# A Rotation of Admixable Operators on Abstract Wiener Space with Applications

Authors: Jae Gil Choi; Seung Jun Chang
Journal: Journal of Function Spaces and Applications
 (2013)

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## Abstract

We investigate certain rotation properties of the abstract Wiener measure. To determine our rotation property for the Wiener measure, we introduce the concept of an admixable operator via an algebraic structure on abstract Wiener space. As for applications, we define the analytic Fourier-Feynman transform and the convolution product associated with the admixable operators and proceed to establish the relationships between this transform and the corresponding convolution product.


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## Body

## 1. Introduction and Preliminaries

Let(C0[0,T],ℳ,mw) denote one-parameter Wiener space. Bearman's rotation theorem [1] for Wiener measure has played an important role in various research areas in mathematics and physics involving Wiener integration theory. Bearman's theorem was further developed by Cameron and Storvick [2] and by Johnson and Skoug [3] in their studies of Wiener integral equations. Recently, in [4], using results in [5], Chang et al. obtained results involving a very general multiple Fourier-Feynman transform on Wiener space.LetH be a real separable infinite-dimensional Hilbert space with inner product 〈·,·〉 and norm |·|=〈·,·〉. Let ∥·∥0 be a measurable norm on H with respect to the Gaussian cylinder set measure ν0 on H. Let B denote the completion of H with respect to ∥·∥0. Let i denote the natural injection from H to B. The adjoint operator i* of i is one to one and maps B* continuously onto a dense subset of H*, where B* and H* are topological duals of B and H, respectively. By identifying H* with H and B* with i*B*, we have a triple B*⊂H*≈H⊂B and 〈x,y〉=(x,y) for all x in H and y in B*, where (·,·) denotes the natural dual pairing between B and B*. By the well-known result of Gross [6], ν0∘i-1 provides a unique countably additive extension, ν, to the Borel σ-algebra ℬ(B) of B. ν is a probability measure on the Borel σ-algebra ℬ(B) of B which satisfies
(1)∫B‍exp{i(y,x)}dν(x)=exp{-12|y|2}foreveryy∈B*.
The triple (B,H,ν) is called an abstract Wiener space. For more details, see [6–9].Let{ej}j=1∞ be a complete orthonormal set in H such that ej's are in B*. For each h∈H and x∈B, we define the stochastic inner product (h,x)~ by
(2)(h,x)~={limn→∞∑j=1n‍〈h,ej〉(ej,x),ifthelimitexists,0,otherwise.
For every h (≠0) in H, (h,x)~ exists for ν-a.e. x∈B, and it is a Gaussian random variable on B with mean zero and variance |h|2; that is, (1) holds with y∈B* replaced with h∈H. In fact, the stochastic inner product (h,x)~ is essentially independent of the choice of the complete orthonormal set used in its definition. Also, if both h and x are in H, then Parseval's identity gives (h,x)~=〈h,x〉. Furthermore, (h,λx)~=(λh,x)~=λ(h,x)~ for any λ∈ℝ, h∈H, and x∈B. We also see that, if {h1,…,hn} is an orthonormal set in H, then the random variables (hj,x)~’s are independent.Let𝒲(B) be the class of ν-measurable subsets of B. A subset E of B is said to be scale-invariant measurable [3, 7] provided ρE is 𝒲(B)-measurable for every ρ>0, and a scale-invariant measurable subset N of B is said to be scale-invariant null provided ν(ρN)=0 for every ρ>0. A property that holds except on a scale-invariant null set is said to hold scale invariant almost everywhere (s-a.e.). A functional F on B is said to be scale-invariant measurable provided F is defined on a scale-invariant measurable set and F(ρ·) is 𝒲(B)-measurable for every ρ>0. If two functionals F and G on B are equal s-a.e., that is, for each ρ>0, ν({x∈B:F(ρx)≠G(ρx)})=0, then we write F≈G.Next, we introduce the concept of anadmixable operator on B.Definition 1.
Let⊙ be an operation between H and B* which satisfies the conditions:(1)
B*×B*∋(g1,g2)↦g1⊙g2=g2⊙g1∈B*.(2)
H×B*∋(h,g)↦h⊙g=g⊙h∈H.(3)
Ifh⊙g=0 for h∈H and g∈B*, then h=0 or g=0.(4)
For everyh∈H and every g1,g2∈B*,
(3)(h⊙g1)⊙g2=h⊙(g1⊙g2).(5)
For everyh1,h2∈H and every g∈B*,
(4)(h1+h2)⊙g=h1⊙g+h2⊙g.(6)
For everyg1,g2∈B*, there exists g3∈B* such that
(5)g1⊙2+g2⊙2=g3⊙2,whereg⊙2=g⊙g. In this case, we write g3=g1⊙2+g2⊙2⊙.(7)
For everyh1,h2∈H and every g∈B*,
(6)〈h1,h2⊙g〉=〈h1⊙g,h2〉.
Giveng∈B*, let Ag:B→B be a linear operator associated with g. The operator Ag is said to be g⊙-admixable provided (h,Agx)~=(h⊙g,x)~ for all h∈H.For a finite subset𝒱={v1,…,vm} of H, let X𝒱:B→ℝm be the random vector given by
(7)X𝒱(x)=((v1,x)~,…,(vm,x)~)≡(v→,x)~.
