# A Conditional Fourier-Feynman Transform and Conditional Convolution Product with Change of Scales on a Function Space II

Authors: Dong Hyun Cho
Journal: Journal of Probability and Statistics
 (2017)

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## Abstract

Using a simple formula for conditional expectations over continuous paths, we will evaluate conditional expectations which are types of analytic conditional Fourier-Feynman transforms and conditional convolution products of generalized cylinder functions and the functions in a Banach algebra which is the space of generalized Fourier transforms of the measures on the Borel class ofL2[0,T]. We will then investigate their relationships. Particularly, we prove that the conditional transform of the conditional convolution product can be expressed by the product of the conditional transforms of each function. Finally we will establish change of scale formulas for the conditional transforms and the conditional convolution products. In these evaluation formulas and change of scale formulas, we use multivariate normal distributions so that the conditioning function does not contain present positions of the paths.


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## Body

## 1. Introduction

LetC[0,T] denote an analogue of Wiener space which is the space of real-valued continuous functions on the interval [0,T] [1]. The conditional expectations which are types of analytic conditional Fourier-Feynman transforms and conditional convolution products on C[0,T] are introduced by the author [2] using a conditioning function which contains present positions of the paths. According to the author’s paper, he investigated the effects that drift has on the conditional Fourier-Feynman transform, the conditional convolution product, and various relationships that occur among them. In particular, he derived several change of scale formulas for the conditional transforms and the conditional convolution products, which simplify evaluating the conditional expectations, because the probability measure used on C[0,T] cannot be scale-invariant [3, 4].Leta be absolutely continuous on [0,T] and let h be of bounded variation with h≠0 a.e. on [0,T]. Define a stochastic process Z:C[0,T]×[0,T]→R by (1)Zx,t=hχ0,t,x+x0+at,for x∈C[0,T] and t∈[0,T], where (·,·) denotes the Paley-Wiener-Zygmund stochastic integral [1]. For a partition 0=t0<t1<⋯<tn<tn+1=T of [0,T], define random vectors Zn:C[0,T]→Rn+1 and Zn+1:C[0,T]→Rn+2 by (2)Znx=Zx,t0,Zx,t1,…,Zx,tn,Zn+1x=Zx,t0,Zx,t1,…,Zx,tn,Zx,tn+1.With the conditioning function Zn+1 which contains the present position Z(x,T) of the path Z(x,·), the author [5] evaluated the conditional Fourier-Feynman transforms and the conditional convolution products of the functions given by(3)∫L20,Texp⁡iv,Zx,·dσv,fv1,Zx,·,…,vr,Zx,·,where σ is a complex Borel measure of bounded variation on L2[0,T], f∈Lp(Rr) with 1≤p≤∞, and {v1,…,vr} is an orthonormal subset of L2[0,T]. He then investigated relationships between the conditional Fourier-Feynman transforms and the conditional convolution products of the functions given by (3).In this paper, using a simple formula for conditional expectations over continuous paths [6], we evaluate conditional expectations of generalized cylinder functions and the functions in a Banach algebra which plays significant roles in Feynman integration theories and quantum mechanics. We then investigate their relationships. Particularly, we prove that the conditional transform of the convolution product can be expressed by the product of the conditional transforms of each function. Finally we establish change of scale formulas for the conditional expectations. In these evaluation formulas and change of scale formulas we use multivariate normal distributions so that the conditioning function does not contain the present positions of the paths. In fact, with the conditioning function Zn which does not contain the present position Z(x,T) of the path Z(x,·), we evaluate conditional expectations, that is, the conditional Fourier-Feynman transforms and the conditional convolution products of the functions given by (3). We then show that the Lp-analytic conditional Fourier-Feynman transform Tq(p)FZ∗GZq∣Zn·,ξ→n∣Zn of the conditional convolution product for the functions FZ and GZ which are described in (3) can be expressed by the formula (4)TqpFZ∗GZq∣Zn·,ξ→n∣Zny,η→n=TqpFZ∣Zn12y+2-1a,12η→n+ξ→n-2-1a→nTqpGZ∣Zn12y-a,12η→n-ξ→n+a→n,for a nonzero real q, almost surely y∈C[0,T], and PZn almost surely ξ→n,η→n∈Rn+1, where a→n=(a(t0),a(t1),…,a(tn)) and PZn is the probability distribution of Zn on the Borel class of Rn+1. Compared with the results in [5, 7], the conditioning function Zn in this paper does not contain the present position Z(x,T) of Z(x,·) and the effects of drift a depend on the polygonal function of a so that we can generalize the theorems in [5, 7] and the results of this paper do not depend on a particular choice of the initial distribution of the paths.


