1. Introduction
Let C0[0,T] denote the classical Wiener space, the space of continuous real-valued functions x on [0,T] with x(0)=0. It is well known that Wiener measure and Wiener measurability are not invariant under change of scale and under translation [1, 2]. As integral transforms, change of scale formulas for Wiener integrals of various functions was developed on the classical and abstract Wiener spaces [3–7]. Further change of scale formulas for conditional Wiener integrals was introduced by the author and his coauthors [8–10]. In fact change of scale formulas for conditional Wiener integrals was established on C0[0,T], on the infinite dimensional Wiener space, and on C[0,T], an analogue of Wiener space [11] which is the space of real-valued continuous paths on [0,T]. Some difficulties in studying the transforms for the conditional Wiener integrals of cylinder functions which play important roles in Feynman integration theories are that they cannot be expressed in terms of the original cylinder functions. To avoid these difficulties, modified cylinder functions expressed by a polygonal function with projection vectors satisfying orthonormality were used to derive the change of scale transforms [8–10]. In this paper we use multivariate normal distributions so that the orthonormalization process of projection vectors which are needed to establish the conditional Wiener integrals can be removed in the existing change of scale transforms. Consequently the transforms in the present paper can be expressed by the cylinder function itself and generalize the results in [8–10].

Let a∈C[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=∫0thsdxs+x0+atfor x∈C[0,T] and t∈[0,T], where the integral denotes a generalized Paley-Wiener-Zygmund stochastic integral. 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.Using simple formulas for generalized conditional Wiener integrals on C[0,T] with the conditioning functions Zn and Zn+1 [12] we evaluate generalized analytic conditional Wiener integrals of the following generalized cylinder function: (3)FZx=f∫0Tv1sdZx,s,…,∫0TvrsdZx,s,where f∈Lp(Rr) with 1≤p≤∞ and v1,…,vr is an orthonormal subset of L2[0,T]. We then establish various change of scale transforms for the generalized analytic conditional Wiener integrals of FZ with Zn and Zn+1. In these evaluation formulas and scale transforms we use multivariate normal distributions so that Gram-Schmidt orthonormalization process of P⊥hv1,…,P⊥hvr can be removed in the existing change of scale transforms for a suitable orthogonal projection P⊥ on L2[0,T]. In contrast with the existing change of scale transforms in [8–10], the transforms in this paper are expressed in terms of the cylinder function FZ itself and generalize some results in those references.

2. An Analogue of Wiener Space and Preliminary Results
In this section we will introduce the analogue of Wiener space C[0,T] and the preliminary results which are needed in the following sections.

For a positive real T 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 (4)Jt→x=xt0,xt1,…,xtn.For Bj (j=0,1,…,n) in B(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 for t→=(t0,t1,…,tn) and u→=(u1,…,un)(6)Wnt→,u→,u0=exp-12∑j=1nuj-uj-12tj-tj-1.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 [11]. Let ek:k=1,2,… be a complete orthonormal subset of L2[0,T] such that each ek is of bounded variation. For v∈L2[0,T] and x in C[0,T] let (7)v,x=limn→∞∑k=1n∫0Tv,ekektdxtif the limit exists, where ·,· denotes the inner product over L2[0,T]. (v,x) is called the Paley-Wiener-Zygmund integral of v associated with x.

Let C 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 X be a random vector on C[0,T] assuming that the value space of X is a normed space with the Borel σ-algebra. Then we have the conditional expectation EF∣X of F given X from a well-known probability theory. Furthermore there exists a PX-integrable complex valued function ψ on the value space of X such that EF∣X(x)=(ψ∘X)(x) for wφ a.e. x∈C[0,T], where PX is the probability distribution of X. The function ψ is called the conditional wφ-integral of F given X and it is also denoted by EF∣X.