A functional F is called a cylinder-type functional on B if there exists a linearly independent subset 𝒱={v1,…,vm} of H such that
(8)F(x)=ψ(X𝒱(x)),x∈B,
where ψ is a complex-valued Lebesgue measurable function on ℝm. It is easy to show that, for the given cylinder-type functional F of the form (8), there exists an orthogonal subset ℋ={h1,…,hn} of H such that F is expressed as
(9)F(x)≡f(Xℋ(x))=f((h1,x)~,…,(hn,x)~),x∈B,
where f is a complex-valued Lebesgue measurable function on ℝn. Thus, we lose no generality in assuming that every cylinder-type functional on B is of the form (9).Lemma 2 (Chung, [7]).
Let(B,H,ν) be an abstract Wiener space, and let ℋ={h1,…,hn} be an orthogonal set in H. Let f be a complex-valued function defined on ℝn. Then, for the cylinder-type functional F given by (9) on B,(i)
F is ℬ(B)-measurable if and only if f is Borel measurable on ℝn,(ii)
F is 𝒲(B)-measurable if and only if f is Lebesgue measurable on ℝn.Forg∈B*, let Ag be the g⊙-admixable operator on B. In this case, for any orthogonal subset ℋ of H,
(10)Xℋ(Agx)=((h1,Agx)~,…,(hn,Agx)~)=((h1⊙g,x)~,…,(hn⊙g,x)~).The seminal results by Bearman in [1] are summarized as follows (see [2]): if F(a2+b2x) is Wiener integrable on C0[0,T] for a,b∈ℝ, then F(ax1+bx2) is integrable on (C0[0,T])2 and
(11)∫(C0[0,T])2‍F(ax1+bx2)d(mw×mw)(x1,x2)=∫C0[0,T]‍F(a2+b2x)dmw(x).The main purpose of this paper is to establish a rotation property for the abstract Wiener integral,(12)∫B2‍F(Ag1x1+Ag2x2)d(ν×ν)(x1,x2)=∫B‍F(Agx)dν(x),
where F is given by (9) and Ag is determined by Ag1 and Ag2.


## 2.  A Typical Example of an Abstract Wiener Space

The classical Wiener spaceC0[0,T], which is one of the most important examples of abstract Wiener spaces (see [9]), is a triple (B,H,ν), where  (i)
B=C0[0,T] is a Banach space consisting of real-valued continuous functions x(t) with x(0)=0 defined on the compact interval [0,T] endowed with uniform norm ∥x∥=supt∈[0,T]|x(t)|,(ii)
H=C0′[0,T] is a real separable infinite dimensional Hilbert space consisting of absolutely continuous functions h(t) with h(0)=0, such that Dh≡dh/dt∈L2[0,T] endowed with the inner product
(13)〈h1,h2〉=∫0T‍Dh1(s)Dh2(s)ds,(iii)
ν=mw is the Wiener measure on the Borel σ-algebra ℬ(C0[0,T]) of C0[0,T] with
(14)mw({x:x(t)≤a})=∫-∞a‍exp{-u22t}du.LetI be the unitary operator from L2[0,T], onto C0′[0,T] given by Iv(t)=∫0t‍v(s)ds for v∈L2[0,T] and let
(15)C0*[0,T]={Iv:viscontinuousexceptforafinitenumberoffinitejumpdiscontinuitiesandisofboundedvariationon[0,T]}.
For any h∈C0′[0,T] and g∈C0*[0,T], let the operation ⊙ between C0′[0,T] and C0*[0,T] be defined by
(16)h⊙g=I(DhDg),
where DhDg denotes the pointwise multiplication of the functions Dh and Dg.It is readily seen that{x(t):(x,t)∈C0[0,T]×[0,T]} is a standard Wiener process on the probability space (C0[0,T],ℬ(C0[0,T]),mw). In this case, we know that, for each h∈C0′[0,T] and s-a.e. x∈C0[0,T],
(17)(h,x)~=∫0T‍Dh(t)d~x(t),
where ∫0T‍Dh(t)d~x(t) denotes the Paley-Wiener-Zygmund stochastic integral [10–12].Remark 3.
Letℳ be the set of all mw-measurable subsets of C0[0,T]. Then, (C0[0,T],ℳ,mw) is a complete measure space. It is well known that ℳ coincides with σ(ℬ(C0[0,T])), the completion of ℬ(C0[0,T]).Letg∈C0′[0,T] with |g|=〈g,g〉>0. Then, the stochastic integral
(18)𝒵g(x,t)=∫0t‍Dg(s)d~x(s),t∈[0,T],
which was introduced by Park and Skoug in [13], is a Gaussian process with mean zero and covariance function
(19)∫C0[0,T]‍𝒵g(x,s)𝒵g(x,t)dmw(x)=∫0min{s,t}‍(Dg(u))2du.
In addition, 𝒵g(x,t) is stochastically continuous in t on [0,T]. For more detailed studies of this process, see [14–17]. Furthermore, if g is an element of C0*[0,T], then, for all x∈C0[0,T], 𝒵g(x,t) is continuous in t, and so is 𝒵g(x,·) in C0[0,T].From [14, Lemma 1], we note that, for each h∈C0′[0,T] and each g∈C0′[0,T] with Dg∈L∞[0,T],
(20)∫0T‍Dh(t)d~𝒵g(x,t)=∫0T‍Dh(t)Dg(t)d~x(t)
for s-a.e. x∈C0[0,T].Giveng∈C0*[0,T], define an operator Ag:C0[0,T]→C0[0,T] by
(21)Agx=𝒵g(x,·).
Then, for all h∈H,
(22)(h,Agx)~=∫0T‍Dh(t)d~𝒵g(x,t)=∫0T‍Dh(t)Dg(t)d~x(t)=(h⊙g,x)~.