## 2. Preliminaries on an Analogue of Wiener Space

We begin this section with introducing a concrete form of an analogue of Wiener space which is our underlying space.For a positive realT let C[0,T] denote the space of real-valued continuous functions on the time interval [0,T] with the supremum norm. For t→=(t0,t1,…,tn) with 0=t0<t1<⋯<tn≤T let Jt→:C[0,T]→Rn+1 be the function given by Jt→(x)=(x(t0),x(t1),…,x(tn)). For Bj  (j=0,1,…,n) in the Borel class B(R) of R, the subset Jt→-1(∏j=0nBj) of C[0,T] is called an interval and let I be the set of all such intervals. For a probability measure φ on B(R), let (5)mφJt→-1∏j=0nBj=∏j=1n12πtj-tj-11/2∫B0∫∏j=1nBjWnt→,u→,u0du→dφu0,where W(t→,u→,u0)=exp⁡{-(1/2)∑j=1n((uj-uj-1)2/(tj-tj-1))} for u→=(u1,…,un). B(C[0,T]), the Borel σ-algebra of C[0,T], coincides with the smallest σ-algebra generated by I and there exists a unique probability measure wφ on C[0,T] such that wφ(I)=mφ(I) for all I∈I. This measure wφ is called an analogue of Wiener measure associated with the probability measure φ [1].LetC and C+ denote the sets of complex numbers and complex numbers with positive real parts, respectively. Let F:C[0,T]→C be integrable and let Xτ be a random vector on C[0,T] assuming that the value space of Xτ is a normed space with the Borel σ-algebra. Then we have the conditional expectation EF∣Xτ of F given Xτ from a well-known probability theory [8]. Furthermore, there exists a PXτ integrable complex-valued function ϕ on the value space of Xτ such that (6)EF∣Xτx=ϕ∘Xτxforwφa.e.x∈C0,T,where PXτ is the probability distribution of Xτ. The function ϕ is called the conditional wφ-integral of F given Xτ and it is also denoted by EF∣Xτ.For an extended real numberp with 1<p≤∞, suppose that p and p′ are related by 1/p+1/p′=1 (possibly p′=1 if p=∞). Let q∈R-{0}. For λ∈C+ let Fλ be a measurable function on C[0,T] such that(7)limλ→-iq⁡EFλ-Fp′=0.Then we write (8)limλ→-iqwp′Fλ=F.Let0=t0<t1<⋯<tn<tn+1=T be a partition of 0,T, where n is a fixed nonnegative integer. Let h be of bounded variation with h≠0 a.e. on [0,T]. Let a be absolutely continuous on [0,T] and define stochastic processes X,Z:C0,T×0,T→R by (9)Xx,t=hχ0,t,x,Zx,t=Xx,t+x0+at,for x∈C[0,T] and t∈[0,T], where χ denotes an indicator function. Define a random vector Zn:C[0,T]→Rn+1 by (10)Znx=Zx,t0,Zx,t1,…,Zx,tn,for x∈C0,T. For t∈0,T, let bt=∫0ths2ds and for any function f on [0,T], define a polygonal function Pb,n+1(f) of f by(11)Pb,n+1ft=∑j=1n+1btj-btbtj-btj-1ftj-1+bt-btj-1btj-btj-1ftjχtj-1,tjt+f0χ0t,for t∈[0,T]. For ξ→n+1=(ξ0,ξ1,…,ξn,ξn+1)∈Rn+2, define a polygonal function Pb,n+1(ξ→n+1) of ξ→n+1 by (11), where f(tj) is replaced by ξj. If ξ→n=(ξ0,ξ1,…,ξn)∈Rn+1, Pb,n(ξ→n) is interpreted as Pb,n+1(ξ→n+1) with ξn+1=ξn. For x∈C[0,T] and t∈[0,T] let(12)At=at-Pb,n+1at,Xb,n+1x,t=Xx,t-Pb,n+1Xx,·t,Zb,n+1x,t=Zx,t-Pb,n+1Zx,·t.For a→, u→∈Rr, λ∈C, and any nonsingular positive r×r matrix Ar on R, let(13)Ψrλ,a→,Ar,u→=λr2πrAr1/2exp⁡-λ2Ar-1u→-a→,u→-a→R,where ·,·R denotes the dot product on Rr. Let Ir be the r×r identity matrix on R. For a measurable function F:C[0,T]→C, let (14)FZx,y=FZx,·+yforx,y∈C0,T,let FZλ(x,y)=FZ(λ-1/2x,y), and let Znλ(x)=Zn(λ-1/2x) for λ>0. Suppose that E[FZλ(·,y)] exists, where the expectation is taken over the first variable. By Theorem 2 in [2] and Theorem 7 in [6](15)EFZλ·,y∣Znλξ→n=∫RΨ1λ,aT-atn,bT-btn,ξn+1-ξnEFy+λ-1/2Xb,n+1x,·+A+Pb,n+1ξ→n+1dξn+1,for PZnλ a.e. ξ→n=(ξ0,ξ1,…,ξn)∈Rn+1, where ξ→n+1=(ξ0,ξ1,…,ξn,ξn+1), PZnλ is the probability distribution of Znλ on (Rn+1,B(Rn+1)) and the expectation is taken over the variable x. Let IFZλ(y,ξ→n) be the right-hand side of (15). If, for wφ a.e. y∈C[0,T] and PZn a.e. ξ→n∈Rn+1, IFZλ(y,ξ→n) has an analytic extension Jλ∗(FZ)(y,ξ→n) on C+, then it is called a generalized analytic conditional Fourier-Wiener transform of F given Zn with the parameter λ and is denoted by (16)TλFZ∣Zny,ξ→n=Jλ∗FZy,ξ→n.Moreover if TλFZ∣Zn(y,ξ→n) has a limit as λ approaches -iq through C+, then it is called a generalized L1-analytic conditional Fourier-Feynman transform of F given Zn with the parameter q and is denoted by (17)Tq1FZ∣Zny,ξ→n=limλ→-iqTλFZ∣Zny,ξ→n.For 1<p≤∞, define a generalized Lp-analytic conditional Fourier-Feynman transform Tq(p)FZ∣Zn of F given Zn by the formula (18)TqpFZ∣Zn·,ξ→n=limλ→-iqwp′TλFZ∣Zn·,ξ→nif  exists.Let G be defined on C[0,T]. For λ>0 and y∈C[0,T], redefine FZλ(·/2,y/2)GZλ(-·/2,y/2)=FZ/2λ(·,y/2)G-Z/2λ(·,y/2) and suppose that E[FZλ(·/2,y/2)GZλ(-·/2,y/2)] exists. By Theorem 2 in [2] and Theorem 7 in [6](19)EFZλ·2,y2GZλ-·2,y2∣Znλξ→n=∫RΨ1λ,aT-atn,bT-btn,ξn+1-ξnEF12y+λ-1/2Xb,n+1x,·+A+Pb,n+1ξ→n+1G12y-λ-1/2Xb,n+1x,·-A-Pb,n+1ξ→n+1dξn+1.Let KFZ,GZλ(y,ξ→n) be the right-hand side of (19). If KFZ,GZλ(y,ξ→n) has an analytic extension Jλ∗(FZ,GZ)(y,ξ→n) on C+, then it is called a generalized conditional convolution product of F and G given Zn with the parameter λ and denoted by (20)FZ∗GZλ∣Zny,ξ→n=Jλ∗FZ,GZy,ξ→n.Moreover if [FZ∗GZ)λ∣Zn(y,ξ→n) has a limit as λ approaches -iq through C+, then it is called a generalized conditional convolution product of F and G given Zn with the parameter q and denoted by (21)FZ∗GZq∣Zny,ξ→n=limλ→-iq⁡FZ∗GZλ∣Zny,ξ→n.Forj=1,…,n+1, let (22)αj=hbtj-btj-1χtj-1,tj,let V be the subspace of L2[0,T] generated by {α1,…,αn,αn+1}, and let V⊥ be the orthogonal complement of V. Let P:L2[0,T]→V be the orthogonal projection given by (23)Pv=∑j=1n+1v,αjαj,and let P⊥:L20,T→V⊥ be the orthogonal projection. For u→∈Rr, λ∈C, v→=(v1,…,vr)∈L2n[0,T] and any nonsingular positive r×r matrix Ar on R, let (24)v→αn+1,αn+1=v1αn+1,αn+1,…,vrαn+1,αn+1,ΛAr=1+bT-btnAr-1v→αn+1,αn+1R2,Φrλ,v→,Ar,u→=1ΛAr1/2exp⁡λ2ΛArbT-btnu→,Ar-1v→αn+1,αn+1R2.The following lemmas are useful to prove the results in the next sections [5].Lemma 1.
Letv∈L2[0,T]. Then for wφ a.e. x∈C[0,T](25)v,Xx,·=Mhv,x,v,Pb,n+1Xx,·=PMhv,x,where Mh:L2[0,T]→L2[0,T] is the multiplication operator defined by (26)Mhu=huforu∈L20,T.Lemma 2.
Letv∈L2[0,T] and ξ→n+1=(ξ0,ξ1,…,ξn,ξn+1)∈Rn+2. Then (27)v,Pb,n+1ξ→n+1=v,Pb,nξ→n+vαn+1,αn+1ξn+1-ξn,where ξ→n=(ξ0,ξ1,…,ξn).Lemma 3.
Let{v1,…,vr} be a subset of L2[0,T] such that {Mhv1,…,Mhvr} is an independent set. Then the random vector v→,Zx,·≡v1,Zx,·,…,vr,Zx,· has the multivariate normal distribution with mean vector (v→,a)≡((v1,a),…,(vr,a)) and covariance matrix ΣMh=[Mhvi,Mhvj]r×r. Moreover, for any Borel measurable function f:Rr+1→C one has (28)∫C0,Tfx0,v→,Zx,·dwφx=∗∫R∫Rrfu0,u→Ψr1,v→,a,ΣMh,u→du→dφu0,where =∗ means that if either side exists, then both sides exist and they are equal.Remark 4.
Because the Borel sets ofC[0,T] are always scale-invariant measurable and we use the Borel class of C[0,T] on which wφ is defined [1], the scale-invariant measurability is not essential in (7).


