# Conditional Fourier-Feynman Transforms with Drift on a Function Space

Authors: Dong Hyun Cho; Suk Bong Park
Journal: Journal of Function Spaces
 (2019)

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## Abstract

In this paper we derive change of scale formulas for conditional analytic Fourier-Feynman transforms and conditional convolution products of the functions which are the products of generalized cylinder functions and the functions in a Banach algebra which is the space of generalized Fourier transforms of the complex Borel measures onL2[0,T] using two simple formulas for conditional expectations with a drift on an analogue of Wiener space. Then 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 establish various changes of scale formulas for the conditional transforms and the conditional convolution products.


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## Body

## 1. Introduction

LetC0[0,T] denote the Wiener space, the space of real-valued continuous functions x on the interval [0,T] with x(0)=0. It has long been known that Wiener measure and Wiener measurability behave badly under the change of scale transformation and under translations on C0[0,T] [1, 2]. Yoo and his coauthors [3] presented a change of scale formula for Wiener integrals of functions on the abstract Wiener space B [4] which generalizes C0[0,T]. We note that the functions used in [3] are the products of generalized cylinder functions on B and the functions on the Fresnel class [5] which is the space of Fourier-Stieltjes transforms of measures on a separable real Hilbert space densely embedded in B, and note that they need not be bounded or continuous.LetC[0,T] denote an analogue of Wiener space which is the space of real-valued continuous functions on the interval [0,T] [6]. Two kinds of integral transforms, which are known as an analytic conditional Fourier-Feynman transform and a conditional convolution product on C[0,T], were introduced by the author and his coauthors [7–9] using some conditioning functions. In fact, the authors [9] investigated the conditional Fourier-Feynman transforms and the conditional convolution products of the cylinder functions on C[0,T] and established various relationships that occur among them using the conditioning function Xn+1:C[0,T]→Rn+2 given by Xn+1(x)=(x(t0),x(t1),…,x(tn),x(tn+1)), where 0=t0<t1<⋯<tn<tn+1=T. Moreover they derived several changes of scale formulas for the conditional transforms and the conditional convolution products, which simplify the evaluations of the conditional expectations, because the probability measure used on C[0,T] may not be scale-invariant. We note that Xn+1 has no drift and contains the present position x(tn+1) of the path x∈C[0,T]. In [7], Cho extended the results in [9] using a more generalized stochastic process Z:C[0,T]×[0,T]→R given by Z(x,t)=(hχ[0,t],x)+x(0)+a(t) for x∈C[0,T] and t∈[0,T], where (·,·) denotes the Paley-Wiener-Zygmund stochastic integral [6], and h(≠0 a.e.) and a are of bounded variation and absolutely continuous, respectively, on [0,T]. Here he used the conditioning function Zn+1(x)=(Z(x,t0),Z(x,t1),…,Z(x,tn+1)) for x∈C[0,T] which generalizes Xn+1, and the functions a and (hχ[0,t],x) have the effect that the normal density defining the premeasure of cylinder sets on C[0,T] contains the mean function a(t) and the variance function ∫0t[h(u)]2du, respectively. Using the conditioning function Zn(x)=(Z(x,t0),Z(x,t1),…,Z(x,tn)) for x∈C[0,T], Cho [8] also derived similar results in [7]. We note that Zn+1 contains the present positions of generalized Wiener paths and Zn does not; that is, the conditional expectation at Zn+1(x)=(ξ0,ξ1,…,ξn,ξn+1)∈Rn+2 generalizes the probability of the Wiener paths x which pass through ξj at time tj for j=0,1,…,n,n+1 while Zn describes the probability of the Wiener paths which pass through ξj at time tj for j=0,1,…,n. In other words, Zn+1 describes the conditional expectation of Wiener paths which pass through ξj at each past time tj(j=0,1,2,…,n) and ξn+1 at the present time tn+1=T, while Zn only describes the conditional expectation of Wiener paths which pass through ξj at each past time tj(j=0,1,2,…,n).In this paper, using two simple formulas for conditional expectations over paths [10, 11], we evaluate conditional expectations of the products of generalized cylinder functions and the functions in a Banach algebra which plays significant roles in Feynman integration theories and quantum mechanics. Then we investigate their relationships. In particular, we establish change of scale formulas for the conditional transforms and the conditional convolution products. In these evaluation formulas and changes of scale formulas, we use multivariate normal distributions so that the evaluation formulas are unaffected by the present positions of paths, despite the existences of the present positions of the paths in the conditioning functions. We also note that the change scale formulas in [7–9] are expressed by finite sums, while the formulas in this paper are expressed by a limit with a complete orthonormal set of L2[0,T]. Moreover, the conditional Fourier-Feynman transforms and convolution products of cylinder functions in [7–9] are still cylinder functions while the transforms and the convolutions of the functions in this paper are not cylinder functions. In fact, with the conditioning functions Zn and Zn+1, we evaluate conditional expectations, namely, the conditional Fourier-Feynman transforms and the conditional convolution products of the functions given by (1)fv1,Zx,·,…,vr,Zx,·∫L20,Texp⁡iv,Zx,·dσv,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]. Then we show that the conditional Fourier-Feynman transform Tq(2)[[(Ψ1∗Ψ2)q∣Zn](·,ξ→n)∣Zn] of the conditional convolution product [(Ψ1∗Ψ2)q∣Zn](·,ξ→n) of the functions Ψi(i=1,2) defined by (2)Ψix=fiv1,Zx,·,…,vr,Zx,·∫L20,Texpiv,Zx,·dσiv,where σi is a complex Borel measure of bounded variation on L2[0,T], fi∈Lp(Rr), can be expressed by the formula (3)Tq2Ψ1∗Ψ2q∣Zn·,ξ→n∣Zny,η→n=Tq1Ψ1∣Zn12y+2-1a,12η→n+ξ→n-2-1a→n×Tq2Ψ2∣Zn12y-a,12η→n-ξ→n+a→nfor 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. Moreover, replacing an by a, we can recover the same formula with Zn+1 and the effects of drift a will be investigated on the polygonal function of a so that the results do not depend on a particular choice of the initial distribution of the paths. We also note that the functions in (1) extend the initial state of the Schrödinger equation [12].