Thus, Ag is g⊙-admixable in view of Definition 1.


## 3. A Rotation of Admixable Operators

In this section, we establish a rotation property for the abstract Wiener integral involving admixable operators. We first introduce an integration formula which plays a key role.Lemma 4.
Let𝒜={α1,…,αn} be an orthogonal set in H and let X𝒜 be given by (7). Let f:ℝn→ℂ be a Lebesgue measurable function. Then
(23)∫B‍f(X𝒜(x))dν(x)=∫B‍f((α1,x)~,…,(αn,x)~)dν(x)=*(∏j=1n‍2π|αj|2)-1/2×∫ℝn‍f(u1,…,un)×exp{-∑j=1n‍uj22|αj|2}du1⋯dun,
where by =* one means that if either side exists, both sides exist and equality holds.The following integration formula is also used several times in this paper:(24)∫ℝ‍exp{-au2+bu}du=πaexp{b24a}
for complex numbers a and b with Re(a)>0.Remark 5.
Letα1 and α2 be elements in H with |α1|=|α2|≡σ2>0. Then, the random variables X1(x)=(α1,x)~ and X2(x)=(α2,x)~ will have the same distribution N(0,σ2).Let𝒜1={α11,…,αn1} and 𝒜2={α12,…,αn2} be orthogonal sets in H with |αj1|=|αj2| for each j∈{1,…,n}. Using the aforementioned facts and applying (23), we see that, for any Lebesgue measurable function f on ℝn,
(25)∫B‍f(X𝒜1(x))dν(x)=*∫B‍f(X𝒜2(x))dν(x).To simplify the expressions in our results, we use the following notations:(26)f(u→)≡f(u1,…,un),f(u→+X𝒜(x))≡f(u1+(α1,x)~,…,un+(αn,x)~)
for u→=(u1,…,un)∈ℝn and X𝒜(x) given by (7).Proposition 6.
Let𝒜1={α11,…,αn1} and 𝒜2={α12,…,αν2} be orthogonal sets in H. Then, for any Lebesgue measurable function f on ℝn,
(27)∫B2‍f(X𝒜1(x1)+X𝒜2(x2))d(ν×ν)(x1,x2)=*∫B‍f(X𝒲(2)(x))dν(x),
where 𝒲(2)={w1,…,wn} is an orthogonal set in H which satisfies the condition
(28)|wj|2=|αj1|2+|αj2|2
for each j∈{1,…,n}. Also, both of the expressions in (27) are given by the last expression in (29).Proof.
First, using (23), the Fubini theorem, and (24), we have that
(29)∫B2‍f(X𝒜1(x1)+X𝒜2(x2))d(ν×ν)(x1,x2)=∫B‍[∫B‍f(X𝒜1(x1)+X𝒜2(x2))dν(x1)]dν(x2)=(∏j=1n‍2π|αj1|2)-1/2×∫ℝn‍[∫B‍f(u→+X𝒜2(x2))dν(x2)]×exp{-∑j=1n‍uj22|αj1|2}du→=(∏j=1n‍2π|αj1|2)-1/2(∏j=1n‍2π|αj2|2)-1/2×∫ℝn‍[{-∑j=1n‍vj22|αj2|2}∫ℝn‍f(u→+v→)×exp{-∑j=1n‍vj22|αj2|2}dv→]×exp{-∑j=1n‍uj22|αj1|2}du→=(∏j=1n‍2π|αj1|2)-1/2(∏j=1n‍2π|αj2|2)-1/2×∬ℝnf(r→)exp{-∑j=1n‍(rj-uj)22|αj2|2}×exp{-∑j=1n‍uj22|αj1|2}du→dr→=(∏j=1n‍2π|αj1|2)-1/2(∏j=1n‍2π|αj2|2)-1/2×∫ℝn‍f(r→)[∫ℝn‍exp{-∑j=1n‍uj22|αj1|2-∑j=1n‍(rj-uj)22|αj2|2}du→]dr→=(∏j=1n‍2π(|αj1|2+|αj2|2))-1/2×∫ℝn‍f(r→)exp{-∑j=1n‍rj22(|αj1|2+|αj2|2)}dr→.
Next, let{e1,…,en} be any orthonormal set in H. For each j∈{1,…,n}, let wj=(|αj1|2+|αj2|2)1/2ej. Then 𝒲(2)={w1,…,wn} is an orthogonal set in H and satisfies (28) above. In this case, using (23), the Fubini theorem, and (24), we see that ∫B‍f(X𝒲(2)(x))dν(x) is given by the last expression of (29). In view of Remark 5, we obtain the desired result.The following corollary follows by the use of mathematical induction.Corollary 7.
For eachk∈{1,…,m}, let 𝒜k={α1k,…,αnk} be an orthogonal set in H. Then for any Lebesgue measurable function f on ℝn,
(30)∫Bm‍f(X𝒜1(x1)+⋯+X𝒜m(xm))dνm(x1,…,xm)=*∫B‍f(X𝒲(m)(x))dν(x),
where 𝒲(m)={w1,…,wn} is an orthogonal set in H which satisfies the condition
(31)|wj|2=∑k=1m‍|αjk|2
for each j∈{1,…,n}. Also, both of the expressions in (30) are given by the expression
(32)(∏j=1n‍2π(∑k=1m‍|αjk|2))-1/2×∫ℝn‍f(r→)exp{-∑j=1n‍rj22(∑k=1m‍|αjk|2)}dr→.The next corollary follows directly from Proposition6.Corollary 8.
