APPMATH ISRN Applied Mathematics 2090-5572 International Scholarly Research Network 578174 10.5402/2012/578174 578174 Research Article A Rotation on Wiener Space with Applications Choi Jae Gil Chang Seung Jun Brünig M. Langthjem M. Yau H. T. Zhang C. Department of Mathematics Dankook University Cheonan 330-714 Republic of Korea dankook.ac.kr 2012 8 7 2012 2012 25 04 2012 26 06 2012 2012 Copyright © 2012 Jae Gil Choi and Seung Jun Chang. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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.

1. Introduction

Let C0[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 denote the class of all Wiener measurable subsets of C0[0,T], and let m denote Wiener measure. Then (C0[0,T],,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 , 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).

Let G(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)=0tcosθ(s)dw(s)-0tsinθ(s)dz(s),z(t)=0tsinθ(s)dw(s)+0tcosθ(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 Theorem 1.1, one can obtain the following corollary.

Corollary 1.2.

Let F be Wiener integrable on C0[0,T]. Then for any θ, F(wsinθ+zcosθ) is integrable on C02[0,T] and (1.4)C02[0,T]F(wsinθ+zcosθ)d(m×m)(w,z)=C0[0,T]F(x)dm(x).

The following more general case of Corollary 1.2 is due to Cameron and Storvick . But we state the theorem with some assumption for our research.

Theorem 1.3.

Let F be Wiener measurable on C0[0,T]. Assume that for any ρ>0, F(ρ·) is Winer integrable. Then for any a,b, 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, Theorem 1.3 is used to study relationships between analytic Fourier-Feynman transforms and convolution products of Feynman integrable functionals on Wiener space, see for instance . 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 .

A subset B of C0[0,T] is said to be scale-invariant measurable  provided ρB 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 FG.

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 vL2[0,T], the Paley-Wiener-Zygmund (PWZ) stochastic integral v,x is defined by the formula (2.1)v,x=limn0Tj=1n(v,ϕj)2ϕj(t)dx(t) for all xC0[0,T] for which the limit exists, where (·,·)2 denotes the L2-inner product.

It was shown in  that for each vL2[0,T], the limit defining the PWZ integral v,x exists for m-a.e. xC0[0,T] and that this limit is essentially independent of the choice of the complete orthonormal set {ϕn}. It was also shown in  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. xC0[0,T]. In fact, the integrals are equal for s-a.e. xC0[0,T] and that for all vL2[0,T], v,x is a Gaussian random variable with mean 0 and variance v22.

For any hL2[0,T] with h2>0, let 𝒵h be the Gaussian process (2.2)Zh(x,t)=0th(s)dx(s)=v,x introduced by Park and Skoug in  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 that 𝒵h 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,h2L2[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, 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 hL2[0,T] and xC0[0,T], we see that (2.6)Zh(x,t)=limn0tPnh(s)dx(s)=limnZPnh(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 functional F:C0[0,T] we consider is scale-invariant measurable and that (2.7)C0[0,T]|F(Zh(x,))|dm(x)<+ for all hL2[0,T].

We are now ready to state the main theorem of this paper.

Theorem 2.1.

Let F be a functional on C0[0,T]. Then for any h1,h2L2[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.

Let F be a functional on C0[0,T], and let ϕ be a function of bounded variation on [0,T]. Then for all a,b, (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 each t[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, (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.

Let F be a functional on C0[0,T]. Then for any h1,h2L2[0,T] and each 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 on C0[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.

Let F be bounded and continuous on C0[0,T]. Then for any h1,h2L2[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 that F 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 and hL2[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(limnZPnh(x,))=limnFn(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)=limnC02[0,T]Fn(Zh1(w,)+Zh2(z,))d(m×m)(w,z)=limnC0[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 Theorem <xref ref-type="statement" rid="thm2.1">2.1</xref>.

Let F 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.

Let F be a functional on C0[0,T]. Then for all hL2[0,T] and all a,b, (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 choose h1=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 Lemmas 3.1, 3.2, and 3.3 and Theorem 2.1 above, we can obtain the following theorems.

Theorem 3.5.

Let F 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=1j=1ν(hj,ϕn)22ϕn(t) for some complete orthonormal set {ϕn} in L2[0,T].

Theorem 3.6.

Let F 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.