## 2. A Function Space and Preliminary Results

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 and let wφ be the analogue of Wiener measure according to the probability measure φ on the Borel class of R [6].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. We will adopt the conditional expectation E[F∣Xτ] of F given Xτ, which is described in [8, 13, 14].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 limλ→-iq‖Fλ-F‖p′=0. Then we write limλ→-iq⁡(wp′)(Fλ)=F.Define a stochastic processX:C[0,T]×[0,T]→R by X(x,t)=(hχ[0,t],x) for x∈C[0,T] and t∈[0,T], where χ denotes an indicator function. For t∈[0,T], let b(t)=∫0t[h(s)]2ds and, for any function f on [0,T], define the polygonal function Pb,n+1(f) of f by (4)Pb,n+1ft=∑j=1n+1btj-btbtj-btj-1ftj-1+bt-btj-1btj-btj-1ftj×χtj-1,tjt+f0χ0tfor t∈[0,T]. For ξ→n+1=(ξ0,ξ1,…,ξn,ξn+1)∈Rn+2, define the polygonal function Pb,n+1(ξ→n+1) of ξ→n+1 by the right-hand side of (4), where f(tj) is replaced by ξj. If ξ→n=(ξ0,ξ1,…,ξn)∈Rn+1, Pb,n+1(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 (5)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 (6)Ψrλ,a→,Ar,u→=λr2πrdet⁡Ar1/2exp⁡-λ2Ar-1u→-a→,u→-a→R,where 〈·,·〉R denotes the dot product on Rr.For a functionF:C[0,T]→C, let FZλ(x,y)=F(Z(λ-1/2x,·)+y), let Zn+1λ(x)=Zn+1(λ-1/2x), and let Znλ(x)=Zn(λ-1/2x) for λ>0 and x,y∈C[0,T]. Suppose that E[FZλ(·,y)] exists, where the expectation is taken over the first variable. By the same method as in Theorem 2.5 of [10] we have (7)EFZλ·,y∣Zn+1λξ→n+1=EFy+λ-1/2Xb,n+1x,·+A+Pb,n+1ξ→n+1for PZn+1λ a.e. ξ→n+1∈Rn+2, and for PZnλ a.e. ξ→n=(ξ0,ξ1,…,ξn)∈Rn+1(8)EFZλ·,y∣Znλξ→n=∫RΨ1λ,aT-atn,bT-btn,ξn+1-ξnEFy+λ-1/2Xb,n+1x,·+A+Pb,n+1ξ→n+1dξn+1,where ξ→n+1=(ξ0,…,ξn,ξn+1), the expectation is taken over the variable x and PZn+1λ, PZnλ are the probability distributions of Zn+1λ, Znλ on B(Rn+2), B(Rn+1), respectively. Let IFλ(y,ξ→n+1) and IFλ(y,ξ→n) be the right-hand sides of (7) and (8), respectively. If, for wφ a.e. y∈C[0,T] and PZn+1 a.e. ξ→n+1∈Rn+2, IFλ(y,ξ→n+1) has an analytic extension Jλ(F)(y,ξ→n+1) on C+, then it is called a generalized analytic conditional Fourier-Wiener transform of F given Zn+1 with the parameter λ and is denoted by (9)TλF∣Zn+1y,ξ→n+1=JλFy,ξ→n+1.Moreover if Tλ[F∣Zn+1](y,ξ→n+1) has a point-wise limit as λ approaches -iq through C+, then it is called a generalized L1-analytic conditional Fourier-Feynman transform of F given Zn+1 with the parameter q and is denoted by (10)Tq1F∣Zn+1y,ξ→n+1=limλ→-iq⁡TλF∣Zn+1y,ξ→n+1.For 1<p≤∞, we define a generalized Lp-analytic conditional Fourier-Feynman transform Tq(p)[F∣Zn+1] of F given Zn+1 by the formula (11)TqpF∣Zn+1·,ξ→n+1=limλ→-iqwp′TλF∣Zn+1·,ξ→n+1ifexists.Tλ[F∣Zn](y,ξ→n) and Tq(p)[F∣Zn](y,ξ→n) are similarly defined with IFλ(y,ξ→n).LetG be defined on C[0,T]. For λ>0 and y∈C[0,T], suppose that E[FZ/2λ(·,y/2)G-Z/2λ(·,y/2)] exists. By the same method as in Theorem 2.5 of [10] we have (12)EFZ/2λ·,y2G-Z/2λ·,y2∣Zn+1λξ→n+1=EF12y+λ-1/2Xb,n+1x,·+A+Pb,n+1ξ→n+1×G12y-λ-1/2Xb,n+1x,·-A-Pb,n+1ξ→n+1,(13)EFZ/2λ·,y2G-Z/2λ·,y2∣Znλξ→n=∫RΨ1λ,aT-atn,bT-btn,ξn+1-ξnEF12y+λ-1/2Xb,n+1x,·+A+Pb,n+1ξ→n+1G12y-λ-1/2×Xb,n+1x,·-A-Pb,n+1ξ→n+1dξn+1.Let KF,Gλ(y,ξ→n+1) and KF,Gλ(y,ξ→n) be the right-hand sides of (12) and (13), respectively. If KF,Gλ(y,ξ→n+1) has an analytic extension Jλ(F,G)(y,ξ→n+1) on C+, then it is called a generalized conditional convolution product of F and G given Zn+1 with the parameter λ and denoted by (14)F∗Gλ∣Zn+1y,ξ→n+1=JλF,Gy,ξ→n+1.Moreover if [(F∗G)λ∣Zn+1](y,ξ→n+1) has a point-wise limit as λ approaches -iq through C+, then it is denoted by (15)F∗Gq∣Zn+1y,ξ→n+1=limλ→-iq⁡F∗Gλ∣Zn+1y,ξ→n+1.[(F∗G)λ∣Zn](y,ξ→n) and [(F∗G)q∣Zn](y,ξ→n) are similarly understood with KF,Gλ(y,ξ→n).Forj=1,…,n,n+1, let αj=h/[b(tj)-b(tj-1)]1/2χ(tj-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 projection given by Pv=∑j=1n+1〈v,αj〉αj and let P⊥:L2[0,T]→V⊥ be the orthogonal projection.The following lemmas are essential to prove the results in the next sections [8].Lemma 1.
Letv∈L2[0,T]. Then for wφ a.e. x∈C[0,T] we have (16)v,Xx,·=Mhv,x,v,Pb,n+1Xx,·=PMhv,x,where Mh:L2[0,T]→L2[0,T] is the multiplication operator defined by Mhu=hu for u∈L2[0,T].Lemma 2.
Letv∈L2[0,T] and ξ→n+1=(ξ0,ξ1,…,ξn,ξn+1)∈Rn+2. Then we have (17)v,Pb,n+1ξ→n+1=v,Pb,n+1ξ→n+vαn+1,αn+1ξn+1-ξn,where ξ→n=(ξ0,ξ1,…,ξn) and 〈·,·〉 denotes the inner product on L2[0,T].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→,Z(x,·))≡((v1,Z(x,·)),…,(vr,Z(x,·))) 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→C we have (18)∫C0,Tfv→,Zx,·dwφx⩮∫Rrfu→Ψr1,v→,a,ΣMh,u→du→,where ⩮ means that if either side exists, then both sides exist and they are equal.


