Evaluation Formulas for Generalized Conditional Wiener Integrals with Drift on a Function Space

Abstract

Let 𝐶[0,𝑡] denote a generalized Wiener space, the space of real-valued continuous functions on the interval [0,𝑡] and define a stochastic process 𝑌∶𝐶[0,𝑡]×[0,𝑡]→ℝ by 𝑌(𝑥,𝑠)=∫^𝑠₀ℎ(𝑢)𝑑𝑥(𝑢)+𝑎(𝑠) for 𝑥∈𝐶[0,𝑡] and 𝑠∈[0,𝑡], where ℎ∈𝐿₂[0,𝑡] with ℎ≠0 a.e. and 𝑎 is continuous on [0,𝑡]. Let random vectors 𝑌_𝑛∶𝐶[0,𝑡]→ℝ^𝑛 and 𝑌_𝑛+1∶𝐶[0,𝑡]→ℝ^𝑛+1 be given by 𝑌_𝑛(𝑥)=(𝑌(𝑥,𝑡₁),…,𝑌(𝑥,𝑡_𝑛)) and 𝑌_𝑛+1(𝑥)=(𝑌(𝑥,𝑡₁),…,𝑌(𝑥,𝑡_𝑛),𝑌(𝑥,𝑡_𝑛+1)), where 0<𝑡₁<⋯<𝑡_𝑛<𝑡_𝑛+1=𝑡 is a partition of [0,𝑡]. In this paper we derive a translation theorem for a generalized Wiener integral and then prove that 𝑌 is a generalized Brownian motion process with drift 𝑎. Furthermore, we derive two simple formulas for generalized conditional Wiener integrals of functions on 𝐶[0,𝑡] with the drift and the conditioning functions 𝑌_𝑛 and 𝑌_𝑛+1. As applications of these simple formulas, we evaluate the generalized conditional Wiener integrals of various functions on 𝐶[0,𝑡].

1. Introduction

Let 𝐶₀[0,𝑡] denote the Wiener space, the space of real-valued continuous functions 𝑥 on [0,𝑡] with 𝑥(0)=0. On the space, Yeh [1] introduced an inversion formula that a conditional expectation can be found by a Fourier-transform. But Yeh’s inversion formula is very complicated in its application when the conditioning function is vector-valued. In [2], Park and Skoug derived a simple formula for conditional Wiener integrals on 𝐶₀[0,𝑡] with a vector-valued conditioning function 𝑋∶𝐶₀[0,𝑡]→ℝ^𝑛 given by 𝑋(𝑥)=(𝑥(𝑡₁),…,𝑥(𝑡_𝑛+1)), where 0=𝑡₀<𝑡₁<⋯<𝑡_𝑛<𝑡_𝑛+1=𝑡 is a partition of the interval [0,𝑡]. In their simple formula, they expressed the conditional Wiener integral directly in terms of an ordinary Wiener integral. Using the simple formula in [2], Chang and Skoug [3] investigated the effect that drift has on the conditional Fourier-Feynman transform, the conditional convolution product, and various relationships that occur between them.

On the other hand, let 𝐶[0,𝑡] denote the space of real-valued continuous functions on the interval [0,𝑡]. Im and Ryu [4] introduced a probability measure 𝑤_𝜑 on 𝐶[0,𝑡], where 𝜑 is a probability measure on the Borel class of ℝ. When 𝜑=𝛿₀, the Dirac measure concentrated at 0, 𝑤_𝜑 is exactly the Wiener measure on 𝐶₀[0,𝑡]. On the space 𝐶[0,𝑡], the author [5, 6] derived two simple formulas for the conditional Wiener 𝑤_𝜑-integral of functions on 𝐶[0,𝑡] with the vector-valued conditioning functions 𝑋_𝑛∶𝐶[0,𝑡]→ℝ^𝑛+1 and 𝑋_𝑛+1∶𝐶[0,𝑡]→ℝ^𝑛+2 given by 𝑋_𝑛(𝑥)=(𝑥(𝑡₀),𝑥(𝑡₁),…,𝑥(𝑡_𝑛)) and 𝑋_𝑛+1(𝑥)=(𝑥(𝑡₀),𝑥(𝑡₁),…,𝑥(𝑡_𝑛),𝑥(𝑡_𝑛+1)) which generalize the Park and Skoug’s formula in [2]. Using these formulas with the conditioning functions 𝑋_𝑛 and 𝑋_𝑛+1, he evaluated the conditional Wiener 𝑤_𝜑-integral of function of the form 𝐹_𝑚(𝑥)=∫^𝑡₀(𝑥(𝑠))^𝑚𝑑𝑠 for any positive integer 𝑚.

