# A Rotation on Wiener Space with Applications

Authors: Jae Gil Choi; Seung Jun Chang
Journal: ISRN Applied Mathematics
 (2012)

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## Abstract

We first investigate a rotation property of Wiener measure on the product of Wiener spaces. Next, using the concept of the generalized analytic Feynman integral, we define a generalized Fourier-Feynman transform and a generalized convolution product for functionals on Wiener space. We then proceed to establish a fundamental result involving the generalized transform and the generalized convolution product.


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## Body

## 1. Introduction

LetC0[0,T] denote one-parameter Wiener space, that is, the space of all real-valued continuous functions x on [0,T] with x(0)=0. Let ℳ denote the class of all Wiener measurable subsets of C0[0,T], and let m denote Wiener measure. Then (C0[0,T],ℳ,m) is a complete measure space, and we denote the Wiener integral of a Wiener integrable functional F by
(1.1)∫C0[0,T]‍F(x)dm(x).In [1], Bearman gave a significant theorem for Wiener integral on product Wiener space. It can be summarized as follows.Theorem 1.1 (Bearman's Rotation Theorem).
LetG(w,z) be an m×m-integrable functional on C02[0,T], the product of 2 copies of C0[0,T], and let θ be a function of bounded variation on [0,T]. Let Tθ:C02[0,T]→C02[0,T] be the transformation defined by Tθ(w,z)=(w′,z′) with
(1.2)w′(t)=∫0t‍cos⁡⁡θ(s)dw(s)-∫0t‍sin⁡⁡θ(s)dz(s),z′(t)=∫0t‍sin⁡⁡θ(s)dw(s)+∫0t‍cos⁡⁡θ(s)dz(s).
Then the transform Tθ is measure preserving and
(1.3)∫C02[0,T]‍G(w,z)d(m×m)(w,z)=∫C02[0,T]‍G(Tθ(w,z))d(m×m)(w,z).As a special case of Theorem1.1, one can obtain the following corollary.Corollary 1.2.
LetF be Wiener integrable on C0[0,T]. Then for any θ∈ℝ, F(wsinθ+zcosθ) is integrable on C02[0,T] and
(1.4)∫C02[0,T]‍F(wsin⁡⁡θ+zcos⁡⁡θ)d(m×m)(w,z)=∫C0[0,T]‍F(x)dm(x).The following more general case of Corollary1.2 is due to Cameron and Storvick [2]. But we state the theorem with some assumption for our research.Theorem 1.3.
LetF be Wiener measurable on C0[0,T]. Assume that for any ρ>0, F(ρ·) is Winer integrable. Then for any a,b∈ℝ, F(aw+bz) is integrable on C02[0,T] and
(1.5)∫C02[0,T]‍F(aw+bz)d(m×m)(w,z)=∫C0[0,T]‍F(a2+b2x)dm(x).In many papers, Theorem1.3 is used to study relationships between analytic Fourier-Feynman transforms and convolution products of Feynman integrable functionals on Wiener space, see for instance [3–6]. In this paper, we will extend the result in Theorem 1.3 to a more general case for functionals of Gaussian processes given by (2.2) below. We then apply our rotation property of Wiener measure to establish a fundamental relationship between the generalized Fourier-Feynman transform and the generalized convolution product.