## 3. Generalized Conditional Fourier-Feynman Transforms

In this section we derive the conditional Fourier-Feynman transforms of functions onC[0,T] and investigate the effects of the drift a.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 Fr having the form (19)Frx=fv→,xfor 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.LetM(L2[0,T]) be the class of all C-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 (20)Fx=∫L20,Texp⁡iv,xdσvfor wφ a.e. x∈C[0,T]. Note that Swφ is a Banach algebra [6].LetC={e1,…,er} be the orthonormal set obtained from C′≡{P⊥Mhv1, …, P⊥Mhvr} by the Gram-Schmidt orthonormalization process. Keeping the orders of the elements in C and C′, respectively, let B be the matrix of change of coordinates from C′ to C. For v∈L2[0,T], λ∈C, and u→∈Rr, let cj(v)=〈v,ej〉 for j=1,…,r, let c→(P⊥Mhv)=(c1(P⊥Mhv),…,cr(P⊥Mhv)), let A1(λ,v)=exp{-1/2λ[‖P⊥Mhv‖2-‖c→(P⊥Mhv)‖R2]}, and let A2(v,u→)=exp{i〈c→(P⊥Mhv),u→〉R}.Theorem 4.
LetΨ(x)=F(x)Fr(x) for wφ a.e. x∈C[0,T], where Fr(∈A(p)(1≤p≤∞)) and F(∈Swφ) are given by (19) and (20), respectively. Then we have for λ∈C+, wφ a.e. y∈C[0,T] and PZn+1 a.e. ξ→n+1∈Rn+2(21)TλΨ∣Zn+1y,ξ→n+1=∫L20,TB1y,v,0,A,ξ→n+1A1λ,v∫Rrfu→B+v→,y+A+Pb,n+1ξ→n+1A2v,u→Ψrλ,0→,Ir,u→du→dσv,where Ir is the r×r identity matrix and, for w1,w2∈L2[0,T], and for x∈C[0,T](22)B1y,w1,w2,x,ξ→n+1=expiw1+w2,y+w1-w2,x+Pb,n+1ξ→n+1.Furthermore Tλ[Ψ∣Zn+1](·,ξ→n+1)∈Lp1(C[0,T]) for 0<p1≤p. If p=1, then, for a nonzero real q, Tq(1)[Ψ∣Zn+1](y,ξ→n+1) is given by the right-hand side of (21) with replacing λ by -iq. In this case, Tq(1)[Ψ∣Zn+1](·,ξ→n+1)∈Lp1(C[0,T]) for 0<p1≤∞.Proof.
Forλ>0, wφ a.e. y∈C[0,T] and PZn+1 a.e. ξ→n+1∈Rn+2, we have (23)IΨλy,ξ→n+1=EΨy+λ-1/2Xb,n+1x,·+A+Pb,n+1ξ→n+1=∫L20,TB1y,v,0,A,ξ→n+1∫C0,Texpiλ-1/2P⊥Mhv,x×fλ-1/2P⊥Mhv→,x+v→,y+A+Pb,n+1ξ→n+1dwφxdσvby Lemma 1, where (P⊥Mhv→,x)=((P⊥Mhv1,x),…,(P⊥Mhvr,x)). For v∈L2[0,T], let er+1=1/cr+1(P⊥Mhv)[P⊥Mhv-∑j=1rcj(P⊥Mhv)ej] if cr+1(P⊥Mhv)≠0, where cr+1(P⊥Mhv)=(‖P⊥Mhv‖2-‖c→(P⊥Mhv)‖R2)1/2. By Lemma 3 we have (24)IΨλy,ξ→n+1=∫L20,TB1y,v,0,A,ξ→n+1∫C0,Texpiλ-1/2∑j=1r+1cjP⊥Mhvej,x×fλ-1/2e→,xB+v→,y+A+Pb,n+1ξ→n+1dwφxdσv=12πr+1/2∫L20,TB1y,v,0,A,ξ→n+1∫Rr+1expiλ-1/2∑j=1r+1cjP⊥Mhvuj-12∑j=1r+1uj2fλ-1/2u→B+v→,y+A+Pb,n+1ξ→n+1dur+1du→dσv,where e→=(e1,…,er) and u→=(u1,…,ur). By simple calculations we have (25)IΨλy,ξ→n+1=∫L20,TB1y,v,0,A,ξ→n+1A1λ,v∫RrA2v,λ-1/2u→fλ-1/2u→B+v→,y+A+Pb,n+1ξ→n+1Ψr1,0→,Ir,u→du→dσv=∫L20,TB1y,v,0,A,ξ→n+1A1λ,v∫Rrfu→B+v→,y+A+Pb,n+1ξ→n+1A2v,u→Ψrλ,0→,Ir,u→du→dσv.Note that if 1≤p<∞, then we have by the change of variable theorem (26)f·B+v→,y+A+Pb,n+1ξ→n+1pp=detB-1fpp<∞,and for λ∈C+ we have A1λ,vA2v,u→≤1 by the Bessel inequality. By the Morera theorem with aid of the Hölder inequality and the dominated convergence theorem, Tλ[Ψ∣Zn+1](y,ξ→n+1) exists for λ∈C+. For λ∈C+ we have by Lemma 3 and the Young inequality [15] (27)TλΨ∣Zn+1·,ξ→n+1pp≤detBσp∫Rr∫Rrfz→-u→B+v→,A+Pb,n+1ξ→n+1×Ψrλ,0→,Ir,u→du→pdz→=detBσpf·B+v→,A+Pb,n+1ξ→n+1∗Ψrλ,0→,Ir,·pp<∞so that Tλ[Ψ∣Zn+1](·,ξ→n+1)∈Lp1(C[0,T]) because wφ(C[0,T])=1<∞. If p=1, then the final results follow from the dominated convergence theorem.Remark 5.
Forv∈L2[0,T] let dj(v)=〈v,P⊥Mhvj〉 for j=1,…,r. Let B-1=[βkl]r×r be the matrix of change of coordinates from C to C′. Now for l=1,…,r, el=∑k=1rβklP⊥Mhvk so that for d→(P⊥Mhv)=(d1(P⊥Mhv),…,dr(P⊥Mhv)) and u→=(u1,…,ur) we have (28)c→P⊥MhvR2=∑l=1r∑k=1rβkldkP⊥Mhv2=d→P⊥MhvB-1R2and 〈c→(P⊥Mhv),u→〉R=〈d→(P⊥Mhv)B-1,u→〉R. Now the expression c→(P⊥Mhv) in A1 and A2 can be replaced by d→(P⊥Mhv)B-1.By Lemma 1.1 in [16] and (27) with minor modifications, we have the following theorem.Theorem 6.
LetΨ be as given in Theorem 4 with 1<p≤2, and let 1/p+1/p′=1. Then for λ∈C+ and PZn+1 a.e. ξ→n+1∈Rn+2, we have Tλ[Ψ∣Zn+1](·,ξ→n+1)∈Lp1(C[0,T]) for 0<p1≤p′.Theorem 7.