For any θ, choosing h1(t)sinθ and h2(t)cosθ in (2.8) yields (1.4).

For any a,b, choosing h1(t)a and h2(t)b in (2.8) or choosing h(t)1 in (3.9) yields (1.5).

For any function of bounded variation θ(·), 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 Generalized <bold />Convolution 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, for λ>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, let + and ~+ denote the complex numbers with positive real part and the nonzero complex numbers with nonnegative real part, respectively.

Let F 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 + 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+ 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+. 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)=limλ-iqC0[0,T]anwλF(Zh(x,))dm(x), where λ approaches -iq through values in +.

Note that if h1 on [0,T], then these definitions agree with the previous definitions of the analytic Wiener integral and the analytic Feynman integral [3, 4, 8, 1214].

Next (see [5, 6, 15]) we state the definition of the generalized Fourier-Feynman transform (GFFT).

Definition 4.1.

For λ+ and yC0[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+), (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)limλ-iqC0[0,T]|Tλ,h(F)(ρy)-Tq,h(p)(F)(ρy)|pdm(y)=0, where 1/p+1/p=1. We define the L1 analytic GFFT, Tq,h(1)(F) of F, by the formula (λ+)(4.8)Tq,h(1)(F)(y)=limλ-iqTλ,h(F)(y) if it exists.

We note that for p[1,2], Tq,h(p)(F) is defined only s-a.e. We also note that if Tq,h(p)(F) exists and if FG, then Tq,h(p)(G) exists and Tq,h(p)(G)Tq,h(p)(F). One can see that for each hL2[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.

Let F and G be scale-invariant measurable functionals on C0[0,T]. For λ~+ and h1,h2L2[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),λC+,C0[0,T]anfqF(y+Zh1(x,)2)G(y-Zh2(x,)2)dm(x),λ=-iq,qR,q0. 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 . 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] 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.

Yg1,g2 and Yg3,g4 are independent processes.

g1g3=g2g4.

Proof.

Since the processes Yg1,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.

Let F and G be functionals on C0[0,T]. Let {h1,h2,h3} be a subset of L2[0,T] such that h32=h1h20 almost everywhere on [0,T], and let (4.14)k1(t)=2-1/2n=1(h1,ϕn)22+(h3,ϕn)22ϕn(t),k2(t)=2-1/2n=1(h2,ϕn)22+(h3,ϕn)22ϕn(t). Furthermore, assume that for all λ+, 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. yC0[0,T].

Proof.

We note that for all λ>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+ by analytic continuation.

In next theorem, we show that the GFFT of the GCP is the product of GFFTs.

Theorem 4.6.

Let F, G, {h1,h2,h3}, k1, and k2 be as in Lemma 4.5. Furthermore, assume that for p[1,2], λ+ and q-{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],limλ-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. yC0[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 Theorem 4.6 above are indeed satisfied by many of the functionals in the following large classes of functionals. These classes of functionals include;

the Banach algebra 𝒮 defined by Cameron and Storvick in : also see [3, 5, 14, 15],

various spaces of functionals of the form (4.20)F(x)=exp{0Tf(t,x(t))dt} for appropriate f:[0,T]× as discussed in [4, 12, 13]; and

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]2×2 as discussed in .

Next five corollaries include the results of  by Huffman et al. The notations used in  are slightly different than ours.

Corollary 4.8.

Refer to Theorem 2.1 in .

Proof.

In our Lemma 4.5, simply choose h1=h2=h3=k1=k2hL2[0,T].

Corollary 4.9.

Refer to Theorem 1 in .

Proof.

In our Theorem 4.6, simply choose h1=h2=h3=k1=k2hL2[0,T].

Corollary 4.10.

Refer to Theorem 3.3 in .

Proof.

In our Theorem 4.6, simply choose h1=h2=h3=k1=k2hL[0,T].

Corollary 4.11.

Refer to Theorem 3.3 in .

Proof.

In our Theorem 4.6, simply choose h1=h2=h3=k1=k21.

Corollary 4.12.

Refer to Lemma 4.1 and Theorems 4.1 and 4.2 in .

Proof.

In our Lemma 4.5 and Theorem 4.6, simply choose h1=h2=h3=k1=k21.

Acknowledgment

The present research was conducted by the research fund of Dankook University in 2010.

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