Let ℎ∈𝐿₂[0,𝑡] with ℎ≠0 a.e. on [0,𝑡], and let 𝑎 be a continuous function on [0,𝑡]. Define a stochastic process 𝑌∶𝐶[0,𝑡]×[0,𝑡]→ℝ by 𝑌(𝑥,𝑠)=∫^𝑠₀ℎ(𝑢)𝑑𝑥(𝑢)+𝑎(𝑠) for 𝑥∈𝐶[0,𝑡] and 𝑠∈[0,𝑡]. Let 𝑌_𝑛∶𝐶[0,𝑡]→ℝ^𝑛 and 𝑌_𝑛+1∶𝐶[0,𝑡]→ℝ^𝑛+1 be given by 𝑌_𝑛(𝑥)=𝑌𝑥,𝑡₁,…,𝑌𝑥,𝑡_𝑛,𝑌_𝑛+1(𝑥)=𝑌𝑥,𝑡₁,…,𝑌𝑥,𝑡_𝑛,𝑌𝑥,𝑡_𝑛+1. (1) In this paper, we derive a translation theorem for a generalized Wiener 𝑤_𝜑-integral, and then prove that 𝑌 is a generalized Brownian motion process with drift 𝑎 and variance ∫^𝑠₀(ℎ(𝑢))²𝑑𝑢 for 𝑠∈[0,𝑡]. Furthermore, we derive two simple formulas for generalized conditional Wiener integrals of functions on 𝐶[0,𝑡] with the drift and the conditioning functions 𝑌_𝑛 and 𝑌_𝑛+1. As applications of these simple formulas, we evaluate the generalized conditional Wiener integrals of functions of the forms 𝐹_𝑚(𝑌(𝑥,⋅)),𝑌𝑥,𝑠₁𝑌𝑥,𝑠₂,_{𝐿2[0,𝑡]}exp{𝑖(𝑣,𝑌(𝑥,⋅))}𝑑𝜎(𝑣) (2) for 𝑥∈𝐶[0,𝑡], where 0<𝑠₁<𝑠₂≤𝑡 and 𝜎 is a complex Borel measure on 𝐿₂[0,𝑡].

2. A Generalized Brownian Motion Process with Drift

In this section, we introduce a generalized Brownian motion process with drift which generalizes the generalized Wiener space 𝐶_𝑎,𝑏[0,𝑇] as given in [7].

Let ℂ and ℂ₊ denote the sets of complex numbers and complex numbers with positive real parts, respectively. Let (𝐶[0,𝑡],ℬ(𝐶[0,𝑡]),𝑤_𝜑) be the analogue of Wiener space associated with a probability measure 𝜑 on the Borel class of ℝ, where ℬ(𝐶[0,𝑡]) denotes the Borel class of 𝐶[0,𝑡] [4]. Let 𝛼∈ℝ, and let 𝜑_𝛼 be a measure on (ℝ,ℬ(ℝ)) such that 𝜑_𝛼(𝐵)=𝜑(𝐵+𝛼) for 𝐵∈ℬ(ℝ). For 𝑣 in 𝐿₂[0,𝑡] and 𝑥 in 𝐶[0,𝑡], let (𝑣,𝑥)=∫^𝑡₀𝑣(𝑠)𝑑𝑥(𝑠) denote the Paley-Wiener-Zygmund integral of 𝑣 according to 𝑥 [4]. Let ℎ∈𝐿₂[0,𝑡] with ℎ≠0 a.e. on [0,𝑡], and let 𝑎 be a continuous function on [0,𝑡]. Define stochastic processes 𝑍,𝑌∶𝐶[0,𝑡]×[0,𝑡]→ℝ by 𝑍(𝑥,𝑠)=∫^𝑠₀ℎ(𝑢)𝑑𝑥(𝑢) and 𝑌(𝑥,𝑠)=∫^𝑠₀ℎ(𝑢)𝑑𝑥(𝑢)+𝑎(𝑠) for 𝑥∈𝐶[0,𝑡] and 𝑠∈[0,𝑡]. Let 𝑏(𝑠)=∫^𝑠₀(ℎ(𝑢))²𝑑𝑢 for 0≤𝑠≤𝑡.

By (3.1) in [4, Theorem 3.1], we have the following translation theorem for a generalized Wiener integral on the analogue of Wiener space.

Theorem 1. Let 𝑔 be of bounded variation on [0,𝑡], let 𝑔₀(𝑠)=∫^𝑠₀ℎ(𝑢)𝑔(𝑢)𝑑𝑢 for 𝑠∈[0,𝑡], and let 𝐹 be a measurable function on 𝐶[0,𝑡]. Then, 𝐹(𝑌(𝑥,⋅)+𝑔₀) is also measurable function of 𝑥 on 𝐶[0,𝑡] and _𝐶[0,𝑡]𝐹(𝑌(𝑥,⋅))𝑑𝑤_𝜑(𝑥)_∗=𝑒^{−‖𝑔‖2/2}_𝐶[0,𝑡]𝐹𝑌(𝑥,⋅)+𝑔₀exp−𝑔,𝑥𝑑𝑤_𝜑𝛼(𝑥), (3) where =^∗ means that if one side exists then both sides exist, and they are equal.

Letting 𝑔=𝜒_[0,𝑠]ℎ, we have the following corollary by Theorem 1.

Corollary 2. Let ℎ be of bounded variation on [0,𝑡] and 𝐹 a measurable function on 𝐶[0,𝑡]. Then, for 𝑠∈[0,𝑡]𝐹(𝑌(𝑥,⋅)+𝑏(min{𝑠,⋅})) is also measurable on 𝐶[0,𝑡] and _𝐶[0,𝑡]𝐹(𝑌(𝑥,⋅))𝑑𝑤_𝜑(𝑥)_∗−1=exp2𝑏(𝑠)_𝐶[0,𝑡]𝐹(𝑌()({{(𝑥,⋅+𝑏min𝑠,⋅}))×exp−𝑍𝑥,𝑠)}𝑑𝑤_𝜑𝛼(𝑥). (4)

Letting 𝐹≡1 in Theorem 1 and Corollary 2, we have the following corollary.

Corollary 3. Under the assumptions as given in Theorem 1 and Corollary 2exp−_𝐶[0,𝑡]𝑔,𝑥𝑑𝑤_𝜑𝛼(𝑥)1=exp2𝑔². (5) In particular for 𝑠∈[0,𝑡]exp{_𝐶[0,𝑡]−𝑍(𝑥,𝑠)}𝑑𝑤_𝜑𝛼(𝑥)1=exp2𝑏(𝑠). (6)

Using the same method as used in the proof of Corollary 3.3 in [4], we have the following corollary.