## 3. Conditional Fourier-Wiener and Fourier-Feynman Transforms

Let1≤p≤∞, let r be any fixed positive integer, and let {v1,v2,…,vr} be an orthonormal subset of L2[0,T] such that both {Mhv1,…,Mhvr} and {P⊥Mhv1,…,P⊥Mhvr} are independent sets. Let A(p) be the space of cylinder functions F having the form(29)Fx=fv→,x,for wφ a.e. x∈C[0,T], where f∈Lp(Rr) and (v→,x)=((v1,x),…,(vr,x)). Without loss of generality we can take f to be Borel measurable.Theorem 5.
Let1≤p≤∞ and let F(∈A(p)) be given by (29). Then for λ∈C+(30)TλFZ∣Zny,ξ→n=∫RrfΣP⊥1/2u→+v→,y+a+Pb,nξ→n-a→nΨrλ,0→,Ir,u→Φrλ,v→,ΣP⊥1/2,u→du→=frξ→n-a→n,a,·∗Ψrλ,0→,Ir,·Φrλ,v→,ΣP⊥1/2,·ΣP⊥-1/2v→,y,for wφ a.e. y∈C[0,T] and PZn a.e. ξ→n∈Rn+1, where ΣP⊥=P⊥Mhvi,P⊥Mhvj, a→n=at0,at1,…,atn, and(31)frξ→n,a,u→=fΣP⊥1/2u→+v→,a+Pb,nξ→n.Moreover TλFZ∣Zn(·,ξ→n)∈A(p).Proof.
Forξ→n=(ξ0,ξ1,…,ξn)∈Rn+1 let ξ→n+1=(ξ0,ξ1,…,ξn,ξn+1), where ξn+1∈R. For λ>0 and wφ a.e. x∈C[0,T] we have by Lemmas 1 and 2(32)IFZλy,ξ→n=∫R∫RrΨ1λ,aT-atn,bT-btn,ξn+1-ξnfΣP⊥1/2u→+v→,y+v→,A+Pb,n+1ξ→n+1Ψrλ,0→,Ir,u→du→dξn+1=∫Rr∫RΨ1λ,0,bT-btn,ξn+1fΣP⊥1/2u→+v→,y+v→,A+v→,Pb,nξ→n+aT-atnv→αn+1,αn+1+v→αn+1,αn+1ξn+1Ψrλ,0→,Ir,u→dξn+1du→,where A and Ψr are given by (12) and (13), respectively. Using the same method as used in the proof of Theorem 3.2 in [7] (33)IFZλy,ξ→n=∫RrfΣP⊥1/2u→+v→,y+v→,A+v→,Pb,nξ→n+aT-atnv→αn+1,αn+1Ψrλ,0→,Ir,u→Φrλ,v→,ΣP⊥1/2,u→du→=∫RrfΣP⊥1/2u→+v→,y+a+Pb,nξ→n-a→nΨrλ,0→,Ir,u→Φrλ,v→,ΣP⊥1/2,u→du→.We note that if 1≤p<∞, then by the change of variable theorem (34)frξ→n-a→n,a,·pp=ΣP⊥-1/2fpp<∞,and for λ∈C+ we have by Schwarz’s inequality(35)Ψrλ,0→,Ir,u→Φrλ,v→,ΣP⊥1/2,u→≤λ2πr/2exp⁡-Reλ2ΛΣP⊥1/2u→R2.Now, by Morera’s theorem with aids of Hölder’s inequality and the dominated convergence theorem, we have (30) for λ∈C+. Since fr(ξ→n,a,·)∈Lp(Rr) and Ψr(λ,0→,Ir,·)Φr(λ,v→,ΣP⊥1/2,·)∈L1(Rr) by (35), the final result follows by the change of variable theorem and Young’s inequality [9].Corollary 6.
Let1≤p≤∞ and let F(∈A(p)) be given by (29). (1)
Ifn=0, then one has for λ∈C+, y∈C[0,T] and ξ0∈R(36)TλFZ∣Z0y,ξ0=∫RrfΣP⊥1/2u→+v→,y+aΨrλ,0→,Ir,u→Φr0λ,v→,ΣP⊥1/2,u→du→,where forΛ0(ΣP⊥1/2)=1+b(T)ΣP⊥-1/2v→α1,α1R2(37)Φr0λ,v→,ΣP⊥1/2,u→=1Λ0ΣP⊥1/21/2exp⁡λbT2Λ0ΣP⊥1/2u→,ΣP⊥-1/2v→α1,α1R2.(2)
Ifr=1, then one has(38)TλFZ∣Zny,ξ→n=∫RfP⊥Mhv1u+v1,y+a+Pb,nξ→n-a→nΔ1λ,udu=f1ξ→n-a→n,a,·∗Δ1λ,·P⊥Mhv1-1v1,y,where foru∈R(39)f1ξ→n,a,u=fP⊥Mhv1u+v1,a+Pb,nξ→n,and forΛ1P⊥Mh=1+bT-btn(P⊥Mhv1-1v1αn+1,αn+1)2(40)Δ1λ,u=λ2πΛ1P⊥Mh1/2exp⁡-λu22Λ1P⊥Mh.(3)
Ifn=0 and r=1, then one has(41)TλFZ∣Z0y,ξ0=∫RfP⊥Mhv1u+v1,y+aΔ10λ,udu=f10a,·∗Δ10λ,·P⊥Mhv1-1v1,y,where foru∈R(42)f10a,u=fP⊥Mhv1u+v1,a,and forΛ10(P⊥Mh)=1+b(T)(P⊥Mhv1-1v1α1,α1)2(43)Δ10λ,u=λ2πΛ10P⊥Mh1/2exp⁡-λu22Λ10P⊥Mh.By the dominated convergence theorem and Theorem5 we have the following theorem.Theorem 7.
LetF(∈A(1)) be given by (29). Then for a nonzero real q, wφ a.e. y∈C[0,T], and PZn a.e. ξ→n∈Rn+1, Tq(1)FZ∣Zn(y,ξ→n) exists and it is given by the right-hand side of (30), where λ is replaced by -iq. Furthermore Tq(1)FZ∣Zn(·,ξ→n)∈A(∞).Using Lemma1.2 in [10] we can prove the following theorem.Theorem 8.
LetF(∈A(p)) be given by (29) with 1<p≤2, let q be a nonzero real number, and let 1/p+1/p′=1. (1)
Suppose thatvjαn+1,αn+1=0 for j=1,…,r. Then for wφ a.e. y∈C[0,T] and PZn a.e. ξ→n∈Rn+1, Tq(p)FZ∣Zn(y,ξ→n) is given by (44)TqpFZ∣Zny,ξ→n=∫RrfΣP⊥1/2u→+v→,y+a+Pb,nξ→n-a→nΨrλ,0→,Ir,u→du→=frξ→n-a→n,a,·∗Ψr-iq,0→,Ir,·ΣP⊥-1/2v→,y,wherefr is given by (31). Furthermore Tq(p)FZ∣Zn(·,ξ→n)∈A(p′).(2)
Ifr=1, then for wφ a.e. y∈C[0,T] and PZn a.e. ξ→n∈Rn+1, Tq(p)FZ∣Zn(y, ξ→n) exists and it is given by the right-hand sides of (38), where λ is replaced by -iq. Furthermore Tq(p)FZ∣Zn(·,ξ→n)∈A(p′).(3)
Ifn=0 and r=1, then for ξ0∈R, Tq(p)FZ∣Z0(y,ξ0) is given by the right-hand sides of (41), where λ is replaced by -iq.Using the same method as used in the proof of Theorem4.2 in [7], we can prove the following theorem.Theorem 9.
LetF(∈A(p)) be given by (29) with 1≤p≤∞, and let q∈R-{0}. (1)
Suppose thatvjαn+1,αn+1=0 for j=1,…,r. For wφ a.e. y∈C[0,T] and PZn a.e. ξ→n,η→n∈Rn+1, let Fr(y,ξ→n,η→n)=f((v→,y)+(v→,2a+Pb,n(ξ→n+η→n-2a→n))). Then(45)∫C0,TTλ¯TλFZ∣Zn·,ξ→n∣Zny,η→n-Fry,ξ→n,η→npdwφy⟶0,for1≤p<∞, and for 1≤p≤∞(46)Tλ¯TλFZ∣Zn·,ξ→n∣Zny,η→n⟶Fry,ξ→n,η→n,asλ approaches -iq through C+.(2)
Suppose thatr=1. Let F1(y,ξ→n,η→n)=f((v1,y)+(v1,2a+Pb,n(ξ→n+η→n-2a→n))). Then one has (45) if 1≤p<∞ and has (46) if 1≤p≤∞, where Fr is replaced by F1.(3)
Suppose thatn=0 and r=1. Let F10(y,ξ0,η0)=f((v1,y)+(v1,2a)). Then one has (45) if 1≤p<∞ and has (46) if 1≤p≤∞, where Fr is replaced by F10.