Let 0=t0<t1<⋯<tn<tn+1=T be a partition of [0,T], where n is a fixed nonnegative integer. Let h∈L2[0,T] be of bounded variation with h≠0 a.e. on [0,T]. For j=1,…,n,n+1 let (8)αj=1χtj-1,tjhχtj-1,tjh,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 Pv=∑j=1n+1v,αjαj and let P⊥:L2[0,T]→V⊥ be an orthogonal projection. Let a∈C[0,T] and let it be absolutely continuous. Define stochastic processes X,Z:C[0,T]×[0,T]→R by (9)Xx,t=hχ0,t,x,Zx,t=hχ0,t,x+x0+atfor x∈C[0,T] and t∈[0,T]. Define random vectors Zn:C[0,T]→Rn+1 and Zn+1:C[0,T]→Rn+2 by (10)Znx=Zx,t0,Zx,t1,…,Zx,tn,Zn+1x=Zx,t0,Zx,t1,…,Zx,tn,Zx,tn+1for x∈C[0,T]. 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(11)Pb,n+1ft=∑j=1n+1btj-btbtj-btj-1ftj-1+bt-btj-1btj-btj-1ftjχtj-1,tjt+f0χ0tfor t∈[0,T], where χ(tj-1,tj] and χ0 denote the indicator functions on the interval [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 (11) with f(tj) replaced by ξj for j=0,1,…,n,n+1. If ξ→n=(ξ0,ξ1,…,ξn)∈Rn+1, Pb,n(ξ→n) is interpreted as Pb,n+1(ξ→n+1)χ[0,tn] on [0,T]. For x∈C[0,T] and t∈[0,T] let(12)At=at-Pb,n+1at,(13)Xb,n+1x,t=Xx,t-Pb,n+1Xx,·t,(14)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(15)Ψrλ,a→,Ar,u→=λr2πrAr1/2exp-λ2Ar-1u→-a→,u→-a→R,where 〈·,·〉R denotes the dot product on Rr. For a function F:C[0,T]→C let FZ(x)=F(Z(x,·)) for x∈C[0,T]. By Theorems 6 and 7 in [12], we have the following theorems.

Theorem 1.
Let F be a complex valued function on C[0,T] and let FZ be integrable over C[0,T]. Then for PZn+1 a.e. ξ→n+1∈Rn+2(16)EFZ∣Zn+1ξ→n+1=EFZb,n+1x,·+Pb,n+1ξ→n+1,where Zb,n+1 is given by (14), PZn+1 is the probability distribution of Zn+1 on (Rn+2,B(Rn+2)), and the expectation on the right hand side of the equation is taken over the variable x.

Theorem 2.
Let FZ be integrable over C[0,T] and let PZn be the probability distribution of Zn on (Rn+1,B(Rn+1)). Then for PZn a.e. ξ→n=(ξ0,ξ1,…,ξn)∈Rn+1(17)EFZ∣Znξ→n=∫RΨ11,aT-atn,bT-btn,ξn+1-ξnEFZb,n+1x,·+Pb,n+1ξ→n+1dξn+1,where ξ→n+1=(ξ0,ξ1,…,ξn,ξn+1) and Ψ1 is given by (15) with r=1.

For λ>0 and x∈C[0,T] let FZλ(x)=FZ(λ-1/2x), Znλ(x)=Zn(λ-1/2x), and Zn+1λ(x)=Zn+1(λ-1/2x). Suppose that E[FZλ] exists. By Theorem 1(18)EFZλ∣Zn+1λξ→n+1=EFλ-1/2Xb,n+1x,·+A+Pb,n+1ξ→n+1for PZn+1λ a.e. ξ→n+1∈Rn+2, where A and Xb,n+1 are given by (12) and (13), respectively, and PZn+1λ is the probability distribution of Zn+1λ on (Rn+2,B(Rn+2)). By Theorem 2 we also have for PZnλ a.e. ξ→n=(ξ0,ξ1,…,ξn) ∈Rn+1(19)EFZλ∣Znλξ→n=∫RΨ1λ,aT-atn,bT-btn,ξn+1-ξnEFλ-1/2Xb,n+1x,·+A+Pb,n+1ξ→n+1dξn+1,where ξ→n+1=(ξ0,ξ1,…,ξn,ξn+1) and PZnλ is the probability distribution of Znλ on (Rn+1,B(Rn+1)).