Corollary 4. Under the assumptions as given in Theorem 1, one has for 𝜆∈ℂ_𝐶[0,𝑡]exp−𝜆𝑔,𝑥𝑑𝑤_𝜑𝛼(𝑥)𝜆=exp²2𝑔². (7) In particular for 𝑠∈[0,𝑡]exp{_𝐶[0,𝑡]−𝜆𝑍(𝑥,𝑠)}𝑑𝑤_𝜑𝛼(𝑥)𝜆=exp²2𝑏(𝑠). (8)

Letting 𝜆=−𝑖𝜉(𝜉∈ℝ) in the equations of Corollary 4, we have the following corollary by Fourier transforms on (𝐶[0,𝑡],𝑤_𝜑𝛼).

Corollary 5. Under the assumptions as given in Theorem 1, let 𝑋(𝑥)=(𝑔,𝑥). Then, 𝑋 is normally distributed with mean zero and variance ‖𝑔‖² on (𝐶[0,𝑡],𝑤_𝜑𝛼) if ‖𝑔‖>0. Moreover, for 𝑠∈(0,𝑡]𝑌(⋅,𝑠) is normally distributed with mean 𝑎(𝑠) and variance 𝑏(𝑠) on (𝐶[0,𝑡],𝑤_𝜑𝛼). In particular, 𝑍(⋅,𝑠) is normally distributed on (𝐶[0,𝑡],𝑤_𝜑𝛼) with mean 0 and variance 𝑏(𝑠).

Using the same method as used in the proof of Theorem 3.5 in [4], we have the following theorem.

Theorem 6. Let {𝑔₁,…,𝑔_𝑚} be an orthonormal subset of 𝐿₂[0,𝑡] such that each 𝑔_𝑗 is of bounded variation. For 𝑗=1,…,𝑚, let 𝑋_𝑗(𝑥)=(𝑔_𝑗,𝑥). Then, 𝑋₁,…,𝑋_𝑚 are independent, and each 𝑋_𝑗 has the standard normal distribution. Moreover if, 𝑓∶ℝ^𝑚→ℂ is Borel measurable, then _𝐶[0,𝑡]𝑓𝑋₁(𝑥),…,𝑋_𝑚(𝑥)𝑑𝑤_𝜑𝛼(𝑥)_∗=12𝜋^𝑚/2_ℝ^𝑚𝑓𝑢₁,𝑢₂,…,𝑢_𝑚−1×exp2^𝑚_𝑗=1𝑢²_𝑗𝑑𝑢₁,𝑢₂,…,𝑢_𝑚. (9)

Theorem 7. Let ℎ be of bounded variation on [0,𝑡], and let 0=𝑠₀<𝑠₁<⋯<𝑠_𝑚≤𝑡. Then, the random vector (𝑌(⋅,𝑠₁),…,𝑌(⋅,𝑠_𝑚)) has a joint density function 𝑊_𝑚 with respect to 𝑤_𝜑𝛼 given by 𝑊_𝑚𝑠₁,…,𝑠_𝑚;𝑢₁,…,𝑢_𝑚=^𝑚_𝑗=112𝜋𝑏𝑠_𝑗−𝑏𝑠_𝑗−1^1/2×exp−^𝑚_𝑗=1𝑢_𝑗−𝑎𝑠_𝑗−𝑢_𝑗−1−𝑎𝑠_𝑗−1²2𝑏𝑠_𝑗−𝑏𝑠_𝑗−1, (10) where 𝑢₀=0, 𝑎(0)=0, and (𝑢₁,…,𝑢_𝑚)∈ℝ^𝑚.

Proof. For 𝑗=1,…,𝑚, let ℎ_𝑗=𝜒_{[𝑠𝑗−1,𝑠𝑗]}ℎ, where 𝜒_{[𝑠𝑗−1,𝑠𝑗]} denotes the indicator function on [𝑠_𝑗−1,𝑠_𝑗]. Let 𝑔_𝑗=(1/𝑏(𝑠_𝑗)−𝑏(𝑠_𝑗−1))ℎ_𝑗. Then, 𝑔₁,…,𝑔_𝑚 are orthonormal, and 𝑌(⋅,𝑠_𝑗)=∑^𝑗_𝑙=1(ℎ_𝑙,⋅)+𝑎(𝑠_𝑗)=∑^𝑗_𝑙=1𝑏(𝑠_𝑙)−𝑏(𝑠_𝑙−1)(𝑔_𝑙,⋅)+𝑎(𝑠_𝑗). Let 𝐵 be a Borel subset of ℝ^𝑚. By Theorem 6, 𝑤_𝜑𝛼𝑌⋅,𝑠₁,…,𝑌⋅,𝑠_𝑚=1∈𝐵2𝜋^𝑚/2_ℝ^𝑚𝜒_𝐵𝑏𝑠₁−𝑏𝑠₀𝑤₁+𝑎𝑠₁,…,_𝑚𝑙=1𝑏𝑠_𝑙−𝑏𝑠_𝑙−1𝑤_𝑙+𝑎𝑠_𝑚−1×exp2^𝑚_𝑗=1𝑤²_𝑗𝑑𝑤₁,…,𝑤_𝑚. (11) For 𝑗=1,…,𝑚, let 𝑢_𝑗=∑^𝑗_𝑙=1𝑏(𝑠_𝑙)−𝑏(𝑠_𝑙−1)𝑤_𝑙+𝑎(𝑠_𝑗). Then, by the change of variable theorem, 𝑤_𝜑𝛼𝑌⋅,𝑠₁,…,𝑌⋅,𝑠_𝑚=∈𝐵_𝐵𝑤_𝑚𝑠₁,…,𝑠_𝑚;𝑢₁,…,𝑢_𝑚𝑑𝑢₁,…,𝑢_𝑚 (12) which completes the proof.