Let the assumptions be as given in Theorem6. Suppose that σ is concentrated on the set Mh-1(V). Then Tq(p)[Ψ∣Zn+1](y,ξ→n+1) is given by the right-hand side of (21), where λ is replaced by -iq and A1(-iq,v)=A2(v,u→)=1 for σ a.e. v∈L2[0,T] and u→∈Rr. In this case, Tq(p)[Ψ∣Zn+1](·,ξ→n+1)∈Lp1(C[0,T]) for 0<p1≤p′.Proof.
Forσ a.e. v∈L2[0,T], u→∈Rr, and λ∈C, we have A1(λ,v)=A2(v,u→)=1 so that it is not difficult to show (29)TλΨ∣Zn+1y,ξ→n+1=∫L20,TB1y,v,0,A,ξ→n+1f·B+v→,A+Pb,n+1ξ→n+1∗Ψrλ,0→,Ir,·B-1v→,ydσv.Now ‖Tq(p)[Ψ∣Zn+1](·,ξ→n+1)‖p′p′ is bounded by (30)detBσp′f·B+v→,A+Pb,n+1ξ→n+1∗Ψr-iq,0→,Ir,·p′p′which is finite from Lemma 1.1 in [16]. By Lemma 3, Theorem 4, and the change of variable theorem, we have for ξ→n+1∈Rn+2(31)TλΨ∣Zn+1·,ξ→n+1-TqpΨ∣Zn+1·,ξ→n+1p′p′≤det⁡Bσp′f·B+v→,A+Pb,n+1ξ→n+1∗Ψrλ,0→,Ir,·-f·B+v→,A+Pb,n+1ξ→n+1∗Ψr-iq,0→,Ir,·p′p′which converges to 0 as λ approaches -iq through C+ by Lemma 1.2 in [16].Forλ∈C let Γλ=λ/b(T)-b(tn), let 〈v→αn+1,αn+1〉=(〈v1αn+1,αn+1〉,…,〈vrαn+1, αn+1〉), and let Λ=1+[b(T)-b(tn)]‖〈v→αn+1,αn+1〉B-1‖R2. For v∈L2[0,T] and u→∈Rr, let (32)Φrλ,v,u→=1Λ1/2exp⁡12ΓλΛλu→,v→αn+1,αn+1B-1R+ivαn+1,αn+1-c→P⊥Mhv,v→αn+1,αn+1B-1R2.Theorem 8.
Suppose that the assumptions and notations are as given in Theorem4. Let a→n=(a(t0),…,a(tn)). Then we have for λ∈C+, wφ a.e. y∈C[0,T] and PZn a.e. ξ→n∈Rn+1(33)TλΨ∣Zny,ξ→n=∫L20,TB1y,v,0,a,ξ→n-a→nA1λ,v∫Rrfu→B+v→,y+a+Pb,n+1ξ→n-a→nA2v,u→Ψrλ,0→,Ir,u→Φrλ,v,u→du→dσv,where B1 and Φr are given by (22) and (32), respectively. Furthermore Tλ[Ψ∣Zn](·, ξ→n)∈Lp1(C[0,T]) for 0<p1≤p. If p=1, then, for a nonzero real q, Tq(1)[Ψ∣Zn](y, ξ→n) is given by the right-hand side of (33) with replacing λ by -iq. In this case, Tq(1)[Ψ∣Zn](·,ξ→n)∈Lp1(C[0,T]) for 0<p1≤∞.Proof.
Forξ→n=(ξ0,…,ξn)∈Rn+1 let ξ→n+1=(ξ0,…,ξn,ξn+1) with ξn+1∈R. For λ>0, wφ a.e. y∈C[0,T] and PZn a.e. ξ→n∈Rn+1 we have by Lemma 2(34)IΨλy,ξ→n=∫RΨ1λ,aT-atn,bT-btn,ξn+1-ξnEΨy+λ-1/2×Xb,n+1x,·+A+Pb,n+1ξ→n+1dξn+1=Γλ2π1/2∫L20,TB1y,v,0,A,ξ→nA1λ,v∫Rr+1exp⁡ivαn+1,αn+1aT-atnfu→B+v→,y+A+Pb,n+1ξ→n+aT-atn+ξv→αn+1,αn+1A2v,u→Ψrλ,0→,Ir,u→exp⁡-Γλ2ξ2+ivαn+1,αn+1ξdu→dξdσv=Γλ2π1/2λ2πr/2∫L20,TB1y,v,0,a,ξ→n-a→nA1λ,v∫Rr+1fu→B+v→,y+a+Pb,n+1ξ→n-a→nexp⁡-Γλ2ξ2+ivαn+1,αn+1ξ-λ2ξv→αn+1,αn+1B-1-u→R2+ic→P⊥Mhv,u→-ξ×v→αn+1,αn+1B-1Rdu→dξdσv=Γλ2π1/2∫L20,TB1y,v,0,a,ξ→n-a→nA1λ,v∫Rr+1fu→B+v→,y+a+Pb,n+1ξ→n-a→nA2v,u→Ψrλ,0→,Ir,u→exp-12Γλ+λv→αn+1,αn+1B-1R2ξ2+λu→,v→αn+1,αn+1B-1R+iv×αn+1,αn+1-c→P⊥Mhv,v→αn+1,αn+1B-1Rξdu→dξdσv=Γλ2π1/2∫L20,TB1y,v,0,a,ξ→n-a→nA1λ,v∫Rr+1fu→B+v→,y+a+Pb,n+1ξ→n-a→nA2v,u→Ψrλ,0→,Ir,u→exp-ΓλΛ2ξ2+λu→,v→αn+1,αn+1B-1R+ivαn+1,αn+1-c→P⊥Mhv,v→αn+1,αn+1B-1Rξdξdu→dσv=∫L20,TB1y,v,0,a,ξ→n-a→nA1λ,v∫Rrfu→B+v→,y+a+Pb,n+1ξ→n-a→nA2v,u→Ψrλ,0→,Ir,u→Φrλ,v,u→du→dσv.For λ∈C+ we have A1λ,vA2v,u→≤1 by the Bessel inequality and (35)Ψrλ,0→,Ir,u→Φrλ,v,u→≤λ2πr/2exp⁡-Reλ2u→R2+Reλ2ΛbT-btnu→,v→αn+1,αn+1B-1R2=λ2πr/2exp⁡-Reλ2Λu→R2+bT-btnu→R2v→αn+1,αn+1B-1R2-u→,v→αn+1,αn+1B-1R2≤λ2πr/2exp⁡-Reλ2Λu→R2by the Schwarz inequality. By the Morera theorem we establish the first part of the theorem. If p=1, then the final results follow from the dominated convergence theorem.Theorem 9.
LetΨ be as given in Theorem 4 with 1<p≤2, and let 1/p+1/p′=1. Then for λ∈C+ and PZn a.e. ξ→n∈Rn+1, we have Tλ[Ψ∣Zn](y,ξ→n)∈Lp1(C[0,T]) for 0<p1≤p′.Proof.
Forλ∈C+, ‖Tλ[Ψ∣Zn](·,ξ→n)‖p′p′ is bounded by (36)detBσp′λΛReλp′r/2×f·B+v→,a+Pb,n+1ξ→n-a→n∗ΨrReλΛ,0→,Ir,·p′p′from (35), which is finite by Lemma 1.1 in [16].Because〈vjαn+1,αn+1〉=0 if and only if 〈Mhvj,αn+1〉=0, we can prove the following theorem using the same method as used in the proof of Theorem 7.Theorem 10.
Let the assumptions be as given in Theorem7. Suppose that 〈Mhvj,αn+1〉=0 for j=1,…,r. Then for a nonzero real q and wφ a.e. y∈C[0,T], Tq(p)[Ψ∣Zn](y,ξ→n) is given by the right-hand side of (33), where λ is replaced by -iq and A1(-iq,v)=A2(v,u→)=1, Φr(-iq,v,u→)=exp{[b(T)-b(tn)]〈vαn+1,αn+1〉2/(2qi)} for σ a.e. v∈L2[0,T] and u→∈Rr. In this case, Tq(p)[Ψ∣Zn](·,ξ→n)∈Lp1(C[0,T]) for 0<p1≤p′.