## 2. A Rotation on Wiener Space

The most important concepts we will employ in the statements and proofs of our results are the concepts of the scale-invariant measurability and the Paley-Wiener-Zygmund stochastic integral [7].A subsetB of C0[0,T] is said to be scale-invariant measurable [8] provided ρB∈ℳ for all ρ>0, and a scale-invariant measurable set N is said to be scale-invariant null provided m(ρN)=0 for all ρ>0. A property that holds except on a scale-invariant null set is said to be hold scale-invariant almost everywhere (s-a.e.). If two functionals F and G are equal s-a.e., we write F≈G.Let{ϕn} be a complete orthonormal set in L2[0,T], each of whose elements is of bounded variation on [0,T]. Then for each v∈L2[0,T], the Paley-Wiener-Zygmund (PWZ) stochastic integral 〈v,x〉 is defined by the formula
(2.1)〈v,x〉=lim⁡n→∞∫0T‍∑j=1n‍(v,ϕj)2ϕj(t)dx(t)
for all x∈C0[0,T] for which the limit exists, where (·,·)2 denotes the L2-inner product.It was shown in [7] that for each v∈L2[0,T], the limit defining the PWZ integral 〈v,x〉 exists for m-a.e. x∈C0[0,T] and that this limit is essentially independent of the choice of the complete orthonormal set {ϕn}. It was also shown in [7] that if v is of bounded variation on [0,T], then the PWZ integral 〈v,x〉 equals the Riemann-Stieltjes integral ∫0T‍v(t)dx(t) for m-a.e. x∈C0[0,T]. In fact, the integrals are equal for s-a.e. x∈C0[0,T] and that for all v∈L2[0,T], 〈v,x〉 is a Gaussian random variable with mean 0 and variance ∥v∥22.For anyh∈L2[0,T] with ∥h∥2>0, let 𝒵h be the Gaussian process
(2.2)Zh(x,t)=∫0t‍h(s)dx(s)=〈v,x〉
introduced by Park and Skoug in [9] and used extensively since; see for example [5, 6, 10, 11]. Of course if h(t)≡1 on [0,T], then 𝒵h(x,t)=x(t).It is easy to see that𝒵h is a Gaussian process with mean zero and covariance function
(2.3)∫C0[0,T]‍Zh(x,s)Zh(x,t)dm(x)=∫0min⁡⁡{s,t}‍h2(u)du.
In addition, 𝒵h(·,t) is stochastically continuous in t on [0,T], and for any h1,h2∈L2[0,T],
(2.4)∫C0[0,T]‍Zh1(x,s)Zh2(x,t)dm(x)=∫0min⁡{s,t}‍h1(u)h2(u)du.For any complete orthonormal set{ϕn} in L2[0,T] and for any n∈ℕ, define the projection map 𝒫n from L2[0,T] into span{ϕ1,…,ϕn} by
(2.5)Pnh(t)=∑j=1n‍(h,ϕj)2ϕj(t).
Then for h∈L2[0,T] and x∈C0[0,T], we see that
(2.6)Zh(x,t)=lim⁡n→∞∫0t‍Pnh(s)dx(s)=lim⁡n→∞ZPnh(x,t),
that is, 𝒵𝒫nh(x,t) converges in L2(C0[0,T])-mean to 𝒵h(x,t).Throughout this paper, we will assume that each functionalF:C0[0,T]→ℂ we consider is scale-invariant measurable and that
(2.7)∫C0[0,T]‍|F(Zh(x,⋅))|dm(x)<+∞
for all h∈L2[0,T].We are now ready to state the main theorem of this paper.Theorem 2.1.
LetF be a functional on C0[0,T]. Then for any h1,h2∈L2[0,T],
(2.8)∫C02[0,T]‍F(Zh1(w,⋅)+Zh2(z,⋅))d(m×m)(w,z)=∫C0[0,T]‍F(Zk(x,⋅))dm(x),
where h1, h2, and k are related by
(2.9)k(t)=∑n=1∞‍(h1,ϕn)22+(h2,ϕn)22ϕn(t)
for some complete orthonormal set {ϕn} in L2[0,T], each of those elements is of bounded variation on [0,T].