Remark 8. The condition 𝑎(0)=0 in Theorem 7 does not mean that the mean function 𝑎 is continuous on [0,𝑡] with 𝑎(0)=0. In fact, it is only needed to express formally the exponential function exp{−(𝑢₁−𝑎(𝑠₁))²/(2𝑏(𝑠₁))} in 𝑊_𝑚.

By Theorem 7 and the change of variable theorem, we have the following corollary.

Corollary 9. 𝑌 is a Gaussian process with respect to 𝑤_𝜑𝛼, and its covariance function is given by 𝐸𝑌⋅,𝑠₁−𝑎𝑠₁𝑌⋅,𝑠₂−𝑎𝑠₂=𝑏min𝑠₁,𝑠₂for𝑠₁,𝑠₂∈[0,𝑡], (13) so that 𝐸𝑌⋅,𝑠₁𝑌⋅,𝑠₂=𝑏min𝑠₁,𝑠₂+𝑎𝑠₁𝑎𝑠₂. (14)

Using Fourier transforms on (𝐶[0,𝑡],𝑤_𝜑𝛼), we have the following corollary by Theorem 7.

Corollary 10. Let 0≤𝑠₁<𝑠₂≤𝑡. Then, 𝑌(⋅,𝑠₂)−𝑌(⋅,𝑠₁) is normally distributed on (𝐶[0,𝑡],𝑤_𝜑𝛼) with mean 𝑎(𝑠₂)−𝑎(𝑠₁) and variance 𝑏(𝑠₂)−𝑏(𝑠₁).

By Corollary 10, we can prove the following corollary.

Corollary 11. Let 0≤𝑠₁<𝑠₂≤𝑠₃<𝑠₄≤𝑡. Then 𝑌(⋅,𝑠₂)−𝑌(⋅,𝑠₁) and 𝑌(⋅,𝑠₄)−𝑌(⋅,𝑠₃) are independent on (𝐶[0,𝑡],𝑤_𝜑𝛼).

Theorem 12. 𝑌 is a generalized Brownian motion process on (𝐶[0,𝑡],𝑤_𝜑𝛼) determined by the mean function 𝑎 and variance function 𝑏.

Remark 13. (1) A generalized Brownian motion process determined by the mean function 𝑎 and variance function 𝑏 is a Gaussian process whose mean and covariance functions are given by 𝑎 and 𝑏(min{𝑠₁,𝑠₂})−𝑏(0), respectively, for 𝑠₁,𝑠₂∈[0,𝑡] [1].
(2) In [8], the author proved that 𝑌−𝑎 is a generalized Brownian motion process using Theorem 6. Comparing with [8], we proved the results in this paper using only the translation theorem [4, Theorem 3.1] which is another approach to prove 𝑌 being a generalized Brownian motion process.

3. Simple Formulas for Generalized Conditional Wiener Integrals

In this section, we derive two simple formulas for generalized conditional Wiener integrals on (𝐶[0,𝑡],𝑤_𝜑𝛼). For this purpose, we start with this section defining a conditional 𝑤_𝜑𝛼-integral.

Let 𝐹∶𝐶[0,𝑡]→ℂ be integrable on 𝑤_𝜑𝛼 and 𝑋 a random vector on 𝐶[0,𝑡] assuming that the value space of 𝑋 is a normed space equipped with the Borel 𝜎-algebra. Then, we have the conditional expectation 𝐸[𝐹∣𝑋] of 𝐹 given 𝑋 from a well-known probability theory. Furthermore, there exists a 𝑃_𝑋-integrable ℂ-valued function 𝜓 on the value space of 𝑋 such that 𝐸[𝐹∣𝑋](𝑥)=(𝜓∘𝑋)(𝑥) for 𝑤_𝜑𝛼 a.e. 𝑥∈𝐶[0,𝑡], where 𝑃_𝑋 is the probability distribution of 𝑋. The function 𝜓 is called the conditional 𝑤_𝜑𝛼-integral of 𝐹 given 𝑋, and it is also denoted by 𝐸[𝐹∣𝑋]. Let 0=𝑡₀<𝑡₁<⋯<𝑡_𝑛<𝑡_𝑛+1=𝑡 be a partition of [0,𝑡]. For 𝑗=1,…,𝑛+1, let 𝛼_𝑗(𝑠)=(𝑏(𝑡_𝑗)−𝑏(𝑠))/(𝑏(𝑡_𝑗)−𝑏(𝑡_𝑗−1)), 𝛽_𝑗(𝑠)=(𝑏(𝑠)−𝑏(𝑡_𝑗−1))/(𝑏(𝑡_𝑗)−𝑏(𝑡_𝑗−1)) and 𝛾_𝑗(𝑠)=𝛼_𝑗(𝑠)𝛽_𝑗(𝑠)(𝑏(𝑡_𝑗)−𝑏(𝑡_𝑗−1)), where 𝑠∈[𝑡_𝑗−1,𝑡_𝑗]. Define random vectors 𝑌_𝑛∶𝐶[0,𝑡]→ℝ^𝑛 and 𝑌_𝑛+1∶𝐶[0,𝑡]→ℝ^𝑛+1 by 𝑌_𝑛(𝑥)=(𝑌(𝑥,𝑡₁),…,𝑌(𝑥,𝑡_𝑛)) and 𝑌_𝑛+1(𝑥)=(𝑌(𝑥,𝑡₁),…,𝑌(𝑥,𝑡_𝑛),𝑌(𝑥,𝑡_𝑛+1)) for 𝑥∈𝐶[0,𝑡]. For any 𝑥 in 𝐶[0,𝑡], define a polygonal function [𝑌(𝑥,⋅)]_𝑎,𝑏 of 𝑌(𝑥,⋅) by [𝑌(𝑥,⋅)]_𝑎,𝑏(𝑠)=^𝑛+1_𝑗=1𝜒_{(𝑡𝑗−1,𝑡𝑗]}(𝑠)𝛼_𝑗(𝑠)𝑌𝑥,𝑡_𝑗−1+𝛽_𝑗(𝑠)𝑌𝑥,𝑡_𝑗+𝜒_{0}(𝑠)𝑎(0) (15) for 𝑠∈[0,𝑡], where 𝜒_{(𝑡𝑗−1,𝑡𝑗]} and 𝜒_{0} denote indicator functions. For 𝑥∈𝐶[0,𝑡] and 𝜉_𝑛+1=(𝜉₁,…,𝜉_𝑛,𝜉_𝑛+1)∈ℝ^𝑛+1, define polygonal functions [𝑍(𝑥,⋅)]_𝑏 and [𝜉_𝑛+1]_𝑏 of 𝑍(𝑥,⋅) and 𝜉_𝑛+1 by (15), where 𝑌(𝑥,𝑡_𝑗) is replaced by 𝑍(𝑥,𝑡_𝑗) and 𝜉_𝑗, respectively, for 𝑗=1,…,𝑛+1 (𝜉₀=0 and 𝑎(0)=0 formally). Moreover, define a polygonal function [𝑎]_𝑏 of 𝑎 by (15), where 𝑌(𝑥,𝑡_𝑗) is replaced by 𝑎(𝑡_𝑗). For 𝜉_𝑛=(𝜉₁,…,𝜉_𝑛)∈ℝ^𝑛, the symbol [𝜉_𝑛]_𝑏 is understood as 𝜒_[0,𝑡𝑛][𝜉_𝑛+1]_𝑏 on [0,𝑡].