## 4. Generalized Conditional Convolution Products

In this section we derive the conditional convolution products of the functions in the previous sections and investigate the effects of the drifta.Theorem 11.
LetFr, Gr and f, g be related by (19), respectively, where Fr∈A(p1) and Gr∈A(p2) with 1≤p1,p2≤∞. Let F,G(∈Swφ) and σF,σG(∈M(L2[0,T])) be related by (20), respectively. Furthermore let Ψ(x)=F(x)Fr(x), H(x)=G(x)Gr(x) for wφ a.e. x∈C[0,T] and let 1/p1+1/p1′=1, 1/p2+1/p2′=1. Then we have for λ∈C+, wφ a.e. y∈C[0,T] and PZn+1 a.e. ξ→n+1∈Rn+2(37)Ψ∗Hλ∣Zn+1y,ξ→n+1=∫L20,T2B1y,w12,w22,A,ξ→n+1A1λ,12w1-w2∫Rrf12v→,y+A+Pb,n+1ξ→n+1+u→Bg12v→,y-A-Pb,n+1ξ→n+1-u→BA212w1-w2,u→Ψrλ,0→,Ir,u→du→dσF×σGw1,w2,where B1 is given by (22). Moreover [(Ψ∗H)λ∣Zn+1](·,ξ→n+1)∈Lq1(C[0,T]), where 0<q1≤1 if either p2≤p1′ or p1≤p2′, 0<q1≤p2 if p2≥p1′ and 0<q1≤p1 if p1≥p2′.Proof.
Forλ>0, wφ a.e. y∈C[0,T] and PZn+1 a.e. ξ→n+1∈Rn+2, we have (38)KΨ,Hλy,ξ→n+1=EΨ12y+λ-1/2Xb,n+1x,·+A+Pb,n+1ξ→n+1H12y-λ-1/2Xb,n+1x,·-A-Pb,n+1ξ→n+1=∫L20,T2∫C0,Texpi2w1,y+λ-1/2Xb,n+1x,·+A+Pb,n+1ξ→n+1+w2,y-λ-1/2Xb,n+1x,·-A-Pb,n+1ξ→n+1×f12v→,y+λ-1/2Xb,n+1x,·+A+Pb,n+1ξ→n+1g12v→,y-λ-1/2Xb,n+1x,·-A-Pb,n+1ξ→n+1dwφxdσF×σGw1,w2=∫L20,T2B1y,w12,w22,A,ξ→n+1∫C0,Texpi2w1-w2,λ-1/2Xb,n+1x,·f12v→,y+λ-1/2Xb,n+1x,·+A+Pb,n+1ξ→n+1g12×v→,y-λ-1/2Xb,n+1x,·-A-Pb,n+1ξ→n+1dwφxdσF×σGw1,w2.Using the same method as used in the proof of Theorem 4, we have(39)KΨ,Hλy,ξ→n+1=∫L20,T2B1y,w12,w22,A,ξ→n+1A1λ,12w1-w2∫Rrf12v→,y+A+Pb,n+1ξ→n+1+u→Bg12v→,y-A-Pb,n+1ξ→n+1-u→BA212w1-w2,u→Ψrλ,0→,Ir,u→du→dσF×σGw1,w2with replacing v by 1/2(w1-w2) in A1 and A2. Now for λ∈C+ we have formally (40)Ψ∗Hλ∣Zn+1·,ξ→n+11≤detBσFσG∫R2rf12z→+u→B+v→,A+Pb,n+1ξ→n+1g12z→-u→B-v→,A+Pb,n+1ξ→n+1×Ψrλ,0→,Ir,u→du→,z→.By the same method as used in the proof of Theorem 3.2 in [9], we have [(Ψ∗H)λ∣Zn+1](·,ξ→n+1)∈L1(C[0,T]) with the existence of [(Ψ∗H)λ∣Zn+1](y,ξ→n+1) for λ∈C+ if either p2≤p1′ or p1≤p2′. Using the same methods as used in the proof of Theorem 3.2 in [9] again, we have the remainder part of the theorem.Theorem 12.
Let the assumptions be as given in Theorem11. Then we have for λ∈C+, wφ a.e. y∈C[0,T] and PZn a.e ξ→n∈Rn+1, (41)Ψ∗Hλ∣Zny,ξ→n=∫L20,T2B1y,w12,w22,a,ξ→n-a→nA1λ,12w1-w2∫Rrf12v→,y+a+Pb,n+1ξ→n-a→n+u→Bg12v→,y-a-Pb,n+1ξ→n-a→n-u→BA212w1-w2,u→Φrλ,12w1-w2,u→Ψrλ,0→,Ir,u→du→dσF×σGw1,w2,where B1 and Φr are given by (22) and (32), respectively. Moreover [(Ψ∗H)λ∣Zn](·, ξ→n)∈Lq1(C[0,T]), where 0<q1≤1 if either p2≤p1′ or p1≤p2′, 0<q1≤p2 if p2≥p1′ and 0<q1≤p1 if p1≥p2′.Proof.
Replacingv by 1/2(w1-w2) in the proof of Theorem 8 with Theorem 11, we have for ξ→n∈Rn+1, λ>0, and y∈C[0,T](42)KΨ,Hλy,ξ→n=∫L20,T2B1y,w12,w22,a,ξ→n-a→nA1λ,12w1-w2∫Rrf12v→,y+a+Pb,n+1ξ→n-a→n+u→Bg12v→,y-a-Pb,n+1ξ→n-a→n-u→BA212w1-w2,u→Φrλ,12w1-w2,u→Ψrλ,0→,Ir,u→du→dσF×σGw1,w2.Now for λ∈C+, the formal expression ‖[(Ψ∗H)λ∣Zn](·,ξ→n)‖1 is bounded by (43)detBσFσGλΛReλr/2∫R2rf12v→,a+Pb,n+1ξ→n-a→nz→+u→B+v→,a+Pb,n+1ξ→n-a→n12g12v→,a+Pb,n+1ξ→n-a→nz→-u→B-v→,a+Pb,n+1ξ→n-a→n12×ΨrReλΛ,0→,Ir,u→du→,z→.By the same process as used in the proof of Theorem 11, the proof is completed.Using the factwφ(C[0,T])<∞ and the same method as used in the proof of Theorem 3.3 in [9], we can prove the following two theorems.Theorem 13.
LetΨ and H be as given in Theorem 11. Let q be a nonzero real number. Then for λ∈C+ or λ=-iq, and ξ→n+1∈Rn+2, we have the following: (1)
ifFr,Gr∈A(1), then [(Ψ∗H)λ∣Zn+1](·,ξ→n+1)∈Lp1(C[0,T]) for 0<p1≤1(2)
ifFr∈A(1) and Gr∈A(2), then [(Ψ∗H)λ∣Zn+1](·,ξ→n+1)∈Lp1(C[0,T]) for 0<p1≤2(3)
ifFr∈A(1) and Gr∈A(∞), then [(Ψ∗H)λ∣Zn+1](·,ξ→n+1)∈Lp1(C[0,T]) for 0<p1≤∞(4)
ifFr,Gr∈A(2), then [(Ψ∗H)λ∣Zn+1](·,ξ→n+1)∈Lp1(C[0,T]) for 0<p1≤∞Theorem 14.
If we replaceZn+1 by Zn in Theorem 13, then the conclusions of the theorem are still true, where ξ→n+1 is replaced by ξ→n∈Rn+1.Remark 15.
The formal expressionsF∈Lp1(C[0,T])(0<p1≤p) in each theorem of this section and of the previous sections include the fact that there exists a cylinder function G∈A(p) with Fx≤Gx for wφ a.e. x∈C[0,T].