Let IFZλ(ξ→n+1) and KFZλ(ξ→n) be the right hand sides of (18) and (19), respectively. If IFZλ(ξ→n+1) has an analytic extension Jλ∗(FZ)(ξ→n+1) on C+, then it is called the conditional analytic Wiener wφ-integral of FZ given Zn+1 with the parameter λ and is denoted by (20)EanwλFZ∣Zn+1ξ→n+1=Jλ∗FZξ→n+1for ξ→n+1∈Rn+2. Moreover if, for nonzero real q, EanwλFZ∣Zn+1(ξ→n+1) has a limit as λ approaches to -iq through C+, then it is called the conditional analytic Feynman wφ-integral of FZ given Zn+1 with the parameter q and is denoted by (21)EanfqFZ∣Zn+1ξ→n+1=limλ→-iqEanwλFZ∣Zn+1ξ→n+1.Replacing IFZλ(ξ→n+1) by KFZλ(ξ→n), we define EanwλFZ∣Zn(ξ→n) and EanfqFZ∣Zn(ξ→n). If E[FZλ] exists for λ>0 and it has an analytic extension Jλ∗(FZ) on C+, then we call Jλ∗(FZ) the analytic Wiener wφ-integral of F over C[0,T] with parameter λ and it is denoted by (22)EanwλFZ=Jλ∗FZ.Eanfq[FZ] is also defined by (23)EanfqFZ=limλ→-iqEanwλFZ,if it exists, where the limit is taken through C+.

Applying Theorem 2.3 in [13], we can easily prove the following theorem.

Theorem 3.
Let h1,h2,…,hr be an orthonormal system of L2[0,T]. Then (h1,·),…,(hr,·) are independent and each (hi,·) has the standard normal distribution. Moreover if f:Rr+1→R is Borel measurable, then (24)∫C0,Tfx0,h1,x,…,hr,xdwφx=∗∫R∫Rrfu0,u→Ψr1,0→,Ir,u→du→ dφu0,where Ir is the identity matrix on Rr and =∗ means that if either side exists then both sides exist and they are equal.

Since φ is a probability measure on R we have the following corollary.

Corollary 4.
Under the assumptions as given in Theorem 3(25)∫C0,Tfh1,x,…,hr,xdwφx=∗∫Rrfu→Ψr1,0→,Ir,u→du→if f:Rr→R is Borel measurable.

The following lemmas are useful to prove the results in the next sections and their proofs are simple.

Lemma 5.
Let v∈L2[0,T]. Then for wφ a.e. x∈C[0,T](26)v,Pb,n+1Xx,·=PMhv,x,where Mh:L20,T→L2[0,T] is the multiplication operator defined by (27)Mhu=hu, for u∈L20,T.

Lemma 6.
Let v∈L2[0,T], ξ→n+1=(ξ0,ξ1,…,ξn,ξn+1)∈Rn+2, and (28)v,Pb,nξ→n=∑j=1nvαj,αjξj-ξj-1,where ξ→n=(ξ0,ξ1,…,ξn). Then (29)v,Pb,n+1ξ→n+1=∑j=1n+1vαj,αjξj-ξj-1=v,Pb,nξ→n+vαn+1,αn+1ξn+1-ξn.

Remark 7.
(1) The multiplication operator Mh in Lemma 5 is well defined because h is of bounded variation which implies the boundedness of h. Mh will denote the operator as given in the lemma unless otherwise specified.

(2) For ξ→n=(ξ0,ξ1,…,ξn)∈Rn+1 it is possible that Pb,n(ξ→n)∉C[0,T] if ξn≠0. In this case the symbol (v,Pb,n(ξ→n)) does not mean the Paley-Wiener-Zygmund integral of v∈L2[0,T]. It is only a formal expression for ∑j=1nvαj,αj(ξj-ξj-1) which is as given in Lemma 6.

3. Multivariate Normal Distributions
In this section we derive a multivariate normal distribution which will be needed in the next section.

Lemma 8.
Let v1,…,vr be a set of independent vectors in L2[0,T]. Then the covariance matrix Σ=[aij]r×r of the random variables (vl,·), l=1,…,r, exists and is positive definite. Moreover aij is given by(30)aij=vi,vjand the determinant Σ of Σ is positive so that Σ is nonsingular and the inverse matrix Σ-1 of Σ is also positive definite.