By Theorems 2.4 and 2.9 in [8], we can easily prove the following theorem.

Theorem 14. Let 𝑡_𝑗−1<𝑠<𝑡_𝑗  for some 𝑗∈{1,…,𝑛+1}. Then, 𝑌(⋅,𝑠)−[𝑌(⋅,⋅)]_𝑎,𝑏(𝑠) is normally distributed on (𝐶[0,𝑡],𝑤_𝜑𝛼) with mean 𝑎(𝑠)−[𝑎]_𝑏(𝑠) and variance 𝛾_𝑗(𝑠).

Theorem 15. The process {𝑌(⋅,𝑠)−[𝑌(⋅,⋅)]_𝑎,𝑏(𝑠)∶0≤𝑠≤𝑡} and 𝑌_𝑛+1 are stochastically independent on (𝐶[0,𝑡],𝑤_𝜑𝛼).

Proof. Let 𝑡_𝑙−1<𝑠<𝑡_𝑙 for some 𝑙∈{1,…,𝑛+1}. Then, for 𝑗=1,…,𝑛+1, 𝐸𝑌(⋅,𝑠)−[𝑌(⋅,⋅)]_𝑎,𝑏(𝑠)𝑌⋅,𝑡_𝑗=𝐸𝑍(⋅,𝑠)−[𝑍(⋅,⋅)]_𝑏(𝑠)(𝑠)−[𝑎]+𝑎_𝑏(𝑠)𝑍⋅,𝑡_𝑗+𝑎𝑡_𝑗=𝐸𝑍(⋅,𝑠)−[𝑍(⋅,⋅)]_𝑏(𝑠)𝐸𝑍⋅,𝑡_𝑗+𝑎(𝑠)−[𝑎]_𝑏(𝑠)𝑎𝑡_𝑗 (16) by Corollary 2.4 and Theorems 2.9 and 2.10 in [8]. By Corollary 5 and Theorem 14 we have 𝐸𝑌(⋅,𝑠)−[𝑌(⋅,⋅)]_𝑎,𝑏(𝑠)𝑌⋅,𝑡_𝑗=𝑎(𝑠)−[𝑎]_𝑏(𝑠)𝑎𝑡_𝑗=𝐸𝑌(⋅,𝑠)−[𝑌(⋅,⋅)]_𝑎,𝑏(𝑠)𝐸𝑌⋅,𝑡_𝑗. (17) Since 𝑌(⋅,𝑠)−[𝑌(⋅,⋅)]_𝑎,𝑏(𝑠) and 𝑌(⋅,𝑡_𝑗) are normally distributed on 𝑤_𝜑𝛼, they are independent. The proofs of remainder cases follow easily.

Theorem 16. The processes {𝑌(⋅,𝑠)−[𝑌(⋅,⋅)]_𝑎,𝑏(𝑠)∶𝑡_𝑗−1≤𝑠≤𝑡_𝑗}, where 𝑗=1,…,𝑛+1, are stochastically independent on (𝐶[0,𝑡],𝑤_𝜑𝛼).