## 5. Relationships between Transforms and Convolution Products

In this section, we investigate the inverse transforms of the conditional Fourier-Feynman transforms of the functions as given in the previous sections. We also show that the analytic conditional Fourier-Feynman transforms of the conditional convolution products for the functions can be expressed as the products of the analytic conditional Fourier-Feynman transforms of each function.Theorem 16.
Letq be a nonzero real number. Then, under the assumptions as given in Theorem 7, we have for PZn+1 a.e. ξ→n+1,η→n+1∈Rn+2(44)∫C0,TTλ-TλΨ∣Zn+1·,ξ→n+1∣Zn+1y,η→n+1-Ψy+2A+Pb,n+1ξ→n+1+η→n+1pdwφy→0for 1≤p<∞, and for 1≤p≤∞(45)Tλ-TλΨ∣Zn+1·,ξ→n+1∣Zn+1y,η→n+1→Ψy+2A+Pb,n+1ξ→n+1+η→n+1for wφ a.e. y∈C[0,T], as λ approaches -iq through C+.Proof.
Forλ∈C+, wφ a.e. y∈C[0,T] and PZn+1 a.e. ξ→n+1,η→n+1∈Rn+2, we have by the change of variable theorem, Theorems 4 and 7(46)Tλ-TλΨ∣Zn+1·,ξ→n+1∣Zn+1y,η→n+1=∫L20,TB1y,v,0,2A,ξ→n+1+η→n+1∫R2rfu→+w→B+v→,y+2A+Pb,n+1ξ→n+1+η→n+1Ψrλ,0→,Ir,u→Ψrλ-,0→,Ir,w→dw→du→dσv=∫L20,TB1y,v,0,2A,ξ→n+1+η→n+1∫Rrfz→B+v→,y+2A+Pb,n+1ξ→n+1+η→n+1Ψrλ22Reλ,0→,Ir,z→dz→dσv=ϵ-rf·B∗Ψr1,0→,Ir,·/ϵv→,y+2A+Pb,n+1ξ→n+1+η→n+1B-1×∫L20,TB1y,v,0,2A,ξ→n+1+η→n+1dσv,where ϵ=(2Reλ/λ2)1/2>0. Let 1≤p<∞. Then we have by Lemma 3 and the change of variable theorem(47)∫C0,TTλ-·,ξ→n+1∣Zn+1TλΨ∣Zn+1·,ξ→n+1∣Zn+1y,η→n+1-ΨPb,n+1ξ→n+1+η→n+1y+2A+Pb,n+1ξ→n+1+η→n+1pdwφy=∫C0,T∫L20,TB1y,v,0,2A,ξ→n+1+η→n+1ξ→n+1dσvpdwφϵ-r1,0→,Ir,·ϵf·B∗Ψr1,0→,Ir,·/ϵξ→n+1+η→n+1v→,y+2A+Pb,n+1ξ→n+1+η→n+1B-1ξ→n+1+η→n+1-fξ→n+1+η→n+1v→,y+2A+Pb,n+1ξ→n+1+η→n+1∫L20,TB1y,v,0,2A,ξ→n+1+η→n+1ξ→n+1dσvpdwφy≤detBσpϵ-rf·B∗Ψr1,0→,Ir,·/ϵ-f·Bpp.Letting λ→-iq through C+ which satisfies ϵ→0, we have the first part of the theorem by Theorem 1.18 in [17]. If 1≤p≤∞, then the remainder part of the theorem follows from Theorem 1.25 in [17].Theorem 17.
Letq be a nonzero real number. Then, under the assumptions as given in Theorem 10, we have for PZn a.e. ξ→n,η→n∈Rn+1(48)∫C0,TTλ-TλΨ∣Zn·,ξ→n∣Zny,η→n-Ψy+2a+Pb,n+1ξ→n+η→n-2a→npdwφy→0for 1≤p<∞, and for 1≤p≤∞(49)Tλ-TλΨ∣Zn·,ξ→n∣Zny,η→n→Ψy+2a+Pb,n+1ξ→n+η→n-2a→nfor wφ a.e. y∈C[0,T], as λ approaches -iq through C+.Proof.
Forλ∈C+, wφ a.e. y∈C[0,T] and PZn a.e. ξ→n,η→n∈Rn+1, we have by the change of variable theorem and Theorems 8 and 10(50)Tλ-TλΨ∣Zn·,ξ→n∣Zny,η→n=ϵ-rf·B∗Ψr1,0→,Ir,·/ϵv→,y+2a+Pb,n+1ξ→n+η→n-2a→nB-1×∫L20,TB1y,v,0,2a,ξ→n+η→n-2a→nA3ϵ,vdσv,where ϵ=(2Reλ/λ2)1/2>0 and A3(ϵ,v)={-ϵ2[b(T)-b(tn)]〈vαn+1, αn+1〉2/2}. Let 1≤p<∞. Then we have by Lemma 3, the Minkowski inequality, and the change of variable theorem (51)∫C0,TTλ-·,ξ→nTλΨ∣Zn·,ξ→n∣Zny,η→n-ΨPb,n+1ξ→n+η→n-2a→ny+2a+Pb,n+1ξ→n+η→n-2a→npdwφy1/p≤∫C0,Tϵ-rΨr1,0→,Ir,·/ϵf·B∗Ψr1,0→,Ir,·/ϵPb,n+1ξ→n+η→n-2a→nv→,y+2a+Pb,n+1ξ→n+η→n-2a→nB-1Pb,n+1ξ→n+η→n-2a→n-fPb,n+1ξ→n+η→n-2a→nv→,y+2a+Pb,n+1ξ→n+η→n-2a→n×∫L20,TB1y,v,0,2a,ξ→n+η→n-2a→nξ→nA3ϵ,vdσvpdwφy1/p+∫C0,TfPb,n+1ξ→n+η→n-2a→nv→,y+2a+Pb,n+1ξ→n+η→n-2a→n∫L20,TB1y,v,0,2a,ξ→n+η→n-2a→nξ→nA3ϵ,v-1dσvPb,n+1ξ→n+η→n-2a→npdwφy∫C0,TfPb,n+1ξ→n+η→n-2a→nv→,y+2a1/p≤detB1/pσϵ-rf·B∗Ψr1,0→,Ir,·/ϵ-f·Bp+fp×∫L20,TA3ϵ,v-1dσv.Letting λ→-iq through C+ which satisfies ϵ→0 and A3(ϵ,v)→1, we have the first part of the theorem by Theorem 1.18 of [17]. If 1≤p≤∞, then the remainder part of the theorem follows from Theorem 1.25 of [17].Theorem 18.