## 3. Proof of the Main Theorem

We begin this section with three lemmas in order to establish (2.8).Lemma 3.1.
LetF be a functional on C0[0,T], and let ϕ be a function of bounded variation on [0,T]. Then for all a,b∈ℝ,
(3.1)∫C02[0,T]‍F(aZϕ(w,⋅)+bZϕ(z,⋅))d(m×m)(w,z)=∫C0[0,T]‍F(a2+b2Zϕ(x,⋅))dm(x).Proof.
We first note that for eacht∈[0,T],
(3.2)aZϕ(w,t)+bZϕ(z,t)=∫0t‍ϕ(s)d(aw(s)+bz(s))=Zϕ(aw+bz,t).
We also note that F(𝒵(x,·)) is Wiener integrable as a functional of x. Hence, by (1.5), we obtain that for all a,b∈ℝ,
(3.3)∫C02[0,T]‍F(aZϕ(w,⋅)+bZϕ(z,⋅))d(m×m)(w,z)=∫C02[0,T]‍F(Zϕ(aw+bz,⋅))d(m×m)(w,z)=∫C0[0,T]‍F(Zϕ(a2+b2x,⋅))dm(x)=∫C0[0,T]‍F(a2+b2Zϕ(x,⋅))dm(x).
Thus (3.1) is established.Lemma 3.2.
LetF be a functional on C0[0,T]. Then for any h1,h2∈L2[0,T] and each n∈ℕ,
(3.4)∫C02[0,T]‍F(ZPnh1(w,⋅)+ZPnh2(z,⋅))d(m×m)(w,z)=∫C0[0,T]‍F(ZPnk(x,⋅))dm(x),
where h1, h2, and k are related by (2.9).Proof.
Since the addition is continuous in the uniform topology onC0[0,T], we can apply (3.1) to the functional F(∑j=1n‍𝒵ϕj(x,·)). Thus using (2.5) and (3.1), we have
(3.5)∫C02[0,T]‍F(ZPnh1(w,⋅)+ZPnh2(z,⋅))d(m×m)(w,z)=∫C02[0,T]‍F(∑j=1n‍[(h1,ϕj)2Zϕj(w,⋅)+(h2,ϕj)2Zϕj(z,⋅)])d(m×m)(w,z)=∫C0[0,T]‍F(∑j=1n‍(h1,ϕj)22+(h2,ϕj)22Zϕj(x,⋅))dm(x)=∫C0[0,T]‍F(ZPnk(x,⋅))dm(x).
Thus (3.4) is established.Lemma 3.3.
LetF be bounded and continuous on C0[0,T]. Then for any h1,h2∈L2[0,T],
(3.6)∫C02[0,T]‍F(Zh1(w,⋅)+Zh2(z,⋅))d(m×m)(w,z)=∫C0[0,T]‍F(Zk(x,⋅))dm(x),
where h1, h2, and k are related by (2.9) above.Proof.
We clearly see thatF is Wiener integrable. We also note that {𝒫nh} is a sequence of functions of bounded variation on [0,T] such that 𝒫nh converges to h in the space L2[0,T] as n→∞. For each n∈ℕ and h∈L2[0,T], let Fn(𝒵h(x,·))=F(𝒵𝒫nh(x,·)). Since 𝒵𝒫nh converges to 𝒵h uniformly and F is continuous in the uniform topology, by (2.6),
(3.7)F(Zh(x,⋅))=F(lim⁡n→∞ZPnh(x,⋅))=lim⁡n→∞Fn(Zh(x,⋅)).
Since F is bounded, by using the dominated convergence theorem and (3.4), we have
(3.8)∫C02[0,T]‍F(Zh1(w,⋅)+Zh2(z,⋅))d(m×m)(w,z)=lim⁡n→∞∫C02[0,T]‍Fn(Zh1(w,⋅)+Zh2(z,⋅))d(m×m)(w,z)=lim⁡n→∞∫C0[0,T]‍Fn(Zk(x,⋅))dm(x)=∫C0[0,T]‍F(Zk(x,⋅))dm(x),
which concludes the proof of Lemma 3.3.We are now ready to prove our main theorem.Proof of Theorem2.1.
LetF be Wiener integrable. Suppose that the left-hand side of (2.8) exists. By usual arguments of integration theory, there exists a sequence {Fn} of bounded and continuous functionals such that Fn converges to F. By Lemma 3.3 and the dominated convergence theorem, we can obtain the desired result.Corollary 3.4.
LetF be a functional on C0[0,T]. Then for all h∈L2[0,T] and all a,b∈ℝ,
(3.9)∫C02[0,T]‍F(aZh(w,⋅)+bZh(z,⋅))d(m×m)(w,z)=∫C0[0,T]‍F(a2+b2Zh(x,⋅))dm(x).Proof.
Simply chooseh1=ah and h2=bh in (2.8) and use the linearity property of the PWZ stochastic integral.Using similar arguments as in the proofs of Lemmas3.1, 3.2, and 3.3 and Theorem 2.1 above, we can obtain the following theorems.Theorem 3.5.
LetF be a functional on C0[0,T], and let {h1,…,hν} be any subset of L2[0,T]. Then
(3.10)∫C0ν[0,T]‍F(∑j=1ν‍Zhj(xj,⋅))dmν(x→)=∫C0[0,T]‍F(Zkν(x,⋅))dm(x),
where mν is the product Wiener measure on C0ν[0,T], the product of ν copies of C0[0,T], and
(3.11)kν(t)=∑n=1∞‍∑j=1ν‍(hj,ϕn)22ϕn(t)
for some complete orthonormal set {ϕn} in L2[0,T].Theorem 3.6.
LetF be a functional on C0[0,T]. Then for any h1 and h2 in L2[0,T],
(3.12)∫C02[0,T]‍F(Zh2(w,⋅)-Zh1(z,⋅),Zh1(w,⋅)+Zh2(z,⋅))d(m×m)(w,z)=*∫C02[0,T]‍F(Zk(w,⋅),Zk(z,⋅))d(m×m)(w,z),
where h1, h2, and k are related by (2.9).Remark 3.7.
Equations (2.8) and (3.12) are indeed very general formulas.(1)
For anyθ∈ℝ, choosing h1(t)≡sinθ and h2(t)≡cosθ in (2.8) yields (1.4).(2)
For anya,b∈ℝ, choosing h1(t)≡a and h2(t)≡b in (2.8) or choosing h(t)≡1 in (3.9) yields (1.5).(3)
For any function of bounded variationθ(·), choosing h1(t)=sinθ(t) and h2(t)=cosθ(t) on [0,T] in (3.12) yields (1.3).