## 4. Relationships between Conditional Fourier-Feynman Transforms and Convolution Products

In this section we evaluate the conditional convolution products and investigate their relationships.Now, we have the following two theorems by (35) and Theorems 3.2 and 3.3 in [7].Theorem 10.
LetF(∈A(p1)), G(∈A(p2)), and f and g be related by (29), respectively, where 1≤p1,p2≤∞. Furthermore let 1/p1+1/p1′=1 and 1/p2+1/p2′=1. Then for λ∈C+, wφ a.e. y∈C[0,T], and PZn a.e. ξ→n∈Rn+1, FZ∗GZλ∣Zn(y,ξ→n) exists and is given by (47)FZ∗GZλ∣Zny,ξ→n=∫Rrf12v→,y+ΣP⊥1/2u→+v→,a+Pb,nξ→n-a→ng12v→,y-ΣP⊥1/2u→-v→,a+Pb,nξ→n-a→nΨrλ,0→,Ir,u→Φrλ,v→,ΣP⊥1/2,u→du→.Moreover one has FZ∗GZλ∣Zn(·,ξ→n)∈A(1) if either p2≤p1′ or p1≤p2′, [FZ∗GZ)λ∣Zn(·,ξ→n)∈A(p2) if p2≥p1′, and [FZ∗GZ)λ∣Zn(·,ξ→n)∈A(p1) if p1≥p2′.Theorem 11.
Letq be a nonzero real number. Then for λ∈C+ or λ=q and PZn a.e. ξ→n∈Rn+1, one has the following:(1)
ifF,G∈A(1), then [FZ∗GZ)λ∣Zn(·,ξ→n)∈A(1),(2)
ifF,G∈A(2), then [FZ∗GZ)λ∣Zn(·,ξ→n)∈A(∞),(3)
ifF∈A(1) and G∈A(2), then [FZ∗GZ)λ∣Zn(·,ξ→n)∈A(2),(4)
ifF∈A(1) and G∈A(1)∩A(2), then [FZ∗GZ)λ∣Zn(·,ξ→n)∈A(1)∩A(2),(5)
ifF∈A(1) and G∈A(∞), then [FZ∗GZ)λ∣Zn(·,ξ→n)∈A(∞).Theorem 12.
LetF,G∈⋃1≤p≤∞A(p) and let a→n=(a(t0),a(t1),…,a(tn)). Then for λ∈C+, wφ a.e. y∈C[0,T], and PZn a.e. ξ→n,η→n∈Rn+1, one has (48)TλFZ∗GZλ∣Zn·,ξ→n∣Zny,η→n=TλFZ∣Zn12y+2-1a,12η→n+ξ→n-2-1a→nTλGZ∣Zn12y-a,12η→n-ξ→n+a→n.Proof.
We note thatTλFZ∗GZλ∣Zn·,ξ→n∣Zn(y,η→n) is well-defined by Theorems 5 and 10. By those theorems as stated above we have for λ∈C+, wφ a.e. y∈C[0,T], and PZn a.e. ξ→n,η→n∈Rn+1(49)TλFZ∗GZλ∣Zn·,ξ→n∣Zny,η→n=∫Rr∫Rrf12v→,y+ΣP⊥1/2u→+z→+v→,2a+Pb,nξ→n+η→n-2a→ng12v→,y+ΣP⊥1/2z→-u→+v→,Pb,nη→n-ξ→nΨrλ,0→,Ir,u→Φrλ,v→,ΣP⊥1/2,u→Ψrλ,0→,Ir,z→Φrλ,v→,ΣP⊥1/2,z→du→dz→.Let α→=(1/2)(z→+u→) and β→=(1/2)(z→-u→). Then we have by the change of variable theorem and Lemma 2(50)TλFZ∗GZλ∣Zn·,ξ→n∣Zny,η→n=∫Rr∫RrfΣP⊥1/2α→+12v→,y+2a+Pb,nξ→n+η→n-2a→ngΣP⊥1/2β→+12v→,y+Pb,nη→n-ξ→nΨrλ,0→,Ir,12α→+β→Φrλ,v→,ΣP⊥1/2,12α→+β→Ψrλ,0→,Ir,12α→-β→Φrλ,v→,ΣP⊥1/2,12α→-β→dα→dβ→=λr2πrΛΣP⊥1/2∫Rr∫RrfΣP⊥1/2α→+v→,12y+2a+Pb,n12ξ→n+η→n-2a→ngΣP⊥1/2β→+v→,12y+Pb,n12ξ→n-η→nexp⁡-λ4α→+β→R2+α→-β→R2+λ4ΛΣP⊥1/2bT-btnα→+β→,ΣP⊥-1/2v→αn+1,αn+1R2+α→-β→,ΣP⊥-1/2v→αn+1,αn+1R2dα→dβ→=TλFZ∣Zn12y+2-1a,12η→n+ξ→n-2-1a→nTλGZ∣Zn12y-a,12η→n-ξ→n+a→n,which completes the proof.We now have the following theorem from Theorems5, 10, 11, and 12.Theorem 13.
Letq be a nonzero real number. Then one has the following: (1)
ifF,G∈A(1), then one has for wφ a.e. y∈C[0,T] and PZn a.e. ξ→n,η→n∈Rn+1(51)Tq1FZ∗GZq∣Zn·,ξ→n∣Zny,η→n=Tq1FZ∣Zn12y+2-1a,12η→n+ξ→n-2-1a→nTq1GZ∣Zn12y-a,12η→n-ξ→n+a→n,(2)
ifF∈A(1) and G∈A(2), then (52)Tq2FZ∗GZq∣Zn·,ξ→n∣Zny,η→n=Tq1FZ∣Zn12y+2-1a,12η→n+ξ→n-2-1a→nTq2GZ∣Zn12y-a,12η→n-ξ→n+a→n.