Under the assumptions as given in Theorem11, we have for λ∈C+, wφ a.e. y∈C[0,T] and PZn+1 a.e. ξ→n+1,η→n+1∈Rn+2(52)TλΨ∗Hλ∣Zn+1·,ξ→n+1∣Zn+1y,η→n+1=TλΨ∣Zn+112y+2-1a,12η→n+1+ξ→n+1-2-1aTλH∣Zn+112y-a,12η→n+1-ξ→n+1+a.Proof.
Forλ∈C+, the existence of Tλ[[(Ψ∗H)λ∣Zn+1](·,ξ→n+1)∣Zn+1](y,η→n+1) follows from Theorems 4 and 11. We have for wφ a.e. y∈C[0,T] and PZn+1 a.e. ξ→n+1,η→n+1∈Rn+2(53)TλΨ∗Hλ∣Zn+1·,ξ→n+1∣Zn+1y,η→n+1=∫L20,T2B1y+A+Pb,n+1η→n+1,w12,w22,A,ξ→n+1A1λ,12w1-w2A1λ,12w1+w2∫R2rf12v→,y+2A+Pb,n+1ξ→n+1+η→n+1+w→+u→Bg12v→,y+Pb,n+1η→n+1-ξ→n+1+w→-u→BA212w1-w2,u→A212w1+w2,w→Ψrλ,0→,Ir,u→Ψrλ,0→,Ir,w→du→,w→dσF×σGw1,w2.Let α→=1/2(w→+u→) and β→=1/2(w→-u→). Then we have(54)TλΨ∗Hλ∣Zn+1·,ξ→n+1∣Zn+1y,η→n+1=∫L20,T2B112y,w1,0,2A,12η→n+1+ξ→n+1B112y,w2,0,0,12η→n+1-ξ→n+1A1λ,12w1-w2A1λ,12w1+w2∫R2rf12v→,y+2A+Pb,n+1ξ→n+1+η→n+1+α→B×g12v→,y+Pb,n+1η→n+1-ξ→n+1+β→BA212w1-w2,12α→-β→A212w1+w2,12α→+β→Ψrλ,0→,Ir,12α→-β→Ψrλ,0→,Ir,12α→+β→dα→,β→dσF×σGw1,w2by the change of variable theorem. Now we have (55)A1λ,w1A1λ,w2=A1λ,w1-w22A1λ,w1+w22,A2w1,α→A2w2,β→=A2w1-w22,α→-β→2×A2w1+w22,α→+β→2,Ψrλ,0→,Ir,α→Ψrλ,0→,Ir,β→=Ψrλ,0→,Ir,α→-β→2Ψrλ,0→,Ir,α→+β→2so that we have by Theorem 4(56)TλΨ∗Hλ∣Zn+1·,ξ→n+1∣Zn+1y,η→n+1=TλΨ∣Zn+112y+2-1a,12η→n+1+ξ→n+1-2-1aTλH∣Zn+112y-a,12η→n+1-ξ→n+1+awhich is the desired result.Theorem 19.
Under the assumptions as given in Theorem12, we have for λ∈C+, wφ a.e. y∈C[0,T] and PZn a.e. ξ→n,η→n∈Rn+1(57)TλΨ∗Hλ∣Zn·,ξ→n∣Zny,η→n=TλΨ∣Zn12y+2-1a,12η→n+ξ→n-2-1a→n×TλH∣Zn12y-a,12η→n-ξ→n+a→n.Proof.
Forλ∈C+, the existence of Tλ[[(Ψ∗H)λ∣Zn](·,ξ→n)∣Zn](y,η→n) follows from Theorems 8 and 12. We have for wφ a.e. y∈C[0,T] and PZn a.e. ξ→n,η→n∈Rn+1(58)TλΨ∗Hλ∣Zn·,ξ→n∣Zny,η→n=∫L20,T2B1y+a+Pb,n+1η→n-a→n,w12,w22,a,ξ→n-a→n×A1λ,12w1-w2A1λ,12w1+w2∫R2rf12v→,y+2a+Pb,n+1η→n+ξ→n-2a→n+w→+u→Bg12v→,y+Pb,n+1η→n-ξ→n+w→-u→BA212w1-w2,u→A212×w1+w2,w→Φrλ,12w1-w2,u→Φrλ,12w1+w2,w→Ψrλ,0→,Ir,u→Ψrλ,0→,Ir,w→du→,w→dσF×σGw1,w2.Let α→=1/2(w→+u→) and β→=1/2(w→-u→). By the change of variable theorem we have (59)TλΨ∗Hλ∣Zn·,ξ→n∣Zny,η→n=∫L20,T2B112y,w1,0,2a,12η→n+ξ→n-2a→nB112y,w2,0,0,12η→n-ξ→nA1λ,w1A1λ,w2∫R2rf12v→,y+2a+Pb,n+1η→n+ξ→n-2a→n+α→Bg12v→,y+Pb,n+1η→n-ξ→n+β→BΦrλ,12w1-w2,12α→-β→Φrλ,12w1+w2,12α→+β→A2w1,α→A2w2,β→Ψrλ,0→,Ir,α→Ψrλ,0→,Ir,β→dα→,β→dσF×σGw1,w2using the same process as used in the proof of Theorem 18. From a long calculation it is not difficult to show (60)Φrλ,w1,α→Φrλ,w2,β→=Φrλ,w1-w22,α→-β→2×Φrλ,w1+w22,α→+β→2so that we have by Theorem 8(61)TλΨ∗Hλ∣Zn·,ξ→n∣Zny,η→n=TλΨ∣Zn12y+2-1a,12η→n+ξ→n-2-1a→n×TλH∣Zn12y-a,12η→n-ξ→n+a→nwhich completes the proof.By Theorems4, 7, 8, 10, 13, 14, 18, and 19, we have the following two theorems.Theorem 20.
LetΨ and H be as given in Theorem 11, where Fr,Gr∈A(1). Let q be a nonzero real number. Then we have for wφ a.e. y∈C[0,T](62)Tq1Ψ∗Hq∣Zn+1·,ξ→n+1∣Zn+1y,η→n+1=Tq1Ψ∣Zn+112y+2-1a,12η→n+1+ξ→n+1-2-1aTq1H∣Zn+112y-a,12η→n+1-ξ→n+1+afor PZn+1 a.e. ξ→n+1,η→n+1∈Rn+2, and for PZn a.e. ξ→n,η→n∈Rn+1(63)Tq1Ψ∗Hq∣Zn·,ξ→n∣Zny,η→n=Tq1Ψ∣Zn12y+2-1a,12η→n+ξ→n-2-1a→n×Tq1H∣Zn12y-a,12η→n-ξ→n+a→n.Theorem 21.
Let the assumptions be as given in Theorem20 except for Gr∈A(2). Moreover suppose that σG is concentrated on Mh-1(V). Then we have (64)Tq2Ψ∗Hq∣Zn+1·,ξ→n+1∣Zn+1y,η→n+1=Tq1Ψ∣Zn+112y+2-1a,12η→n+1+ξ→n+1-2-1aTq2H∣Zn+112y-a,12η→n+1-ξ→n+1+a.In addition, suppose that 〈Mhvj,αn+1〉=0 for j=1,…,r. Then we have (65)Tq2Ψ∗Hq∣Zn·,ξ→n∣Zny,η→n=Tq1Ψ∣Zn12y+2-1a,12η→n+ξ→n-2-1a→n×Tq2H∣Zn12y-a,12η→n-ξ→n+a→n.


