un2k⋅exp−12∑j=1nuj2du⟶=12πn/22π⋅2k!k!⋅12kn=2k!k!2kn.Remark 2.
By Lemma2, we have interesting Wiener integrals about the unbounded function: for an orthonormal set α1,α2,…,αn in L20,T,(13)∫C00,T∫0Tα1tdxt2⋅∫0Tα2tdxt2⋅⋯⋅∫0Tαntdxt2dmx=1,∫C00,T∫0Tα1tdxt4⋅∫0Tα2tdxt4⋅⋯⋅∫0Tαntdxt4dmx=3n,∫C00,T∫0Tα1tdxt6⋅∫0Tα2tdxt6⋅⋯⋅∫0Tαntdxt6dmx=15n,….
BecauseFx is a Wiener integrable function even though it is unbounded, we can challenge to prove the change of scale formula for the Wiener integral about the unbounded function Fx in (6) on the Wiener space C00,T.
First, we prove the existence of the analytic Wiener integral and the analytic Feynman integral about the unbounded functionFx in (6) on the Wiener space C00,T.Theorem 2.
LetF:C00,T⟶C be the unbounded function defined by (6). Then, for z∈C+ and for k=1,2,…, the analytic Wiener integral and the analytic Feynman integral of Fx exist and are given by(14)1∫C00,TanwzFxdmx=2k!k!⋅zk⋅2kn,2∫C00,TanfqFxdmx=2k!k!⋅−iqk⋅2kn,whenever z⟶−iq through C+.Proof.
By (4) and by Lemma 2, we have that for real z>0 and for k=1,2,…,(15)∫C00,TFz−1/2xdmx=∫C00,Tz−1/2∫0Tα1tdxt2k⋅z−1/2∫0Tα2tdxt2k⋅⋯⋅z−1/2∫0Tαntdxt2kdmx=z−kn∫C00,T∫0Tα1tdxt2k⋅∫0Tα2tdxt2k⋅⋯⋅∫0Tαntdxt2kdmx=z−kn2k!k!⋅2kn=2k!k!⋅zk⋅2kn.
By the analytic continuation ofz∈C+, we can deduce the desired analytic Wiener integral and the analytic Feynman integral of Fx on the Wiener space C00,T.Remark 3.
In Theorem2, we prove that the analytic Wiener integral and the analytic Feynman integral about the unbounded function F can exist, even though f∉LpRn, 1≤p≤∞, on the Wiener space C00,T.
To investigate the behavior of a change of scale formula for the Wiener integral, we prove some relationships between the Wiener integral and the analytic Wiener integral about the unbounded functionF in (6) on the Wiener space C00,T.Lemma 3.
LetF:C00,T⟶C be the unbounded function defined by (6). For z∈C+,(16)exp1−z2∑j=1n∫0Tαjtdxt2Fx,is a Wiener integrable function of x∈C00,T.Proof.
By (4) and by (7), we have that for z∈C+ and for k=1,2,…,(17)∫C00,Texp1−z2∑j=1n∫0Tαjtdxt2Fxdmx=12πn/2∫Rnexp1−z2∑j=1nuj2⋅u12k⋅u22k⋅⋯⋅un2k⋅exp−12∑j=1nuj2du⟶=12πn/2∫Rnu12k⋅u22k⋅⋯⋅un2k⋅exp−z2∑j=1nuj2du⟶=12πn/2⋅2πzn/2⋅2k!k!⋅2k⋅zkn=z−n/2⋅2k!k!⋅2k⋅zkn<∞.
Therefore, we have the desired result.
We prove the relationship between the analytic Wiener integral and the Wiener integral for the unbounded functionFx in (6). That is, we prove that the analytic Wiener integral of Fx can be successfully expressed as the sequence of Wiener integrals on the Wiener space C00,T.Theorem 3.
LetF:C00,T⟶C be the unbounded function defined by (6). Then, for z∈C+, the analytic Wiener integral of Fx can be successfully expressed as the sequence of Wiener integrals:(18)∫C00,TanwzFxdmx=zn/2∫C00,Texp1−z2∑j=1n∫0Tαjtdxt2Fxdmx.Proof.
By the proof of Lemma3, we have that for z∈C+ and for k=1,2,…,(19)∫C00,Texp1−z2∑j=1n∫0Tαjtdxt2Fxdmx=z−n/2·2k!k!⋅2k⋅zkn=z−n/2∫C00,TanwzFxdmx.
We prove that the unbounded functionFx in (6) successfully satisfies the change of scale formula for the Wiener integral on the Wiener space C00,T.Theorem 4.
LetF:C00,T⟶C be the unbounded function defined by (6). Then, for a positive real ρ>0,(20)∫C00,TFρxdmx=ρ−n∫C00,Texpρ2−12ρ2∑j=1n∫0Tαjtdxt2Fxdmx.Proof.
By Theorem3, we have that for real z>0,(21)∫C00,TFz−1/2xdmx=zn/2∫C00,Texp1−z2∑j=1n∫0Tαjtdxt2Fxdmx.
If we letz=ρ−2 in the above equation, we have the desired result.
Finally, we prove the relationship between the analytic Feynman integral and the Wiener integral. That is, we prove that the analytic Feynman integral of the unbounded functionFx can be successfully expressed as the limit of the sequence of Wiener integrals on the Wiener space C00,T.Theorem 5.
LetF:C00,T⟶C be the unbounded function defined by (6). Then, the analytic Feynman integral of Fx can be successfully expressed as the limit of the sequence of analytic Wiener integrals:(22)∫C00,TanfqFxdmx=lims⟶∞n/2sn/2∫C00,Texp1−zs2∑j=1n∫0Tαjtdxt2Fxdmx,whenever zss∈N⟶−iq through C+ with N=1,2,3,….Proof.
By Theorem3 and by the definition of the analytic Feynman integral, we have that(23)∫C00,TanfqFxdmx=limzs⟶−iq∫C00,TanwzsFxdmx=lims⟶∞n/2sn/2∫C00,Texp1−zs2∑j=1n∫0Tαjtdxt2Fxdmx,whenever zss∈N⟶−iq through C+.Remark 4.
The motivation of this paper follows from the notationfx=<x,α1>n1,…,<x,αr>nr and by some properties on the Hilbert space in [18, 19]. To check the existence of the analytic Wiener integral and the analytic Feynman integral of Fx, we take n1=n2=⋯=nr=2k and there are no other reasons about this choice.



---

*Source: 2671474-2020-10-17.xml*