Let𝒜1 and 𝒜2 be as in Proposition 6. Assume that 𝒜1∪𝒜2 is an orthogonal set in H. Then, for any Borel measurable function f on ℝn, (27) is valid. In this case, 𝒲(2) is given by the set {α11+α12,…,αn1+αn2}.Throughout the rest of this section, for convenience, we use the following notation: for a finite sequence𝒢={g1,…,gm} in B*, let
(33)s(𝒢)≡s(g1,…,gm)=∑k=1m‍gk⊙2⊙.For our rotation property presented, namely, Theorem11, we will consider the pair of finite subsets ℋ={h1,…,hn} of H-{0} and 𝒢={g1,…,gm} of B*-{0} (this allows |𝒢|≤m) such that  (c1)
{h1,…,hn} is orthogonal in H,(c2)
{h1⊙gk,…,hn⊙gk} is orthogonal in H for all k∈{1,…,m}.Let us return to the classical Wiener space(C0[0,T],C0′[0,T],mw). See Section 2. For each l∈{1,2,…}, let βl(t)=sin(((l-1/2)π/T)t) on [0,T]. Then, Θ≡{βl}l=1∞ is an orthogonal sequence in C0′[0,T]. Additionally, Θ~={(2T/(l-1/2)π)βl}l=1∞ is a complete orthonormal set in C0′[0,T]. In fact, βl∈C0*[0,T] for all l∈{1,2,…}. We can take different pair (ℋ,𝒢) from Θ, which satisfies conditions (c1) and (c2) above. For instance, let ℋ={βl1,…,βln} and let 𝒢={βk1,…,βkm}, where l1<l2<⋯<ln<k1≤k2≤⋯≤km.Remark 9.
Letℋ={h1,…,hn} and 𝒢={g1,…,gm} satisfy the conditions (c1) and (c2). Then, the stochastic inner products
(34)(hj,As(𝒢)x)~=(hj⊙s(𝒢),x)~,j=1,…,n,
form a set of independent Gaussian random variables on B with mean 0 and variance
(35)|hj⊙s(𝒢)|2=〈hj⊙s(𝒢),hj⊙s(𝒢)〉=〈hj⊙s(𝒢)⊙s(𝒢),hj〉=〈hj⊙(∑k=1m‍gk⊙2),hj〉=〈∑k=1m‍(hj⊙gk⊙2),hj〉=∑k=1m‍〈hj⊙gk⊙2,hj〉=∑k=1m‍〈hj⊙gk,hj⊙gk〉=∑k=1m‍|hj⊙gk|2,
where s(𝒢)≡s(g1,…,gm) is given by (33).Remark 10.
Given an orthogonal subsetℋ, let Bℋ* be the space of every element g in B*-{0} such that {h⊙g:h∈ℋ} is orthogonal in H. Then, for any finite subset 𝒢 of Bℋ*, the pair (ℋ,𝒢) satisfies conditions (c1) and (c2) above.Given an orthogonal setℋ in H, let 𝔊ℋ(1) be the class of all cylinder-type functionals, F, given by (9) for ν-a.e. x∈B, where the corresponding function f of F satisfies the condition
(36)∫ℝn‍|f(v→)|exp{-∑j=1n‍vj22|hj⊙g|2}dv→<+∞
for all g∈Bℋ*.We are now ready to present our rotation property of abstract Wiener measure associated with admixable operators.Theorem 11.
Letℋ={h1,…,hn} be an orthogonal set in H and let F∈𝔊ℋ(1) be given by (9). Let 𝒢={g1,…,gm} be a finite subset of Bℋ*, and, for each k∈{1,…,m}, let Agk be the gk⊙-admixable operator on B. Then,
(37)∫Bm‍F(∑k=1m‍Agkxk)dνm(x1,…,xm)=∫B‍F(As(𝒢)x)dν(x),
where s(𝒢) is given by (33).Proof.
For eachj∈{1,…,n} and each k∈{1,…,m}, let αjk=hj⊙gk. For each k∈{1,…,m}, 𝒜k={α1k,…,αnk} is orthogonal in H by condition (c2). Hence, for each k∈{1,…,m}, the stochastic inner products
(38)(hj,Agkx)~=(αjk,x)~,j=1,…,n,
form a set of independent Gaussian random variables with mean 0 and variance |αjk|2.
We observe that(39)Xℋ(∑k=1m‍Agkxk)=((h1,∑k=1m‍Agkxk)~,…,(hn,∑k=1m‍Agkxk)~)=(∑k=1m‍(h1,Agkxk)~,…,∑k=1m‍(hn,Agkxk)~)=∑k=1m‍((h1,Agkxk)~,…,(hn,Agkxk)~)=∑k=1m‍((α1k,xk)~,…,(αnk,xk)~)=∑k=1m‍X𝒜k(xk).
Thus, using (9), (39), and (30), we obtain
(40)∫Bm‍F(∑k=1m‍Agkxk)dνm(x1,…,xm)=∫Bm‍f(Xℋ(∑k=1m‍Agkxk))dνm(x1,…,xm)=∫Bm‍f(∑k=1m‍X𝒜k(xk))dνm(x1,…,xm)=∫B‍f(X𝒲(m)(x))dν(x),
where 𝒲(m)={w1,…,wn} is an orthogonal set in H, such that
(41)|wj|2=∑k=1m‍|αjk|2=∑k=1m‍|hj⊙gk|2
for each j∈{1,…,n}.