## 4. Generalized Fourier-Feynman Transform and GeneralizedConvolution Product

In this section, we will apply our main theorem to the generalized analytic Fourier-Feynman transform and the convolution product theories.In defining various analytic Feynman integrals, one usually starts, forλ>0, with the Wiener integral
(4.1)∫C0[0,T]‍F(λ-1/2x)dm(x)
and then extends analytically in λ to the right-half complex plane. Here we start with the (generalized) Wiener integral
(4.2)∫C0[0,T]‍F(λ-1/2Zh(x,⋅))dm(x)=J(h;λ),
where 𝒵h is the Gaussian process given by (2.2) above.Throughout this section, letℂ+ and ℂ~+ denote the complex numbers with positive real part and the nonzero complex numbers with nonnegative real part, respectively.LetF be a complex-valued scale-invariant measurable functional on C0[0,T] such that J(h;λ) given by (4.2) exists and is finite for all λ>0. If there exists a function J*(h;λ) analytic on ℂ+ such that J*(h;λ)=J(h;λ) for all λ>0, then J*(h;λ) is defined to be the generalized analytic Wiener integral (with respect to the process 𝒵h) of F over C0[0,T] with parameter λ, and for λ∈ℂ+ we write
(4.3)∫C0[0,T]anwλ‍F(Zh(x,⋅))dm(x)=J*(h;λ).
Let q be a nonzero real number and let F be a functional such that ∫C0[0,T]anwλ‍F(𝒵h(x,·))dm(x) exists for all λ∈ℂ+. If the following limit exists, we call it the generalized analytic Feynman integral of F with parameter q and we write
(4.4)∫C0[0,T]anfq‍F(Zh(x,⋅))dm(x)=lim⁡λ→-iq∫C0[0,T]anwλ‍F(Zh(x,⋅))dm(x),
where λ approaches -iq through values in ℂ+.Note that ifh≡1 on [0,T], then these definitions agree with the previous definitions of the analytic Wiener integral and the analytic Feynman integral [3, 4, 8, 12–14].Next (see [5, 6, 15]) we state the definition of the generalized Fourier-Feynman transform (GFFT).Definition 4.1.
Forλ∈ℂ+ and y∈C0[0,T], let
(4.5)Tλ,h(F)(y)=∫C0[0,T]anwλ‍F(y+Zh(x,⋅))dm(x).
Let q be a non-zero real number. For p∈(1,2], we define the Lp analytic GFFT with respect to 𝒵h, Tq,h(p)(F) of F, by the formula (λ∈ℂ+),
(4.6)Tq,h(p)(F)(y)=l.i.m.λ→-iqTλ,h(F)(y)
if it exists; that is, for each ρ>0,
(4.7)lim⁡λ→-iq∫C0[0,T]‍|Tλ,h(F)(ρy)-Tq,h(p)(F)(ρy)|p′dm(y)=0,
where 1/p+1/p′=1. We define the L1 analytic GFFT, Tq,h(1)(F) of F, by the formula (λ∈ℂ+)(4.8)Tq,h(1)(F)(y)=lim⁡λ→-iqTλ,h(F)(y)
if it exists.We note that forp∈[1,2], Tq,h(p)(F) is defined only s-a.e. We also note that if Tq,h(p)(F) exists and if F≈G, then Tq,h(p)(G) exists and Tq,h(p)(G)≈Tq,h(p)(F). One can see that for each h∈L2[0,T], Tq,h(p)(F)≈Tq,-h(p)(F) since