Proof. Let 𝑠₁∈[𝑡_𝑙−1,𝑡_𝑙] and 𝑠₂∈[𝑡_𝑗−1,𝑡_𝑗] with 𝑙<𝑗. If 𝑠₁∈{0,𝑡₁,…,𝑡_𝑛+1} or 𝑠₂∈{0,𝑡₁,…,𝑡_𝑛+1}, we can prove easily the independence of 𝑌(⋅,𝑠₁)−[𝑌(⋅,⋅)]_𝑎,𝑏(𝑠₁) and 𝑌(⋅,𝑠₂)−[𝑌(⋅,⋅)]_𝑎,𝑏(𝑠₂). Suppose that 𝑡_𝑙−1<𝑠₁<𝑡_𝑙≤𝑡_𝑗−1<𝑠₂<𝑡_𝑗. By Theorems 3.1 and 2.11 in [8], we have 𝐸𝑌⋅,𝑠₁−[𝑌(⋅,⋅)]_𝑎,𝑏𝑠₁𝑌⋅,𝑠₂−[𝑌(⋅,⋅)]_𝑎,𝑏𝑠₂=𝐸𝑍⋅,𝑠₁−[𝑍(⋅,⋅)]_𝑏𝑠₁+𝑎𝑠₁−[𝑎]_𝑏𝑠₁×𝑍⋅,𝑠₂−[𝑍(⋅,⋅)]_𝑏𝑠₂+𝑎𝑠₂−[𝑎]_𝑏𝑠₂=𝐸𝑍⋅,𝑠₁−[𝑍(⋅,⋅)]_𝑏𝑠₁𝐸𝑍⋅,𝑠₂−[𝑍(⋅,⋅)]_𝑏𝑠₂+𝑎𝑠₁−[𝑎]_𝑏𝑠₁𝑎𝑠₂−[𝑎]_𝑏𝑠₂=𝐸𝑌⋅,𝑠₁−[𝑌(⋅,⋅)]_𝑎,𝑏𝑠₁𝐸𝑌⋅,𝑠₂−[𝑌(⋅,⋅)]_𝑎,𝑏𝑠₂ (18) which completes the proof.

Applying the same method as used in the proof of Theorem 2 in [2, page 383] with Problem 4 of [9, page 216], we have the following theorem by Theorem 15.

Theorem 17. Let 𝐹∶𝐶[0,𝑡]→ℂ be a function, and let 𝐹(𝑌(𝑥,⋅)) be 𝑤_𝜑𝛼-integrable over the variable 𝑥. Then, for a Borel subset 𝐵 of ℝ^𝑛_{𝑌−1𝑛+1(𝐵)}𝐹(𝑌(𝑥,⋅))𝑑𝑤_𝜑𝛼(𝑥)=_𝐵𝐸𝐹𝑌(𝑥,⋅)−[𝑌(𝑥,⋅)]_𝑎,𝑏+𝜉_𝑛+1𝑏𝑑𝑃_𝑌𝑛+1𝜉_𝑛+1, (19) where the expectation is taken over 𝑤_𝜑𝛼 and 𝑃_𝑌𝑛+1 is the probability distribution of  𝑌_𝑛+1 on (ℝ^𝑛+1,ℬ(ℝ^𝑛+1)). Moreover, for 𝑃_𝑌𝑛+1 a.e. 𝜉_𝑛+1∈ℝ^𝑛+1 (hence for a.e. 𝜉_𝑛+1∈ℝ^𝑛+1) 𝐸𝐹(𝑌(𝑥,⋅))∣𝑌_𝑛+1𝜉_𝑛+1=𝐸𝐹𝑌(𝑥,⋅)−[𝑌(𝑥,⋅)]_𝑎,𝑏+𝜉_𝑛+1𝑏. (20)

Using the same method as used in the proof of Theorem 2.5 in [6], we can prove the following theorem.

Theorem 18. Let 𝐹∶𝐶[0,𝑡]→ℂ be a function and 𝐹(𝑌(𝑥,⋅))𝑤_𝜑𝛼-integrable over the variable 𝑥. Moreover, let 𝑃_𝑌𝑛 be a probability distribution of 𝑌_𝑛 on (ℝ^𝑛,ℬ(ℝ^𝑛)). Then, for 𝑃_𝑌𝑛 a.e. 𝜉_𝑛=(𝜉₁,…,𝜉_𝑛)∈ℝ^𝑛 (hence for a.e. 𝜉_𝑛∈ℝ^𝑛) 𝐸𝐹∣𝑌_𝑛𝜉_𝑛=12𝜋𝑏(𝑡)−𝑏𝑡_𝑛^1/2×_ℝ𝐸𝐹𝑌(𝑥,⋅)−[𝑌(𝑥,⋅)]_𝑎,𝑏+𝜉_𝑛+1𝑏−𝜉×exp^𝑛+1−𝑎(𝑡)−𝜉_𝑛−𝑎𝑡_𝑛²2𝑏(𝑡)−𝑏𝑡_𝑛𝑑𝜉_𝑛+1, (21) where  𝜉_𝑛+1=(𝜉₁,…,𝜉_𝑛,𝜉_𝑛+1) and the expectation is taken over 𝑤_𝜑𝛼.

Note that the conditioning functions 𝑌_𝑛+1 and 𝑌_𝑛 describe the positions of paths at the times 𝑡₁,…,𝑡_𝑛,𝑡_𝑛+1=𝑡 (the present time). 𝑌_𝑛+1 contains the present position 𝑌(𝑥,𝑡) of the path 𝑌(𝑥,⋅) for 𝑥∈𝐶[0,𝑡], while 𝑌_𝑛 does not. Moreover if we let ℎ=1 a.e., 𝛼=0, and 𝜑=𝛿₀, the Dirac measure concentrated at 0, then we can obtain Theorems 1 and 2 in [10] by the translation theorem (Theorem 1). If 𝑎(0)=0, ℎ=𝑏^′, 𝜑_𝛼=𝛿₀, and 𝑡=𝑇, then we can obtain the space 𝐶_𝑎,𝑏[0,𝑇] in [7] by Theorem 12. Furthermore, if 𝑌 is replaced by the generalized Brownian motion process 𝑥(𝑠) on 𝐶_𝑎,𝑏[0,𝑇]×[0,𝑇], and we let 𝜑_𝛼=𝛿₀, then we can also obtain Theorem 3.4 in [7] by Theorem 17. If we let 𝑎≡0 and 𝜑_𝛼=𝛿₀, then we can obtain Theorem 3 in [11] by Theorem 17. If we let 𝑎≡0, then we can obtain Theorem 2.12 in [8] by Theorem 17. If we let 𝑛=0 and 𝜑_𝛼=𝛿₀, then we can obtain Remark 2.2 in [3] by Theorem 17. Finally, if we let ℎ=1 a.e., 𝜑_𝛼=𝛿₀ and 𝑎≡0, then we can obtain Theorem 2 in [2] by Theorem 17 which is among the first result expressing the conditional Wiener integrals of functions on 𝐶₀[0,𝑡] as ordinary Wiener integrals.