Next, we note that, for eachj,l∈{1,…,n} with j≠l,
(42)〈hj⊙s(𝒢),hl⊙s(𝒢)〉=〈hj⊙s(𝒢)⊙s(𝒢),hl〉=〈hj⊙∑k=1m‍gk⊙2,hl〉=〈∑k=1m‍hj⊙gk⊙2,hl〉=∑k=1m‍〈hj⊙gk⊙2,hl〉=∑k=1m‍〈hj⊙gk,hl⊙gk〉=0.
Hence, from (35) and (42), we see that {h1⊙s(𝒢),…,hn⊙s(𝒢)} is an orthogonal set in H with |hj⊙s(𝒢)|2=∑k=1m‍|hj⊙gk|2 for all j∈{1,…,n}, and so we can choose 𝒲(m) to be the orthogonal set {h1⊙s(𝒢),…,hn⊙s(𝒢)}. In this case, we see that
(43)∫B‍f(X𝒲(m)(x))dν(x)=∫B‍f((h1⊙s(𝒢),x)~,…,(hn⊙s(𝒢),x)~)dν(x)=∫B‍f((h1,As(𝒢)x)~,…,(hn,As(𝒢)x)~)dν(x)=∫B‍F(As(𝒢)x)dν(x).
Equation (37) follows from (40) and (43).Example 12.
Let us return to the classical Wiener space(C0[0,T],C0′[0,T],mw) again. We introduce the family of functions Γ≡{γτ:τ∈[0,T]} from C0′[0,T]:
(44)γτ(s)={s,s∈[0,τ]τ,s∈[τ,T].
These functions have the reproducing property
(45)〈h,γτ〉=∫0T‍Dh(s)Dγτ(s)ds=∫0T‍Dh(s)χ[0,τ](s)ds=∫0τ‍Dh(s)ds=h(τ)
for all h∈C0′[0,T]. In fact, Γ⊂C0*[0,T].
Forγτ∈Γ, we consider the admixable operator Aγτ given by (21). We note that, for each g∈C0*[0,T],
(46)(g,Aγτx)~=(g⊙γτ,x)~=∫0T‍Dg(s)Dγτ(s)d~x(s)=∫0τ‍Dg(s)d~x(s)=𝒵g(x,τ)=Agx(τ).
From this, we see that, for gI(t)=t=∫0t‍ds∈C0*[0,T] and a∈ℝ,
(47)(gI,Aaγτx)~=ax(τ),
and so, using (37), we find that, for any f∈L1(ℝ) and every a,b∈ℝ,
(48)∫(C0[0,T])2‍f(ax1(τ)+bx2(τ))d(mw×mw)(x1,x2)=∫(C0[0,T])2‍f((gI,Aaγτx1)~+(gI,Abγτx2)~)d(mw×mw)(x1,x2)=∫C0[0,T]‍f((gI,As(aγτ,bγτ)x)~)dmw(x).
Using (16), we now have
(49)s(aγτ,bγτ)(t)=(aγτ)⊙2+(bγτ)⊙2⊙(t)=(aγτ)⊙2+(bγτ)⊙2(t)=a2+b2γτ(t)
on [0,T]. Thus, by (48), (49), and (47), we have
(50)∫(C0[0,T])2‍f(ax1(τ)+bx2(τ))d(mw×mw)(x1,x2)=∫C0[0,T]‍f(a2+b2x(τ))dmw(x).Proposition 13 (Cameron and Storvick, [2]).
LetF(a2+b2x) be Wiener integrable on C0[0,T] for a,b∈ℝ. As a result, F(ax1+bx2) is integrable on C02[0,T], and
(51)∫C02[0,T]‍F(ax1+bx2)d(mw×mw)(x1,x2)=∫C0[0,T]‍F(a2+b2x)dmw(x).Proof.
Letn be any positive integer, and let 0=τ0<τ1<⋯<τn≤T be any partition of [0,T]. It suffices to show that (51) holds for any tame function
(52)F(x)=f(x(τ1),x(τ2)-x(τ1),…,x(τn)-x(τn-1))
with f∈L1(ℝn).
Let⊙ be defined by (16) between C0′[0,T] and C0*[0,T]. For gI(t)=t=∫0t‍ds∈C0*[0,T], let the gI⊙-admixable operator AgI be given by (21), and, for each j∈{1,…,n}, let
(53)hj(t)=∫0t‍χ[τj-1,τj](s)ds
on [0,T]. For any a,b∈ℝ-{0}, (c1)
{h1,…,hn} is an orthogonal set in C0′[0,T], and(c2)
{h1⊙agI,…,hn⊙agI} and {h1⊙bgI,…,hn⊙bgI} are orthogonal sets in C0′[0,T].
Also, for anya,b∈ℝ-{0}, As(agI,bgI)=Aa2+b2gI, and, for each j∈{1,…,n},
(54)(hj,AagIx)~=a∫0T‍χ[τj-1,τj](s)dx(s)=a(x(τj)-x(τj-1)).
Thus, the left side of (51) with F given by (52) is rewritten by
(55)∫(C0[0,T])2‍F(ax1+bx2)d(mw×mw)(x1,x2)=∫(C0[0,T])2‍f((h1,AagIx1)~+(h1,AbgIx2)~,…,(hn,AagIx1)~+(hn,AbgIx2)~)d(mw×mw)(x1,x2).
Thus, from Theorem 11, we obtain
(56)∫(C0[0,T])2‍F(ax1+bx2)d(mw×mw)(x1,x2)=∫C0[0,T]‍f((h1,Aa2+b2gIx)~,…,(hn,Aa2+b2gIx)~)dmw(x)=∫C0[0,T]‍F(a2+b2x)dmw(x).