(4.9)∫C0[0,T]‍F(x)dm(x)=∫C0[0,T]‍F(-x)dm(x).Next we give the definition of the generalized convolution product (GCP).Definition 4.2.
LetF and G be scale-invariant measurable functionals on C0[0,T]. For λ∈ℂ~+ and h1,h2∈L2[0,T], we define their GCP with respect to {𝒵h1,𝒵h2} (if it exists) by
(4.10)(F*G)λ(h1,h2)(y)={∫C0[0,T]anwλ‍F(y+Zh1(x,⋅)2)G(y-Zh2(x,⋅)2)dm(x),λ∈C+,∫C0[0,T]anfq‍F(y+Zh1(x,⋅)2)G(y-Zh2(x,⋅)2)dm(x),λ=-iq,q∈R,q≠0.
When λ=-iq, we denote (F*G)λ(h1,h2) by (F*G)q(h1,h2).Remark 4.3.
Our definition of the GCP is different than the definition given by Huffman et al. in [5, 6] and used by Chang et al. in [15]. But if we choose h1=h2 in (4.10), our GCP (F*G)q(h1,h2) is the GCP used in [5, 6, 15].We begin this section with a key lemma for a relationship between the GFFT and the GCP.Lemma 4.4.
Let{g1,g2,g3,g4} be a subset of L2[0,T], and let Yg1,g2,Yg3,g4:C02[0,T]×[0,T]→ℝ be given by
(4.11)Yg1,g2(w,z;t)=Zg1(w,t)+Zg2(z,t),Yg3,g4(w,z;t)=Zg3(w,t)-Zg4(z,t),
respectively. Then the following assertions are equivalent.  (i)
Yg1,g2 and Yg3,g4 are independent processes.(ii)
g1g3=g2g4.Proof.
Since the processesYg1,g2 and Yg3,g4 are Gaussian with mean zero, we know that Yg1,g2 and Yg3,g4 are independent processes if and only if
(4.12)∫C02[0,T]‍Yg1,g2(w,z;s)Yg3,g4(w,z;t)d(m×m)(w,z)=0
for every s,t∈[0,T]. But, using (2.4), we have
(4.13)∫C02[0,T]‍Yg1,g2(w,z;s)Yg3,g4(w,z;t)d(m×m)(w,z)=∫C02[0,T]‍{Zg1(w,s)Zg3(w,t)-Zg1(w,s)Zg4(z,t)+Zg2(z,s)Zg3(w,t)-Zg2(z,s)Zg4(z,t)}×d(m×m)(w,z)=∫0min⁡{s,t}‍g1(u)g3(u)du-∫0min⁡{s,t}‍g2(u)g4(u)du.
From this, we can obtain the desired result.We are now ready to establish fundamental relationships between the GFFT and the GCP.Lemma 4.5.
LetF and G be functionals on C0[0,T]. Let {h1,h2,h3} be a subset of L2[0,T] such that h32=h1h2≥0 almost everywhere on [0,T], and let
(4.14)k1(t)=2-1/2∑n=1∞‍(h1,ϕn)22+(h3,ϕn)22ϕn(t),k2(t)=2-1/2∑n=1∞‍(h2,ϕn)22+(h3,ϕn)22ϕn(t).
Furthermore, assume that for all λ∈ℂ+, Tλ,h3((F*G)λ(h1,h2)), Tλ,k1(F) and Tλ,k2(G) all exist. Then
(4.15)Tλ,h3((F*G)λ(h1,h2))(y)=Tλ,k1(F)(y2)Tλ,k2(G)(y2)
for s-a.e. y∈C0[0,T].Proof.
We note that for allλ>0,
(4.16)Tλ,h3((F*G)λ(h1,h2))(y)=∫C0[0,T]‍(F*G)λ(h1,h2)(y+λ-1/2Zh3(w,⋅))dm(w)=∬C0[0,T]F(y2+12λ(Zh3(w,⋅)+Zh1(z,⋅)))×G(y2+12λ(Zh3(w,⋅)-Zh2(z,⋅)))d(m×m)(w,z)=∫C02[0,T]‍F(y2+λ-1/2(Zh3(w,⋅)+Zh1(z,⋅)2))×G(y2+λ-1/2(Zh3(w,⋅)-Zh2(z,⋅)2))d(m×m)(w,z).