Proof.
By Theorem 3(31)vi2+2∫C0,Tvi,xvj,xdwφx+vj2=∫C0,Tvi+vj,x2dwφx=vi2+2vi,vj+vj2so that the covariance aij of (vi,·) and (vj,·) is given by aij=vi,vj which proves (30). We have for c→=(c1,…,cr)∈Rr(32)Σc→,c→R=∑l=1r ∑j=1raljclcj=∑j=1rcjvj2≥0.Moreover if 〈Σc→,c→〉R=0, then ∑j=1rcjvj=0 which implies cj=0 for j=1,…,r by the assumption. Thus the covariance matrix Σ is positive definite. Since Σ is symmetric and positive definite, the eigenvalues λ1,…,λr of Σ are real and positive. Since Σ=∏j=1rλj>0, Σ is invertible. Since (33)Σ-1c→,c→R=c→,Σ-1c→R=ΣΣ-1c→,Σ-1c→R≥0,〈Σr-1c→,c→〉R=0 implies Σr-1c→=0→; that is, c→=0→ and Σ-1 is positive definite.

For simplicity let (34)v→,x=v1,x,…,vr,xfor x∈C[0,T] and v1,…,vr⊆L2[0,T].

By Lemma 8, Theorem 4 of [14], and the change of variable theorem, we have the following theorem.

Theorem 9.
Let the assumptions and notations be as given in Lemma 8. Then for every Borel measurable function f:Rr+1→C(35)∫C0,Tfx0,v→,xdwφx=∗∫R∫Rrfu0,u→Ψr1,0→,Σ,u→du→ dφu0=∗∫R∫Rrfu0,Σ1/2u→Ψr1,0→,Ir,u→du→ dφu0,where Ψr is given by (15), Ir is the identity matrix on R, and Σ1/2 is the positive definite matrix satisfying (Σ1/2)2=Σ.

By the same process as used in Lemma 2.1 of [15], Theorem 9, and the change of variable theorem, we have the following corollary.

Corollary 10.
Let v1,…,vr be a subset of L2[0,T] and suppose that Mhv1,…,Mhvr is an independent set. Then the random vector (v→,Z(x,·)) has the multivariate normal distribution with mean vector (v→,a) and covariance matrix ΣMh=[〈Mhvi,Mhvj〉]r×r. Moreover, for any Borel measurable function f:Rr+1→C, we have (36)∫C0,Tfx0,v→,Zx,·dwφx=∗∫R∫Rrfu0,u→Ψr1,v→,a,ΣMh,u→du→ dφu0=∗∫R∫Rrfu0,ΣMh1/2u→+v→,aΨr1,0→,Ir,u→du→ dφu0.

4. Analytic Feynman Integrals and Conditional Analytic Feynman Integrals
We begin this section with introducing the cylinder function on the analogue of Wiener space. Let v1,v2,…,vr be an orthonormal subset of L2[0,T], let r be any positive integer, let 1≤p≤∞, and let F be given by(37)Fx=fv→,xfor wφ a.e. x∈C[0,T], where f∈Lp(Rr). Without loss of generality we can take f to be Borel measurable. In the following theorem we evaluate the Wiener and Feynman integrals of FZ.

Theorem 11.
Let F be given by (37) with 1≤p≤∞ and suppose that Mhv1,…,Mhvr is an independent subset of L2[0,T]. Then for λ∈C+(38)EanwλFZ=∫Rrfu→Ψrλ,v→,a,ΣMh,u→du→=∫RrfΣMh1/2u→+v→,aΨrλ,0→,Ir,u→du→,where ΣMh=[〈Mhvi,Mhvj〉]r×r and Ψr is given by (15). Moreover if p=1, then, for a nonzero real q, Eanfq[FZ] is given by the right hand side of (38) with replacing λ by -iq.

Proof.
Let λ>0. Replacing h by λ-1/2h we have by Corollary 10(39)∫C0,TFZλxdwφx=∫Rrfu→Ψrλ,v→,a,ΣMh,u→du→=∫RrfΣMh1/2u→+v→,aΨrλ,0→,Ir,u→du→.For any real N>0(40)exp-N2ΣMh-1u→-v→,a,u→-v→,aR=exp-N2ΣMh-1/2u→-v→,aR2which is integrable over u→, so that we have the theorem by the change of variable theorem, Morera’s theorem, the uniqueness of analytic continuation, and the dominated convergence theorem.