Using standard methods, similar to those in [1], we can obtain the result for general functionals F on C0[0,T].


## 4. Fourier-Feynman Transforms and Convolutions Associated with Admixable Operators

In this section, to apply our results from the previous section, we first define anLp analytic Fourier-Feynman transform associated with admixable operators on B. Then, we establish the existence theorem and the inverse transform theorem of this transform for some classes of cylinder-type functionals on B having the form (9) for s-a.e. x∈B. Moreover, we present various relationships involving the convolution and the transforms.Throughout the rest of this paper, letℂ, ℂ+, and ℂ~+ denote, respectively, the complex numbers, the complex numbers with positive real part, and the nonzero complex numbers with nonnegative real part.Letg∈B*, and let Ag be the corresponding admixable operator on B. Let F:B→ℂ be a scale-invariant measurable functional such that
(57)JF(g;λ)=∫B‍F(λ-1/2Agx)dν(x)
exists as a finite number for all λ>0. If there exists a function JF*(g;λ) analytic on ℂ+ such that JF*(g;λ)=JF(g;λ) for all λ>0, then JF*(g;λ) is defined to be the analytic Wiener integral (associated with the g⊙-admixable operator Ag) of F over B with parameter λ. For λ∈ℂ+, we write
(58)Iganwλ[F]=JF*(g;λ).
Let q≠0 be a real number, and let F be a functional such that ∫Banwλ‍F(Agx)dν(x)=JF*(g;λ) exists for all λ∈ℂ+. If the following limit exists, we call it the analytic Feynman integral of F with parameter q, and we write
(59)Iganfq[F]=limλ→-iqIganwλ[F],
where λ approaches -iq through values in ℂ+.Note that ifAg is the identity operator on B, then these definitions agree with the previous definitions of the analytic Wiener integral and the analytic Feynman integral [18–20].We are now ready to state the definition of the analytic Fourier-Feynman transform associated with admixable operator (admix-FFT).Definition 14.
Let(B,H,ν) be an abstract Wiener space. For g∈B*, λ∈ℂ+, and y∈B, let
(60)Tλ,g(F)(y)=∫Banwλ‍F(y+Agx)dν(x),
where Ag is the g⊙-admixable operator on B. Let q be a nonzero real number. For p∈(1,2], we define the Lp analytic g⊙-admix-FFT, Tq,g(p)(F) of F, by the formula (λ∈ℂ+),
(61)Tq,g(p)(F)(y)=l.i.m.λ→-iqTλ,g(F)(y),
if it exists; that is, for each ρ>0,
(62)limλ→-iq∫B‍|Tλ,g(F)(ρy)-Tq,g(p)(F)(ρy)|p′dν(y)=0,
where 1/p+1/p′=1. We define the L1 analytic g⊙-admix-FFT, Tq,g(1)(F) of F, by the formula (λ∈ℂ+),
(63)Tq,g(1)(F)(y)=limλ→-iqTλ,g(F)(y),
if it exists.We note that, forp∈[1,2], Tq,g(p)(F) is defined only s-a.e.. We also note that if Tq,g(p)(F) exists and if F≈G, then Tq,g(p)(G) exists and Tq,g(p)(G)≈Tq,g(p)(F).Next, we give the definition of the convolution product (CP).Definition 15.
LetF and G be scale-invariant measurable functionals on B. For λ∈ℂ~+ and g∈B*, we define their CP with respect to Ag (if it exists) by
(64)(F*G)λ,g(y)={∫Banwλ‍F(y+Agx2)G(y-Agx2)dν(x),λ∈ℂ+,∫Banfq‍F(y+Agx2)G(y-Agx2)dν(x),λ=-iq,q∈ℝ,q≠0.
When λ=-iq, we denote (F*G)λ,g by (F*G)q,g.For any scale-invariant measurable functionalF, we see that, for λ>0,
(65)∫Banwλ‍F(y+Agx)dν(x)=∫B‍F(y+λ-1/2Agx)dν(x)
if it exists.Letℳ(ℝn) denote the space of complex-valued, countably additive (and hence finite) Borel measures on ℬ(ℝn), the Borel σ-algebra of ℝn. It is well known that a complex-valued Borel measure τ necessarily has a finite total variation ∥τ∥, and ℳ(ℝn) is a Banach algebra under the norm ∥·∥ and with convolution as multiplication.Forτ∈ℳ(ℝn), the Fourier transform τ^ of τ is a complex-valued function defined on ℝn by the formula
(66)τ^(u→)=∫ℝn‍exp{i∑j=1n‍ujvj}dτ(v→),
where u→=(u1,…,un) and v→=(v1,…,vn) are in ℝn.Letℋ={h1,…,hn} be an orthonormal set in H. Define the functional F:B→ℂ by
(67)F(x)=τ^(Xℋ(x))=τ^((h1,x)~,…,(hn,x)~)
for s-a.e. x∈B, where τ^ is the Fourier transform of τ in ℳ(ℝn). Then F is a bounded cylinder-type functional because |τ^(u→)|≤∥τ∥<+∞.Let𝔗ℋ be the set of all functionals F on B having the form (67). Note that F∈𝔗ℋ implies that F is scale-invariant measurable on B. Throughout this section, we fix the orthogonal set ℋ.We now state the existence theorem for the analytic Feynman integral of the functionals in𝔗ℋ.Theorem 16.