## 5. Evaluation Formulas for the Functions in a Banach Algebra

Letψ be the function defined by(53)ψu→=∫Rrexp⁡iu→,z→Rdρz→foru→∈Rr,where ρ is a complex-valued Borel measure of bounded variation over Rr. For wφ a.e. x∈C[0,T], let Π be given by(54)Πx=ψv→,x.By Theorem4.1 in [11], we can prove the following theorem.Theorem 14.
Let1≤p≤∞ and let Π be given by (54). Then for λ∈C+, wφ a.e. y∈C[0,T], and PZn a.e. ξ→n∈Rn+1, TλΠZ∣Zn(y,ξ→n) is given by(55)TλΠZ∣Zny,ξ→n=∫Rrexp⁡-12λbT-btnz→,v→αn+1,αn+1R2+ΣP⊥z→,z→R+iz→,v→,y+a+Pb,nξ→n-a→nRdρz→.For a nonzero real q, Tq(p)ΠZ∣Zn(y,ξ→n) exists and it is given by the right-hand side of (55), where λ is replaced by -iq. Furthermore TλΠZ∣Zn(·,ξ→n)∈A(∞) and Tq(p)ΠZ∣Zn(·,ξ→n)∈A(∞).Theorem 15.
Let1≤p≤∞ and let Π be given by (54). Then one has that for PZn a.e. ξ→n,η→n∈Rn+1(56)Tλ-TλΠZ∣Zn·,ξ→n∣Zn·,η→n-Π·+2a+Pb,nξ→n+η→n-2a→npconverges to 0 and for wφ a.e. y∈C[0,T](57)Tλ¯TλΠZ∣Zn·,ξ→n∣Zny,η→n⟶Πy+2a+Pb,nξ→n+η→n-2a→n,as λ approaches -iq through C+.Theorem 16.
Letψ1 and ψ2 and ρ1 and ρ2 be related by (53), respectively. Let Π1(x)=ψ1((v→,x)) and Π2(x)=ψ2((v→,x)) for wφ a.e. x∈C[0,T]. Then for λ∈C+, wφ a.e. y∈C[0,T], and PZn a.e. ξ→n∈Rn+1, Π1Z∗Π2Zλ∣Zn(y,ξ→n) exists and it is given by (58)
Π
1
Z
∗
Π
2
Z
λ
∣
Z
n
y
,
ξ
→
n
=
∫
R
r
∫
R
r
e
x
p
-
1
4
λ
Σ
P
⊥
1
/
2
u
→
-
w
→
R
2
b
T
-
b
t
n
·
v
→
α
n
+
1
,
α
n
+
1
,
u
→
-
w
→
R
2
+
Σ
P
⊥
1
/
2
u
→
-
w
→
R
2
+
i
2
v
→
,
a
+
P
b
,
n
ξ
→
n
-
a
→
n
v
→
,
y
,
u
→
+
w
→
R
+
v
→
,
a
+
P
b
,
n
ξ
→
n
-
a
→
n
,
u
→
-
w
→
R
Σ
P
⊥
1
/
2
u
→
-
w
→
R
2
d
ρ
1
u
→
d
ρ
2
w
→
.For a nonzero real q, Π1Z∗Π2Zq∣Zn(y,ξ→n) is given by the above equation, where λ is replaced by -iq. Furthermore, Π1Z∗Π2Zq∣Zn(·,ξ→n)∈A(∞).Theorem 17.
Letq be a nonzero real number and let 1≤p≤∞. Furthermore let Π1 and Π2 be as given in Theorem 16. Then one has for wφ a.e. y∈C[0,T] and PZn a.e. ξ→n,η→n∈Rn+1(59)TqpΠ1Z∗Π2Zq∣Zn·,ξ→n∣Zny,η→n=TqpΠ1Z∣Zn12y+2-1a,12η→n+ξ→n-2-1a→nTqpΠ2Z∣Zn12y-a,12η→n-ξ→n+a→n.LetM(L2[0,T]) be the class of all complex-valued Borel measures of bounded variation over L2[0,T] and let Swφ be the space of all functions F which for σ∈M(L2[0,T]) have the form(60)Fx=∫L20,Texp⁡iv,xdσv,for wφ a.e. x∈C[0,T]. Note that Swφ is a Banach algebra [1].Theorem 18.
Let1≤p≤∞ and let F(∈Swφ) be given by (60). Then for λ∈C+, wφ a.e. y∈C[0,T], and PZn a.e. ξ→n∈Rn+1, TλFZ∣Zn(y,ξ→n) is given by (61)TλFZ∣Zny,ξ→n=∫L20,Texp⁡-12λbT-btnvαn+1,αn+12+P⊥Mhv2+iv,y+a+Pb,nξ→n-a→ndσv.For a nonzero real q, Tq(p)FZ∣Zn(y,ξ→n) is given by the right-hand side of the above equality, where λ is replaced by -iq. Furthermore, Tq(p)FZ∣Zn(·,ξ→n)∈A(∞).Proof.
Forξ→n=(ξ0,ξ1,…,ξn)∈Rn+1 let ξ→n+1=(ξ0,ξ1,…,ξn,ξn+1), where ξn+1∈R. Then for λ>0, wφ a.e. y∈C[0,T], and PZn a.e. ξ→n∈Rn+1, we have by Lemmas 1 and 2(62)IFZλy,ξ→n=∫RΨ1λ,aT-atn,bT-btn,ξn+1-ξnEFy+λ-1/2Xb,n+1x,·+A+Pb,n+1ξ→n+1dξn+1=λ2πbT-btn1/2∫L20,T∫Rexp-λ2bT-btnξn+12+iξn+1vαn+1,αn+1-12λP⊥Mhv2+iv,y+A+Pb,nξ→n+aT-atnvαn+1,αn+1dξn+1dσv=∫L20,Texp-12λbT-btnvαn+1,αn+12+P⊥Mhv2+iv,y+a+Pb,nξ→n-a→ndσv.By Morera’s theorem and the dominated convergence theorem we have the theorem.Theorem 19.
Under the assumptions as given in Theorem18, one has that for PZn a.e. ξ→n,η→n∈Rn+1(63)Tλ-TλFZ∣Zn·,ξ→n∣Zn·,η→n-F·+2a+Pb,nξ→n+η→n-2a→npconverges to 0 and for wφ a.e. y∈C[0,T](64)Tλ¯TλFZ∣Zn·,ξ→n∣Zny,η→n⟶Fy+2a+Pb,nξ→n+η→n-2a→n,as λ approaches -iq through C+.Proof.