Let e11,…,e1r be the orthonormal set obtained from Mhv1,…,Mhvr by the Gram-Schmidt orthonormalization process. For l=1,…,r let Mhvl=∑j=1rαlje1j be the linear combination of e1j’s and let B1=[αlj]r×r be the coefficient matrix of the combinations. Since(41)Mhvi,Mhvj=∑l=1r ∑k=1rαile1l,αjke1k=∑l=1rαilαjlwe have ΣMh=B1B1T, where B1T is the transpose of B1.

We now have the following corollary by (41), Theorem 11, and the change of variable theorem.

Corollary 12.
Under the assumptions as given in Theorem 11(42)EanwλFZ=∫Rrfu→Ψrλ,v→,a,B1B1T,u→du→=∫RrfB1u→+v→,aΨrλ,0→,Ir,u→du→.Moreover if p=1, then Eanfq[FZ] is given by the right hand side of (42) with replacing λ by -iq.

Theorem 13.
Let F be given by (37) with 1≤p≤∞ and suppose that P⊥Mhv1,…,P⊥Mhvr is an independent subset of L2[0,T]. Then for λ∈C+(43)EanwλFZ∣Zn+1ξ→n+1=∫Rrfu→Ψrλ,v→,A+Pb,n+1ξ→n+1,ΣP⊥,u→du→=∫RrfΣP⊥1/2u→+v→,A+Pb,n+1ξ→n+1Ψrλ,0→,Ir,u→du→for a.e. ξ→n+1∈Rn+2, where ΣP⊥=[〈P⊥Mhvi,P⊥Mhvj〉]r×r and A is given by (12). Moreover if p=1, then, for a nonzero real q, EanfqFZ∣Zn+1 is given by the right hand side of (43) with replacing λ by -iq.

Proof.
For j=1,…,r and wφ a.e. x∈C[0,T] we have by Lemma 5(44)vj,Xb,n+1x,·=vj,Xx,·-Pb,n+1Xx,·=Mhvj-PMhvj,x=P⊥Mhvj,xso that for λ>0 and ξ→n+1∈Rn+2 we have by Corollary 10(45)IFZλξ→n+1=∫C0,TFλ-1/2Xb,n+1x,·+A+Pb,n+1ξ→n+1dwφx=∫C0,Tfλ-1/2P⊥Mhv1,x,…,P⊥Mhvr,x+v→,A+Pb,n+1ξ→n+1dwφx=∫Rrfu→Ψrλ,v→,A+Pb,n+1ξ→n+1,ΣP⊥,u→du→=∫RrfΣP⊥1/2u→+v→,A+Pb,n+1ξ→n+1Ψrλ,0→,Ir,u→du→,where Ψr is given by (15). By the Morera and the dominated convergence theorem we have the theorem.

Let e21,…,e2r be the orthonormal set obtained from P⊥Mhv1,…,P⊥Mhvr by the Gram-Schmidt orthonormalization process. For l=1,…,r let P⊥Mhvl =∑j=1rβlje2j be the linear combination of e2j’s and let B2=[βlj]r×r be the coefficient matrix of the combinations.

We now have the following corollary by a similar calculation of (41).

Corollary 14.
Under the assumptions as given in Theorem 13(46)EanwλFZ∣Zn+1ξ→n+1=∫Rrfu→Ψrλ,v→,A+Pb,n+1ξ→n+1,B2B2T,u→du→=∫RrfB2u→+v→,A+Pb,n+1ξ→n+1Ψrλ,0→,Ir,u→du→.Moreover if p=1, then EanfqFZ∣Zn+1 is given by the right hand side of (46) with replacing λ by -iq.

For v→=(v1,…,vr), u→∈Rr, λ∈C, and any nonsingular positive r×r matrix Ar on R let (47)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→=1ΛAr1/2expλ2ΛArbT-btnu→,Ar-1v→αn+1,αn+1R2.With the above notations, we have the following theorem.