LetF∈𝔗ℋ be given by (67). Then, for all g∈Bℋ* and all nonzero real numbers q, the analytic Feynman integral Iganfq[F] of F exists and is given by the formula
(68)Iganfq[F]=∫ℝn‍exp{-i2q∑j=1n‍|hj⊙g|2vj2}dτ(v→).Proof.
By (67), (66), the Fubini theorem, (23), and (24), we see that, for all λ>0,
(69)JF(g;λ)=∫B‍F(λ-1/2Agx)dν(x)=∫ℝn‍[∫B‍exp{iλ-1/2∑j=1n‍(hj⊙g,x)~vj}dν(x)]dτ(v→)=(∏j=1n‍λ2π|hj⊙g|2)1/2×∫ℝn‍[∫ℝn‍exp{i∑j=1n‍ujvj-λ2∑j=1n‍uj2|hj⊙g|2}du→]dτ(v→)=(∏j=1n‍λ2π|hj⊙g|2)1/2×∫ℝn‍[∏j=1n‍∫ℝ‍exp{iujvj-λ2uj2|hj⊙g|2}duj∏j=1n‍]dτ(v→)=∫ℝn‍exp{-12λ∑j=1n‍|hj⊙g|2vj2}dτ(v→).
Now, letJF*(g;λ)=∫ℝn‍exp{-(1/2λ)∑j=1n‍|hj⊙g|2vj2}dτ(v→) for λ∈ℂ+. Then, JF*(g;λ)=JF(g;λ) for all λ>0 and |JF*(g;λ)|≤∫ℝn‍d|τ|(v→)≤∥τ∥<∞ for all λ∈ℂ+. Thus, applying the dominated convergence theorem, we see that JF*(g;λ) is continuous on ℂ+. Also, because k(λ)≡exp{-(1/2λ)∑j=1n‍|hj⊙g|2vj2} is analytic on ℂ+, applying the Fubini theorem, we have
(70)∫Δ‍JF*(g;λ)dλ=∫ℝn‍∫Δ‍k(λ)dλdν(v→)=0
for all rectifiable simple closed curve Δ lying in ℂ+. Thus, by the Morera theorem, JF*(g;λ) is analytic on ℂ+. Therefore, the analytic Wiener integral Iganwλ[F]=JF*(g;λ) exists. Finally, applying the dominated convergence theorem, we know that Iganfq[F]=limλ→-iqIganwλ[F] is given by the right side of (68).Next, we establish the existence of the admix-FFT for functionals in𝔗ℋ.Theorem 17.
LetF∈𝔗ℋ be given by (67). Then, for all p∈[1,2] and all g∈B*, the analytic Lpg⊙-admix-FFT, Tq,g(p)(F), exists for all nonzero real numbers q, belongs to 𝔗ℋ, and is given by the formula
(71)Tq,g(p)(F)(y)=∫ℝnexp{i∑j=1n‍(hj,y)~vj}dτgq(v→)
for s-a.e. y∈B, where τgq is the complex measure on ℝn given by
(72)τgq(E)=∫Eexp{-i2q∑j=1n‍|hj⊙g|2vj2}dτ(v→)
for E∈ℬ(ℝn).Proof.
Proceeding as in the proof of Theorem16, we see that, for all λ∈ℂ+, all g∈Bℋ*, and for s-a.e. y∈B,
(73)Tλ,g(F)(y)=∫B‍F(y+λ-1/2Agx)dν(x)=∫ℝnexp{i∑j=1n‍(hj,y)~vj-12λ∑j=1n‍|hj⊙g|2vj2}dτ(v→),
is an analytic function of λ on ℂ+, and that for any q∈ℝ-{0} and s-a.e. y∈B,
(74)Tq,g(1)(F)(y)=limλ→-iqTλ,g(F)(y)=∫ℝnexp{i∑j=1n‍(hj,y)~vj-i2q∑j=1n‍|hj⊙g|2vj2}dτ(v→).
Clearly, the set function τgq given by (72) is a complex measure on ℬ(ℝn), and so the right side of (74) can be rewritten as the right side of (71).
Next, we note that(75)|Tλ,g(F)(y)|≤∫ℝn‍d|τ|(v→)=∥τ∥
for all λ∈ℂ+ and
(76)|Tq,g(1)(F)(y)|≤∫ℝn‍d|τ|(v→)=∥τ∥.
Using these, we see that, for all p∈(1,2], all ρ>0, and all λ∈ℂ+,
(77)|Tλ,g(F)(ρy)-Tq,g(1)(F)(ρy)|p′≤(2∥τ∥)p′,
and so, by the dominated convergence theorem, we see that, for any nonzero real q,
(78)limλ→-iq∫B‍|Tλ,g(F)(ρy)-Tq,g(1)(F)(ρy)|p′dν(y)=0.
Hence, Tq,g(p)(F)(y) exists and is given by the right side of (74) for all desired values of p and q and all g∈Bℋ*. Thus, the theorem is proved.Theorem 18.
LetF∈𝔗ℋ be given by (67). Then, for all p∈[1,2], all g∈Bℋ*, and all nonzero real q,
(79)T-q,g(p)(Tq,g(p)(F))≈F.
As such, the admix-FFT, Tq,g(p), has the inverse transform {Tq,g(p)}-1=T-q,g(p).Theorem 19.