By Theorem18, Tλ¯TλFZ∣Zn·,ξ→n∣Zn(y,η→n) is well-defined so that we have for λ∈C+(65)Tλ¯TλFZ∣Zn·,ξ→n∣Zny,η→n=∫L20,Texp-Reλλ2bT-btnvαn+1,αn+12+P⊥Mhv2+iv,y+2a+Pb,nξ→n+η→n-2a→ndσv.Applying a similar method as used in the proof of Theorem 4.2 in [11] with minor modifications we can obtain the remainder part of the proof.Theorem 20.
LetF1 and F2 and σ1 and σ2 be related by (60), respectively. Then for λ∈C+, wφ a.e. y∈C[0,T], and a.e. ξ→n∈Rn+1(66)F1Z∗F2Zλ∣Zny,ξ→n=∫L20,T∫L20,Texp-14λbT-btnv1-v2αn+1,αn+12+P⊥Mhv1-v22+i2v1+v2,y+v1-v2,a+Pb,nξ→n-a→ndσ1v1dσ2v2.For a nonzero real q, F1Z∗F2Zq∣Zn(y,ξ→n) is given by the above equation, where λ is replaced by -iq. Furthermore, F1Z∗F2Zq∣Zn(·,ξ→n)∈A(∞).Proof.
Forξ→n=(ξ0,ξ1,…,ξn)∈Rn+1 let ξ→n+1=(ξ0,ξ1,…,ξn,ξn+1), where ξn+1∈R. Then for λ>0, wφ a.e. y∈C[0,T], and PZn a.e. ξ→n∈Rn+1, we have by Lemmas 1 and 2(67)
K
F
1
Z
,
F
2
Z
λ
y
,
ξ
→
n
=
λ
2
π
b
T
-
b
t
n
1
/
2
·
∫
L
2
0
,
T
∫
L
2
0
,
T
∫
R
∫
C
0
,
T
exp
⁡
-
λ
2
b
T
-
b
t
n
·
ξ
n
+
1
-
ξ
n
-
a
T
-
a
t
n
2
+
i
2
v
1
,
y
+
λ
-
1
/
2
X
b
,
n
+
1
x
,
·
+
A
+
P
b
,
n
+
1
ξ
→
n
+
1
+
ξ
→
n
+
1
v
2
,
y
-
λ
-
1
/
2
X
b
,
n
+
1
x
,
·
-
A
-
P
b
,
n
+
1
ξ
→
n
+
1
λ
-
1
/
2
λ
2
b
T
-
b
t
n
d
w
φ
x
d
ξ
n
+
1
d
σ
1
v
1
d
σ
2
v
2
(68)
=
λ
2
π
b
T
-
b
t
n
1
/
2
·
∫
L
2
0
,
T
∫
L
2
0
,
T
∫
R
exp
⁡
-
λ
ξ
n
+
1
2
2
b
T
-
b
t
n
+
i
2
v
1
+
v
2
,
y
+
v
1
-
v
2
,
A
+
v
1
-
v
2
,
P
b
,
n
ξ
→
n
+
a
T
-
a
t
n
+
ξ
n
+
1
v
1
-
v
2
α
n
+
1
,
α
n
+
1
ξ
→
n
-
1
4
λ
P
⊥
M
h
v
1
-
v
2
2
λ
ξ
n
+
1
2
2
b
T
-
b
t
n
d
ξ
n
+
1
d
σ
1
v
1
d
σ
2
v
2
=
∫
L
2
0
,
T
∫
L
2
0
,
T
exp
⁡
-
1
4
λ
b
T
-
b
t
n
·
v
1
-
v
2
α
n
+
1
,
α
n
+
1
2
+
P
⊥
M
h
v
1
-
v
2
2
+
i
2
P
b
,
n
ξ
→
n
v
1
+
v
2
,
y
+
v
1
-
v
2
,
a
+
P
b
,
n
ξ
→
n
-
a
→
n
1
λ
d
σ
1
v
1
d
σ
2
v
2
.By Morera’s theorem and the dominated convergence theorem we have the theorem.Theorem 21.
Letq be a nonzero real number and let 1≤p≤∞. Furthermore let F1 and F2 be as given in Theorem 20. Then one has for wφ a.e. y∈C[0,T] and PZn a.e. ξ→n,η→n∈Rn+1(69)TqpF1Z∗F2Zq∣Zn·,ξ→n∣Zny,η→n=TqpF1Z∣Zn12y+2-1a,12η→n+ξ→n-2-1a→nTqpF2Z∣Zn12y-a,12η→n-ξ→n+a→n.Proof.
By Theorems18 and 20 we have for λ∈C+, wφ a.e. y∈C[0,T], and PZn a.e. ξ→n,η→n∈Rn+1(70)
T
λ
F
1
Z
∗
F
2
Z
q
∣
Z
n
·
,
ξ
→
n
∣
Z
n
y
,
η
→
n
=
∫
L
2
0
,
T
∫
L
2
0
,
T
exp
⁡
1
4
q
i
b
T
-
b
t
n
·
v
1
+
v
2
α
n
+
1
,
α
n
+
1
2
+
P
⊥
M
h
v
1
+
v
2
2
-
1
4
λ
P
⊥
M
h
2
b
T
-
b
t
n
·
v
1
+
v
2
α
n
+
1
,
α
n
+
1
2
+
P
⊥
M
h
v
1
+
v
2
2
+
i
2
v
1
+
v
2
,
y
+
a
+
P
b
,
n
η
→
n
-
a
→
n
+
v
1
-
v
2
,
a
+
P
b
,
n
ξ
→
n
-
a
→
n
d
σ
1
v
1
d
σ
2
v
2
.Let Tq(p)F1Z∗F2Zq∣Zn·,ξ→n∣Zn(y,η→n) be given by the right-hand side of the above equality, where λ is replaced by -iq. The existence of Tq(1)F1Z∗F2Zq∣Zn·,ξ→n∣Zn(y,η→n) follows from the dominated convergence theorem. Now let 1<p≤∞ and 1/p+1/p′=1. Then we have by the dominated convergence theorem (71)
∫
C
0
,
T
T
λ
F
1
Z
∗
F
2
Z
q
∣
Z
n
·
,
ξ
→
n
∣
Z
n
y
,
η
→
n
-
T
q
p
F
1
Z
∗
F
2
Z
q
∣
Z
n
·
,
ξ
→
n
∣
Z
n
y
,
η
→
n
p
′
d
w
φ
y
≤
∫
L
2
0
,
T
∫
L
2
0
,
T
exp
⁡
-
1
4
λ
b
T
-
b
t
n
·
v
1
+
v
2
α
n
+
1
,
α
n
+
1
2
+
P
⊥
M
h
v
1
+
v
2
2
1
4
λ
-
exp
⁡
1
4
q
i
b
T
-
b
t
n
v
1
+
v
2
α
n
+
1
,
α
n
+
1
2
+
P
⊥
M
h
v
1
+
v
2
2
1
4
q
i
d
σ
1
v
1
d
σ
2
v
2
p
′
⟶
0
,as λ approaches -iq through C+, which shows the existence of Tq(p)F1Z∗F2Zq∣Zn·,ξ→n∣Zn(y,η→n). The equality in the theorem follows from Theorem 18 with simple calculations.Remark 22.
( 1 ) Theorems 15 and 17 can be obtained from Theorems 9 and 12, respectively. They also can be proved directly with aids of Theorems 14 and 16.
( 2 ) Comparing Theorems 17 and 21 with Theorem 13, the results in Theorems 17 and 21 hold for 1≤p≤∞.