Theorem 15.
Let F be given by (37) with 1≤p≤∞. Then for λ∈C+(48)EanwλFZ∣Znξ→n=∫RrfΣP⊥1/2u→+v→,A+v→,Pb,nξ→n+aT-atnv→αn+1,αn+1Ψrλ,0→,Ir,u→Φrλ,v→,ΣP⊥1/2,u→du→for a.e. ξ→n∈Rn+1, where (v→,Pb,n(ξ→n))=(v1,Pb,n(ξ→n),…,(vr,Pb,n(ξ→n))). If p=1, then, for a nonzero real q, EanfqFZ∣Zn is given by the right hand side of the above equality with replacing λ by -iq. Moreover the matrix ΣP⊥1/2 can be replaced by B2 in the above results.

Proof.
For ξ→n=(ξ0,ξ1,…,ξn)∈Rn+1 let ξ→n+1=(ξ0,ξ1,…,ξn,ξn+1), where ξn+1∈R. For λ>0 we have by Lemma 6, Theorem 13, and the change of variable theorem (49)KFZλξ→n=∫R∫RrΨ1λ,aT-atn,bT-btn,ξn+1-ξnfΣP⊥1/2u→+v→,A+Pb,n+1ξ→n+1Ψrλ,0→,Ir,u→du→ dξn+1=∫Rr∫RΨ1λ,aT-atn,bT-btn,ξn+1-ξnfΣP⊥1/2u→+v→,A+v→,Pb,nξ→n+ξn+1-ξnv→αn+1,αn+1Ψrλ,0→,Ir,u→dξn+1du→=∫Rr∫RΨ1λ,0,bT-btn,ξn+1fΣP⊥1/2u→+v→,A+v→,Pb,nξ→n+aT-atnv→αn+1,αn+1+v→αn+1,αn+1ξn+1Ψrλ,0→,Ir,u→dξn+1du→.Using the same method as used in the proof of Theorem 3.2 in [9] (50)EanwλFZ∣Znξ→n=∫RrfΣP⊥1/2u→+v→,A+v→,Pb,nξ→n+aT-atnv→αn+1,αn+1Ψrλ,0→,Ir,u→Φrλ,v→,ΣP⊥1/2,u→du→by the Schwarz inequality and the Morera theorem. The final results follow from the dominated convergence theorem and by a similar calculation of (41).

Remark 16.
(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.

(2) It does not mean that B1=ΣMh1/2 in the equations of Corollary 12 and B2=ΣP⊥1/2 in the equations of Corollary 14 and of Theorem 15. They satisfy only the following equations: (51)B1B1T=ΣMh=ΣMh1/22,B2B2T=ΣP⊥=ΣP⊥1/22.

5. Integral Transforms with Change of Scales
In this section we derive change of scale transforms for the generalized conditional Wiener integrals of the function FZ which is introduced in the previous section. To derive these scale transforms we use multivariate normal distributions so that the orthonormalization process of projection vectors can be removed from the change of scale transforms in [8–10] and the transforms are expressed in terms of FZ itself.

For λ∈C and x∈C[0,T] let(52)K1λ,x=exp1-λ2ΣMh-1/2v→,Xx,·R2=exp1-λ2B1-1v→,Xx,·R2.

We now have the following theorem.

Theorem 17.
Let 1≤p≤∞ and let F be given by (37). Then for λ∈C+(53)EanwλFZ=λr/2∫C0,TK1λ,xFZxdwφx,where K1 is given by (52). If p=1 and q is a nonzero real number, then (54)EanfqFZ=limm→∞λmr/2∫C0,TK1λm,xFZxdwφxfor any sequence {λm}m=1∞ in C+ converging to -iq as m approaches to ∞.

Proof.
For λ∈C+ we have by (41), Corollary 10, and Theorem 11(55)λr/2∫C0,TK1λ,xFZxdwφx=λr/2∫C0,Tfv→,Zx,·exp1-λ2ΣMh-1v→,Xx,·,v→,Xx,·Rdwφx=λr/2∫C0,Tfv→,Zx,·exp1-λ2ΣMh-1v→,Zx,·-v→,a,v→,Zx,·-v→,aRdwφx=λr/2∫Rrfu→Ψr1,v→,a,ΣMh,u→exp1-λ2ΣMh-1u→-v→,a,u→-v→,aRdu→=∫Rrfu→Ψrλ,v→,a,ΣMh,u→du→=EanwλFZ,where Ψr is given by (15). If p=1, the final result immediately follows from the dominated convergence theorem.