LetF and G be elements of 𝔗ℋ with corresponding finite Borel measures τ and μ in ℳ(H). Then, for all k∈Bℋ*, the CP, (F*G)q,k, exists for all nonzero real numbers q, belongs to 𝔗ℋ, and is given by the formula
(80)(F*G)q,k(y)=∬ℝnexp{i∑j=1n‍(hj,y)~rj}d(ϖkq∘φ-1)(h)
for s-a.e. y∈B, where φ:ℝn×ℝn→ℝn is a continuous function defined by
(81)φ(v→,w→)=(v→+w→)2,
and ϖkq is a complex measure on ℬ(ℝn) given by
(82)ϖkq(E)=∫E‍exp{-i4q∑j=1n‍|hj⊙k|2(vj-wj)2}dτ(v→)dμ(w→)
for E∈ℬ(ℝn).Proof.
Using (64), the Fubini theorem, (23), and (24), we have that, for all λ>0 and s-a.e. y∈B,
(83)(F*G)λ,k(y)=∫B‍F(y+Akx2)G(y-Akx2)dν(x)=∬ℝnexp{i2∑j=1n‍(hj,y)~(vj+wj)}×[∫B‍exp{i2λ∑j=1n‍(hj⊙k,x)~×(vj-wj)∑j=1n‍}dν(x)]dτ(v→)dμ(w→)=∬ℝnexp{i2∑j=1n‍(hj,y)~(vj+wj)-14λ∑j=1n‍|hj⊙k|2(vj-wj)2}dτ(v→)dμ(w→).
Using the same argument as in the proof of Theorem 17, we can show that the last expression in the previous equation is an analytic function of λ on ℂ+ and is a bounded continuous function of λ on ℂ~+ because τ and μ are finite Borel measures. Hence, (F*G)q,k exists and is given by
(84)(F*G)q,k(y)=∬ℝnexp{i2∑j=1n‍(hj,y)~(vj+wj)-i4q∑j=1n‍|hj⊙k|2(vj-wj)2}dτ(v→)dμ(w→)
for all q∈ℝ-{0} and s-a.e. y∈B.
Consider the set functionϖkq and the continuous function φ given by (82) and (81), respectively. Clearly, the set function ϖkq is a complex measure on ℬ(ℝn). Hence, ϖkq∘φ is an element of ℳ(ℝn), and so the right side of (84) can be rewritten as the right side of (80). Thus, the theorem is proved.Lemma 20.
Letℋ={h1,…,hn} be any orthogonal set in H. For every g∈Bℋ*, every v→=(v1,…,vn) and w→=(w1,…,wn) in ℝn, let Yg,v→ℋ,Zg,w→ℋ:B2→ℝ be given by
(85)Yg,v→ℋ(x1,x2)=∑j=1n‍(hj,Agx1+Agx2)~2vj,Zg,w→ℋ(x1,x2)=∑j=1n‍(hj,Agx1-Agx2)~2wj,
respectively. As a result, Yg,v→ℋ and Zg,w→ℋ are independent random variables.Proof.
Since the random variablesYg,v→ℋ and Zg,w→ℋ are Gaussian with mean zero, it suffices to show that
(86)∫B2‍Yg,v→ℋ(x1,x2)Zg,w→ℋ(x1,x2)d(ν×ν)(x1,x2)=0.
We know that{hj⊙g,…,hn⊙g} is an orthogonal set in H; thus, {(hj⊙g,x)~,…,(hn⊙g,x)~} is a set of independent Gaussian random variables with mean zero on B. However, using the Fubini theorem, we obtain
(87)∫B2‍Yg,v→ℋ(x1,x2)Zg,w→ℋ(x1,x2)d(ν×ν)(x1,x2)=12∑j=1n‍∑k=1n‍{vjwk∬B(hj⊙g,x1)~×(hk⊙g,x1)~dν(x1)dν(x2)+vjwk∫B‍(hj⊙g,x1)~dν(x1)×∫B‍(hk⊙g,x2)~dν(x2)+vjwk∫B‍(hj⊙g,x2)~dν(x2)×∫B‍(hk⊙g,x1)~dν(x1)-vjwk∬B(hj⊙g,x2)~×(h⊙g,x2)~dν(x1)dν(x2)∬bx}=12∑j=1n‍∑k=1n‍{vjwk∫B‍(hj⊙g,x1)~(hk⊙g,x1)~dν(x1)-vjwk∫B‍(hj⊙g,x2)~×(hk⊙g,x2)~dν(x2)∫B‍(hj⊙g,x2)~}=12∑j=1n‍{vjwj∫B‍[(hj⊙g,x1)~]2dν(x1)-vjwj∫B‍[(hj⊙g,x2)~]dν(x2)}+12∑j≠kn‍{vjwk∫B‍(hj⊙g,x1)~(hk⊙g,x1)~dν(x1)-vjwk∫B‍(hj⊙g,x2)~(h⊙g,x2)~dν(x2)}=0,
which concludes the proof of Lemma 20.Remark 21.
For eachv→∈ℝn, let
(88)ψ(v→;x)=exp{i∑j=1n‍(hj,x)~vj},x∈B.
As such, (67) is rewritten as
(89)F(x)=∫ℝn‍ψ(v→;x)dτ(v→),
and the functional ψ(v→;·) is an element of 𝔊ℋ(1) for each v→.Applying the same method as used in the proof of Theorem 3.5 in [4], we have the following theorem. For the proof of Theorem 22, we can apply Lemma 20 and Theorem 11 to the functional ψ given by (88).Theorem 22.
LetF, G, τ, and μ be as in Theorem 19, and let g be an element of Bℋ*. Then, for all p∈[1,2] and all nonzero real q,
(90)Tq,g(p)((F*G)q,g)(y)=Tq,g(p)(F)(y2)Tq,g(p)(G)(y2)
for s-a.e. y∈B.



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*Source: 671909-2013-07-16.xml*