But h32=h1h2, and so (𝒵h3(w,·)+𝒵h1(z,·))/2 and (𝒵h3(w,·)-𝒵h2(z,·))/2 are independent processes by Lemma 4.4. Hence by (2.8), we obtain that for all λ>0,
(4.17)Tλ,h3((F*G)λ(h1,h2))(y)=∫C02[0,T]‍F(y2+12λ(Zh3(w,⋅)+Zh1(z,⋅)))d(m×m)(w,z)×∫C02[0,T]‍G(y2+12λ(Zh3(w,⋅)-Zh2(z,⋅)))d(m×m)(w,z)=∫C0[0,T]‍F(y2+Zk1(x,⋅)λ)dm(x)∫C0[0,T]‍G(y2+Zk2(x,⋅)λ)dm(x)=Tλ,k1(F)(y2)Tλ,k2(G)(y2).
Equation (4.15) holds for all λ∈ℂ+ by analytic continuation.In next theorem, we show that the GFFT of the GCP is the product of GFFTs.Theorem 4.6.
LetF, G, {h1,h2,h3}, k1, and k2 be as in Lemma 4.5. Furthermore, assume that for p∈[1,2], λ∈ℂ+ and q∈ℝ-{0}, Tλ,h3((F*G)q(h1,h2)), Tq,h3(p)((F*G)q(h1,h2)), Tq,k1(p)(F), and Tq,k2(p)(G) all exist and that
(4.18)Tq,h3(p)((F*G)q(h1,h2))={l.i.m.λ→-iqTλ,h3((F*G)λ(h1,h2))(y),p∈(1,2],lim⁡λ→-iqTλ,h3((F*G)λ(h1,h2))(y),p=1.
Then
(4.19)Tq,h3(p)((F*G)q(h1,h2))(y)=Tq,k1(p)(F)(y2)Tq,k2(p)(G)(y2)
for s-a.e. y∈C0[0,T].Proof.
Equation (4.19) follows from (4.15) by letting λ→-iq, since all transforms in (4.18) and (4.19) exist.Remark 4.7.
We note that the hypotheses (and hence the conclusions) of Theorem4.6 above are indeed satisfied by many of the functionals in the following large classes of functionals. These classes of functionals include; (i)
the Banach algebra𝒮 defined by Cameron and Storvick in [16]: also see [3, 5, 14, 15],(ii)
various spaces of functionals of the form(4.20)F(x)=exp⁡{∫0T‍f(t,x(t))dt}
for appropriate f:[0,T]×ℝ→ℂ as discussed in [4, 12, 13]; and(iii)
various spaces of functionals of the form(4.21)F(x)=exp⁡{∬0Tf(s,t,x(s),x(t))dsdt}
for appropriate f:[0,T]2×ℝ2→ℂ as discussed in [3].Next five corollaries include the results of [3–6] by Huffman et al. The notations used in [3–6] are slightly different than ours.Corollary 4.8.
Refer to Theorem 2.1 in [5].Proof.
In our Lemma4.5, simply choose h1=h2=h3=k1=k2≡h∈L2[0,T].Corollary 4.9.
Refer to Theorem 1 in [6].Proof.
In our Theorem4.6, simply choose h1=h2=h3=k1=k2≡h∈L2[0,T].Corollary 4.10.
Refer to Theorem 3.3 in [5].Proof.
In our Theorem4.6, simply choose h1=h2=h3=k1=k2≡h∈L∞[0,T].Corollary 4.11.
Refer to Theorem 3.3 in [3].Proof.
In our Theorem4.6, simply choose h1=h2=h3=k1=k2≡1.Corollary 4.12.
Refer to Lemma 4.1 and Theorems 4.1 and 4.2 in [4].Proof.
In our Lemma4.5 and Theorem 4.6, simply choose h1=h2=h3=k1=k2≡1.



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*Source: 578174-2012-08-05.xml*