For λ∈C, x∈C[0,T], and ξ→n+1∈Rn+2 let(56)K2λ,ξ→n+1,x=1ΣP⊥1/2exp-λ2ΣP⊥-1/2v→,Zx,·-A-Pb,n+1ξ→n+1R2=1B2exp-λ2B2-1v→,Zx,·-A-Pb,n+1ξ→n+1R2.

We now have the following theorem.

Theorem 18.
Let 1≤p≤∞ and let F be given by (37). Then for λ∈C+(57)EanwλFZ∣Zn+1ξ→n+1=λrΣMh1/2∫C0,TK10,xK2λ,ξ→n+1,xFZxdwφxfor a.e. ξ→n+1∈Rn+2, where K2 is given by (56). If p=1 and q is a nonzero real number, then (58)EanfqFZ∣Zn+1ξ→n+1=limm→∞λmrΣMh1/2∫C0,TK10,xK2λm,ξ→n+1,xFZxdwφxfor any sequence λmm=1∞ in C+ converging to -iq as m approaches to ∞. Moreover Mh can be replaced by B12 in the above equalities.

Proof.
For λ>0 and a.e. ξ→n+1∈Rn+2 we have by Corollary 10(59)λrΣMh1/2∫C0,TK10,xK2λ,ξ→n+1,xFZxdwφx=λrΣMhΣP⊥1/2∫C0,Tfv→,Zx,·exp12ΣMh-1v→,Zx,·-v→,a,v→,Zx,·-v→,aR-λ2ΣP⊥-1v→,Zx,·-A-Pb,n+1ξ→n+1,v→,Zx,·-A-Pb,n+1ξ→n+1Rdwφx=λrΣMhΣP⊥1/2∫Rrfu→Ψr1,v→,a,ΣMh,u→exp12ΣMh-1u→-v→,a,u→-v→,aR-λ2ΣP⊥-1u→-v→,A+Pb,n+1ξ→n+1,u→-v→,A+Pb,n+1ξ→n+1Rdu→=∫Rrfu→Ψrλ,v→,A+Pb,n+1ξ→n+1,ΣP⊥,u→du→.By (41), the analytic continuation, the dominated convergence theorem, and Theorem 13 we have the theorem.

By Theorems 15 and 18 we have the final theorem.

Theorem 19.
Let 1≤p≤∞ and let F be given by (37). Then for λ∈C+(60)EanwλFZ∣Znξ→n=λrΣMh1/2∫R∫C0,TΨ1λ,aT-atn,bT-btn,ξn+1-ξnK10,xK2λ,ξ→n+1,xFZxdwφxdξn+1for a.e. ξ→n=(ξ0,ξ1,…,ξn)∈Rn+1, where ξ→n+1=(ξ0,ξ1,…,ξn,ξn+1) and Ψ1, K2 are given by (15) and (56), respectively. If p=1 and q is a nonzero real number, then (61)EanfqFZ∣Znξ→n=limm→∞λmrΣMh1/2∫R∫C0,TΨ1λm,aT-atn,bT-btn,ξn+1-ξnK10,xK2λm,ξ→n+1,xFZxdwφxdξn+1for any sequence λmm=1∞ in C+ converging to -iq as m approaches to ∞. Moreover Mh can be replaced by B12 in the above equalities.

Remark 20.
(1) Letting λ=ρ-2 in the theorems of this section, where ρ>0, we have the change of scale formulas for E[FZ(ρ·)], EFZρ·∣Zn(ρ·), and EFZρ·∣Zn+1(ρ·) as integral transforms.

(2) If a=0 and h=1 a.e., then we can obtain Theorems 5.1 and 5.2 in [9].

(3) If a=0, h=1 a.e., and φ=δ0 which is the Dirac measure concentrated at 0, then we can obtain the change of scale transforms in [10].

(4) The results in this paper are independent of a particular choice of the initial distribution φ.