## 6. Change of Scale Formulas for the Transforms and Convolutions

In this section, we derive two types of change of scale formulas for the conditional expectations as described in the previous sections.Forλ∈C, x∈C[0,T] and ξ→n+1∈Rn+2 let(72)K1λ,ξ→n+1,x=ΣMhΣP⊥1/2exp⁡12ΣMh-1/2v→,Xx,·R2-λ2ΣP⊥-1/2v→,Zx,·-A-Pb,n+1ξ→n+1R2.For a nonzero real q, let {λm} be any sequence in C+ converging to -iq.By simple calculations, we have the following two theorems.Theorem 23.
Let1≤p≤∞ and let F∈A(p). Then for λ∈C+, wφ a.e. y∈C[0,T], and PZn a.e. ξ→n=(ξ0,ξ1,…,ξn)∈Rn+1(73)TλFZ∣Zny,ξ→n=λr/2∫RΨ1λ,aT-atn,bT-btn,ξn+1-ξnEK1λ,ξ→n+1,·FZ·,ydξn+1,where ξ→n+1=(ξ0,ξ1,…,ξn,ξn+1) and K1 is given by (72). If p=1, then(74)Tq1FZ∣Zny,ξ→n=limm→∞λmr/2∫RΨ1λm,aT-atn,bT-btn,ξn+1-ξnEK1λm,ξ→n+1,·FZ·,ydξn+1.Theorem 24.
LetF∈A(p1) and G∈A(p2) with 1≤p1,p2≤∞. Then for λ∈C+, wφ a.e. y∈C[0,T], and PZn a.e. ξ→n∈Rn+1, FZ∗GZλ∣Zn(y,ξ→n) is given by the right-hand side of (73) with replacing FZ(·,y) by FZ(·/2,y/2)GZ(-·/2,y/2). Moreover if p1=p2=1, then FZ∗GZq∣Zn(y,ξ→n) is given by the right-hand side of (74) with replacing FZ(·,y) by FZ(·/2,y/2)GZ(-·/2,y/2).By Theorems14 and 23 and the dominated convergence theorem, we have the following theorem.Theorem 25.
Let1≤p≤∞ and let Π be given by (54). Then, for λ∈C+, wφ a.e. y∈C[0,T], and PZn a.e. ξ→n∈Rn+1, TλΠZ∣Zn(y,ξ→n) and Tq(p)ΠZ∣Zn(y,ξ→n) are given by the right-hand sides of (73) and (74), respectively, with replacing FZ by ΠZ.By Theorems16 and 24 and the dominated convergence theorem, we have the following theorem.Theorem 26.
Let the assumptions be as given in Theorem16. Then, for λ∈C+, wφ a.e. y∈C[0,T], and PZn a.e. ξ→n∈Rn+1, Π1Z∗Π2Zλ∣Zn(y,ξ→n) and Π1Z∗Π2Zq∣Zn(y,ξ→n) are given by the right-hand sides of (73) and (74), respectively, with replacing FZ by (Π1)Z(·/2,y/2)(Π2)Z(-·/2,y/2).Forλ∈C, x∈C[0,T] and ξ→n∈Rn+1 let(75)K2λ,ξ→n,x=ΣMhΛΣP⊥ΣP⊥1/2exp⁡12ΣMh-1/2v→,Xx,·R2-λ2ΣP⊥-1/2v→,Zx,·-a-Pb,nξ→n-a→nR2+λ2ΛΣP⊥bT-btnΣP⊥-1v→,Zx,·-a-Pb,nξ→n-a→n,v→αn+1,αn+1R2.Theorem 27.
Let1≤p≤∞ and let F(∈A(p)) be given by (29). Then for λ∈C+, wφ a.e. y∈C[0,T], and PZn a.e. ξ→n∈Rn+1(76)TλFZ∣Zny,ξ→n=λr/2EK2λ,ξ→n,·FZ·,y,where K2 is given by (75). If p=1, then(77)Tq1FZ∣Zny,ξ→n=limm→∞⁡λmr/2EK2λm,ξ→n,·FZ·,y.Proof.
Forλ>0, wφ a.e. y∈C[0,T], and PZn a.e. ξ→n∈Rn+1, we have by Lemma 3(78)λr/2EK2λ,ξ→n,·FZ·,y=λrΣMhΛΣP⊥ΣP⊥1/2∫C0,TFZx,·+yexp⁡12ΣMh-1/2v→,Zx,·-aR2-λ2ΣP⊥-1/2v→,Zx,·-a-Pb,nξ→n-a→nR2+λ2ΛΣP⊥bT-btnΣP⊥-1v→,Zx,·-a-Pb,nξ→n-a→n,v→αn+1,αn+1R2dwφx=λr2πrΛΣP⊥ΣP⊥1/2∫Rrfu→+v→,yexp⁡-λ2ΣP⊥-1/2u→-v→,a+Pb,nξ→n-a→nR2+λ2ΛΣP⊥bT-btnΣP⊥-1/2u→-v→,a+Pb,nξ→n-a→n,ΣP⊥-1/2v→αn+1,αn+1R2du→=λr2πrΛΣP⊥1/2∫RrfΣP⊥1/2u→+v→,y+a+Pb,nξ→n-a→nexp⁡-λ2u→R2+λ2ΛΣP⊥bT-btnu→,ΣP⊥-1/2v→αn+1,αn+1R2du→.By the analytic continuation, the dominated convergence theorem, and Theorem 5, we have the theorem.Theorem 28.
LetF∈A(p1) and G∈A(p2) with 1≤p1,p2≤∞. Then for λ∈C+, wφ a.e. y∈C[0,T], and PZn a.e. ξ→n∈Rn+1, FZ∗GZλ∣Zn(y,ξ→n) is given by the right-hand side of (76) with replacing FZ(·,y) by FZ(·/2,y/2)GZ(-·/2,y/2). Moreover if p1=p2=1, then FZ∗GZq∣Zn(y,ξ→n) is given by the right-hand side of (77) with replacing FZ(·,y) by FZ(·/2,y/2)GZ(-·/2,y/2).By Theorems14, 16, and 27 and the dominated convergence theorem, we have the following two theorems.Theorem 29.