Remark 19. Note that Theorems 17 and 18 are not generalizations of Theorem 2.9 in [5] and Theorem 2.5 in [6]. In Theorem 2.9 of [5] and Theorem 2.5 of [6], the conditioning functions have initial distributions 𝜑, while 𝑌_𝑛+1 and 𝑌_𝑛 in Theorems 17 and 18 have no initial distributions.

4. Evaluation Formulas Using the Simple Formulas

In this section, we derive evaluation formulas for the generalized conditional Wiener integrals of various functions which are of interest in Feynman integration theories and quantum mechanics. For a function 𝐹∶𝐶[0,𝑡]→ℂ let 𝐹_𝑌(𝑥)=𝐹(𝑌(𝑥,⋅)) for 𝑥∈𝐶[0,𝑡].

Lemma 20. Let 𝐹_𝑚(𝑥)=∫^𝑡₀(𝑥(𝑠))^𝑚𝑑𝑠(𝑚∈ℕ) for 𝑥∈𝐶[0,𝑡]. Then, (𝐹_𝑚)_𝑌 is 𝑤_𝜑𝛼-integrable and 𝐸𝐹_𝑚𝑌=^[𝑚/2]_𝑙=0𝑚!2^𝑙𝑙!(𝑚−2𝑙)!^𝑡₀(𝑎(𝑠))^𝑚−2𝑙(𝑏(𝑠))^𝑙𝑑𝑠, (22) where [⋅] denotes the greatest integer function.

Proof. By Corollary 5, the change of variable theorem and the binomial expansion, ^𝑡_{0𝐶[0,𝑡]}|𝑌(𝑥,𝑠)|^𝑚𝑑𝑤_𝜑𝛼(𝑥)≤1𝑑𝑠𝜋^𝑚1/2_𝑙=02^𝑙/2Γ𝑙+12𝑚𝑙^(0,𝑡]|𝑎(𝑠)|^𝑚−𝑙(𝑏(𝑠))^𝑙/2𝑑𝑠, (23) where Γ denote the gamma function, so that (𝐹_𝑚)_𝑌 is integrable over (𝐶[0,𝑡],𝑤_𝜑𝛼). Furthermore, we have _𝐶[0,𝑡]^𝑡₀(𝑌(𝑥,𝑠))^𝑚𝑑𝑠𝑑𝑤_𝜑𝛼(𝑥)=2^[𝑚/2]_𝑙=0𝑚2𝑙^(0,𝑡](𝑎(𝑠))^𝑚−2𝑙12𝜋𝑏(𝑠)^1/2×^∞₀𝑢^2𝑙−𝑢exp²(𝑠)=2𝑏𝑑𝑢𝑑𝑠_{[𝑚/2]𝑙=0}𝑚!2^𝑙𝑙!(𝑚−2𝑙)!^𝑡₀(𝑎(𝑠))^𝑚−2𝑙(𝑏(𝑠))^𝑙𝑑𝑠 (24) which completes the proof.

Theorem 21. Let the assumptions be as given in Lemma 20. Then, for a.e. 𝜉_𝑛+1∈ℝ^𝑛+1 (hence, 𝑃_𝑌𝑛+1 a.e. 𝜉_𝑛+1∈ℝ^𝑛+1) 𝐸𝐹_𝑚𝑌∣𝑌_𝑛+1𝜉_𝑛+1=^𝑛+1_𝑗=1^[𝑚/2]_𝑙=0𝑚!2^𝑙()!×𝑙!𝑚−2𝑙_{𝑡𝑗𝑡𝑗−1}𝑎(𝑠)−[𝑎]_𝑏(𝑠)+𝜉_𝑛+1𝑏(𝑠)^𝑚−2𝑙𝛾_𝑗(𝑠)^𝑙𝑑𝑠, (25) where the expectation is taken over 𝑤_𝜑𝛼.

Proof. For 𝑃_𝑌𝑛+1 a.e. 𝜉_𝑛+1∈ℝ^𝑛+1, we have by Theorems 14 and 17𝐸𝐹_𝑚𝑌∣𝑌_𝑛+1𝜉_𝑛+1=^𝑛+1_{𝑗=1(𝑡𝑗−1,𝑡𝑗)}12𝜋𝛾_𝑗(𝑠)^1/2×_ℝ𝑢+𝑎(𝑠)−[𝑎]_𝑏(𝑠)+𝜉_𝑛+1𝑏(𝑠)^𝑚−𝑢×exp²2𝛾_𝑗(𝑠)=𝑑𝑢𝑑𝑠_{𝑛+1𝑗=1[𝑚/2]𝑙=0}𝑚!2^𝑙()!×𝑙!𝑚−2𝑙_{𝑡𝑗𝑡𝑗−1}𝑎(𝑠)−[𝑎]_𝑏(𝑠)+𝜉_𝑛+1𝑏(𝑠)^𝑚−2𝑙𝛾_𝑗(𝑠)^𝑙𝑑𝑠 (26) by the same method as used in the proof of Lemma 20.