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

In this section we investigate change of scales for evaluating the conditional expectations of the functions as described in the previous sections.Let{e1,e2,…} be a complete orthonormal basis of L2[0,T] containing C. For m∈N, λ∈C, and x∈C[0,T] let Hm(λ,x)=exp{1-λ/2∑j=1m(ej,x)2}. Let q be a nonzero real number and let {λm}m=1∞ be a sequence in C+ converging to -iq as m approaches ∞.Theorem 22.
Let1≤p≤∞ and let Ψ be as given in Theorem 4. Then we have for λ∈C+, wφ a.e. y∈C[0,T] and PZn+1 a.e. ξ→n+1∈Rn+2(66)TλΨ∣Zn+1y,ξ→n+1=limm→∞⁡λm/2∫C0,THmλ,xΨy+Zb,n+1x,·+Pb,n+1ξ→n+1dwφx.Moreover if p=1, then Tq(1)[Ψ∣Zn+1](y,ξ→n+1) is given by the right-hand side of (66) with replacing λ by λm.Proof.
Form>r, λ>0 and a.e. ξ→n+1∈Rn+2 we have by the same process as used in the proof of Theorem 4(67)∫C0,THmλ,xΨy+Zb,n+1x,·+Pb,n+1ξ→n+1dwφx=∫L20,TB1y,v,0,A,ξ→n+1∫C0,Texp1-λ2∑j=1mej,x2+iP⊥Mhv,x1-λ2∑j=1mej,x2×fP⊥Mhv→,x+v→,y+A+Pb,n+1ξ→n+1dwφxdσv=12πm+1/2∫L20,TB1y,v,0,A,ξ→n+1∫Rm+1exp⁡-12um+12-λ2∑j=1muj2+i∑j=1m+1cjP⊥Mhvujfu→B+v→,y+A+Pb,n+1ξ→n+1du→,ur+1,…,um+1dσv,where u→=(u1,…,ur) and B1 is given by (22). By a simple calculation we have (68)λm/2∫C0,THmλ,xΨy+Zb,n+1x,·+Pb,n+1ξ→n+1dwφx=∫L20,TB1y,v,0,A,ξ→n+1∫Rrexp-12∑j=1mcjP⊥Mhv2P⊥Mhv2-∑j=1mcjP⊥Mhv2-12λ∑j=r+1mcjP⊥Mhv2fu→B+v→,y+A+Pb,n+1ξ→n+1A2v,u→Ψrλ,0→,Ir,u→du→dσvwhich holds for λ∈C+ by an analytic continuation. Now, by the dominated convergence theorem and the Parseval identity, we have (66). The remainder part of the theorem follows from the dominated convergence theorem.Using a similar process as used in the proofs of Theorems8, 22, and 23, we can prove the following theorems by the dominated convergence theorem.Theorem 23.
Let1≤p≤∞ and let Ψ be as given in Theorem 4. Then we have for λ∈C+, wφ a.e. y∈C[0,T] and PZn a.e. ξ→n=(ξ0,…,ξn)∈Rn+1(69)TλΨ∣Zny,ξ→n=limm→∞⁡λm/2∫R∫C0,TΨ1λ,aT-atn,bT-btn,ξn+1-ξn×Hmλ,xΨy+Zb,n+1x,·+Pb,n+1ξ→n+1dwφxdξn+1,where ξ→n+1=(ξ0,…,ξn,ξn+1). Moreover if p=1, then Tq(1)[Ψ∣Zn](y,ξ→n) is given by the right-hand side of (69) with replacing λ by λm.Theorem 24.
Let the assumptions be as given in Theorem11. Then we have for λ∈C+, wφ a.e. y∈C[0,T] and PZn+1 a.e. ξ→n+1∈Rn+2(70)Ψ∗Hλ∣Zn+1y,ξ→n+1=limm→∞⁡λm/2∫C0,THmλ,xΨ12y+Zb,n+1x,·+Pb,n+1ξ→n+1×H12y-Zb,n+1x,·-Pb,n+1ξ→n+1dwφx.Moreover if p1=p2=1, then [(Ψ∗H)q∣Zn+1](y,ξ→n+1) is given by the right-hand side of (70) with replacing λ by λm.Theorem 25.
Let the assumptions be as given in Theorem11. Then we have for λ∈C+, wφ a.e. y∈C[0,T] and PZn a.e. ξ→n=(ξ0,…,ξn)∈Rn+1(71)Ψ∗Hλ∣Zny,ξ→n=limm→∞⁡λm/2∫R∫C0,TΨ1λ,aT-atn,bT-btn,ξn+1-ξn×Hmλ,xΨ12y+Zb,n+1x,·+Pb,n+1ξ→n+1H12y-Zb,n+1x,·-Pb,n+1ξ→n+1dwφxdξn+1,where ξ→n+1=(ξ0,…,ξn,ξn+1). Moreover if p1=p2=1, then [(Ψ∗H)q∣Zn](y, ξ→n) is given by the right-hand side of (71) with replacing λ by λm.Remark 26.
(1)
In the evaluation ofTλ[Ψ∣Zn+1] for λ>0, the scale λ-1/2 in Ψ is moved into Hm on (66). This is the reason why (66) is called a change of scale formula for the transform. (2)
All the results of this paper are independent for a particular choice of the initial distributionφ. (3)
The results of this paper generalize most of theorems in [8, 9, 13, 18].



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