Let1≤p≤∞ and let Π be given by (54). Then, for λ∈C+, wφ a.e. y∈C[0,T], and PZn a.e. ξ→n∈Rn+1, TλΠZ∣Zn(y,ξ→n) and Tq(p)ΠZ∣Zn(y,ξ→n) are given by the right-hand sides of (76) and (77), respectively, with replacing FZ by ΠZ.Theorem 30.
Let the assumptions be as given in Theorem16. Then, for λ∈C+, wφ a.e. y∈C[0,T], and PZn a.e. ξ→n∈Rn+1, Π1Z∗Π2Zλ∣Zn(y,ξ→n) and Π1Z∗Π2Zq∣Zn(y,ξ→n) are given by the right-hand sides of (76) and (77), respectively, with replacing FZ by (Π1)Z(·/2,y/2)(Π2)Z(-·/2,y/2).Let{e1,e2,…} be a complete orthonormal basis of L2[0,T]. For m∈N, λ∈C and x∈C[0,T] let (79)Hmλ,x=exp⁡1-λ2∑j=1mej,x2.We now have the following relationships among the conditional Fourier-Feynman transform, the conditional convolution products, and the generalized Wiener integrals of the functions in Swφ. Their proofs are similar to the proofs of the results in [12] with additional calculations.Theorem 31.
Let1≤p≤∞ and let F be given by (60). Then for λ∈C+, wφ a.e. x∈C[0,T], and PZn a.e. ξ→n=(ξ0,ξ1,…,ξn)∈Rn+1(80)TλFZ∣Zny,ξ→n=limm→∞⁡λm/2∫RΨ1λ,aT-atn,bT-btn,ξn+1-ξnEHmλ,xFy+Zb,n+1x,·+Pb,n+1ξ→n+1dξn+1,where ξ→n+1=(ξ0,…,ξn,ξn+1) for ξn+1∈R. Moreover, Tq(p)FZ∣Zn(y,ξ→n) is given by the right-hand side of the above equality with replacing λ by λm.Theorem 32.
LetF1, F2, and σ1, σ2 be related by (60), respectively. Then for λ∈C+, wφ a.e. y∈C[0,T], and PZn a.e. ξ→n∈Rn+1(81)F1Z∗F2Zλ∣Zny,ξ→n=limm→∞⁡λm/2∫RΨ1λ,aT-atn,bT-btn,ξn+1-ξnEHmλ,xF112y+Zb,n+1x,·+Pb,n+1ξ→n+1F212y-Zb,n+1x,·-Pb,n+1ξ→n+1dξn+1.Moreover, F1Z∗F2Zq∣Zn(y,ξ→n) is given by the right-hand side of the above equality with replacing λ by λm.Remark 33.
Lettingλ=γ-2 in the theorems of this section, where γ>0, we can obtain the change of scales for EFZγ·,y∣Znγ· and EFZγ·/2,y/2GZ-γ·/2,y/2∣Znγ·; that is, the scale γ in the functions FZ(γ·,y), GZ(-γ·/2,y/2), and Zn(γ·) is moved to K1, K2, Ψ1, and Hm in evaluating the conditional expectations as above.Remark 34.
( 1 ) An orthonormal subset {v1,v2,…,vr} of L2[0,T] such that both {Mhv1,…,Mhvr} and {P⊥Mhv1,…,P⊥Mhvr} are independent sets exists [12].
( 2 ) Let {e11,…,e1r} and {e21,…,e2r} be the orthonormal sets obtained from {Mhv1,…,Mhvr} and {P⊥Mhv1,…,P⊥Mhvr}, respectively, by the Gram-Schmidt orthonormalization process. For l=1,…,r, let (82)Mhvl=∑j=1rαlje1j,P⊥Mhvl=∑j=1rβlje2jbe the linear combinations. Let B1=[αlj]r×r and B2=[βlj]r×r be the coefficient matrices of the combinations. Then Mh1/2 and ΣP⊥1/2 can be replaced by B1 and B2, respectively, in each expression of the theorems.
( 3 ) It does not mean that B1=ΣMh1/2 or B2=ΣP⊥1/2 in (2). They only satisfy the following equations: (83)B1B1T=ΣMh=ΣMh1/22,B2B2T=ΣP⊥=ΣP⊥1/22.Remark 35.
( 1 ) If y=0, then we can obtain the change of scale formulas in [2] with Zn.
( 2 ) If a=0 and y=0, then we can obtain the results in [12] with the cylinder functions.
( 3 ) The results of this paper are independent of a particular choice of the initial distribution φ.



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