Theorem 22. Let the assumptions be as given in Theorem 21, and for 𝜉_𝑛=(𝜉₁,…,𝜉_𝑛)∈ℝ^𝑛 let Ξ𝜉_𝑛=^𝑛_𝑗=1^[𝑚/2]_𝑙=0𝑚!2^𝑙()!×𝑙!𝑚−2𝑙_{𝑡𝑗𝑡𝑗−1}𝑎(𝑠)−[𝑎]_𝑏(𝑠)+𝜉_𝑛𝑏(𝑠)^𝑚−2𝑙𝛾_𝑗(𝑠)^𝑙𝑑𝑠. (27) Then, for a.e. 𝜉_𝑛∈ℝ^𝑛 (hence 𝑃_𝑌𝑛 a.e. 𝜉_𝑛∈ℝ^𝑛) 𝐸𝐹_𝑚𝑌∣𝑌_𝑛𝜉_𝑛=Ξ𝜉_𝑛×+𝑚!_{[𝑚/2]𝑙=0𝑚−2𝑙𝑘𝑘=0𝑝=0[𝑝/2]𝑞=0}𝜉_𝑛^𝑘−𝑝𝑎(𝑡)−𝑎𝑡_𝑛^𝑝−2𝑞𝑏(𝑡)−𝑏𝑡_𝑛^𝑞2^𝑙+𝑞!!()!×𝑙!𝑞!𝑘−𝑝𝑝−2𝑞𝑚−2𝑙−𝑘_𝑡𝑡𝑛𝛾_𝑛+1(𝑠)^𝑙𝛽_𝑛+1(𝑠)^𝑝𝑎(𝑠)−[𝑎]_𝑏(𝑠)^{𝑚−2𝑙−𝑘}𝑑𝑠, (28) where the expectation is taken over 𝑤_𝜑𝛼.

Proof. For 𝜉_𝑛=(𝜉₁,…,𝜉_𝑛)∈ℝ^𝑛, let 𝜉_𝑛+1=(𝜉₁,…,𝜉_𝑛,𝜉_𝑛+1). By the binomial expansion, we have for 𝑡_𝑛≤𝑠≤𝑡𝑎(𝑠)−[𝑎]_𝑏(𝑠)+𝜉_𝑛+1𝑏(𝑠)^𝑚−2𝑙=^{𝑚−2𝑙𝑘}_𝑘=0𝑝=0𝜉_𝑛^𝑘−𝑝(𝑚−2𝑙)!!()!𝛽𝑝!𝑘−𝑝𝑚−2𝑙−𝑘^𝑛+1(𝑠)^𝑝×𝑎(𝑠)−[𝑎]_𝑏(𝑠)^{𝑚−2𝑙−𝑘}𝜉_𝑛+1−𝜉_𝑛^𝑝, (29) so that by Theorems 18 and 21, we have for a.e. 𝜉_𝑛=(𝜉₁,⋯,𝜉_𝑛)∈ℝ^𝑛𝐸𝐹_𝑚𝑌∣𝑌_𝑛𝜉_𝑛=Ξ𝜉_𝑛+^[𝑚/2]_𝑙=0^{𝑚−2𝑙𝑘}_𝑘=0𝑝=0^[𝑝/2]_𝑞=0𝑚!𝜉_𝑛^𝑘−𝑝𝑎(𝑡)−𝑎𝑡_𝑛^𝑝−2𝑞2^𝑙−1!!!()!×𝑙!2𝑞𝑘−𝑝𝑝−2𝑞𝑚−2𝑙−𝑘_𝑡𝑡𝑛𝛾_𝑛+1(𝑠)^𝑙𝛽_𝑛+1(𝑠)^𝑝×𝑎(𝑠)−[𝑎]_𝑏(𝑠)^{𝑚−2𝑙−𝑘}12𝜋𝑏(𝑡)−𝑏𝑡_𝑛^1/2×^∞₀𝑢^2𝑞−𝑢exp²2𝑏(𝑡)−𝑏𝑡_𝑛𝑑𝑢𝑑𝑠 (30) by the change of variable theorem. Using the same method as used in the proof of Lemma 20, we have the desired result.

From now on, we assume that every expectation is taken over 𝑤_𝜑𝛼 unless otherwise specified. By Corollary 9, Theorems 14 and 17, and Theorem 3.4 in [8], we have the following theorem.

Theorem 23. Let 0<𝑠₁<𝑠₂≤𝑡, 𝑠₁∈[𝑡_𝑙−1−𝑡_𝑙] and 𝑠₂∈[𝑡_𝑗−1−𝑡_𝑗]. For 𝑥∈𝐶[0,𝑡], let 𝐺(𝑥)=𝑥(𝑠₁)𝑥(𝑠₂). (1)If  𝑙≠𝑗, then for a.e. 𝜉_𝑛+1∈ℝ^𝑛+1 (hence, 𝑃_𝑌𝑛+1 a.e. 𝜉_𝑛+1∈ℝ^𝑛+1) 𝐸𝐺_𝑌∣𝑌_𝑛+1𝜉_𝑛+1=𝑎𝑠₁−[𝑎]_𝑏𝑠₁+𝜉_𝑛+1𝑏𝑠₁×𝑎𝑠₂−[𝑎]_𝑏𝑠₂+𝜉_𝑛+1𝑏𝑠₂. (31)(2)If  𝑙=𝑗, then for a.e. 𝜉_𝑛+1∈ℝ^𝑛+1 (hence, 𝑃_𝑌𝑛+1 a.e. 𝜉_𝑛+1∈ℝ^𝑛+1) 𝐸𝐺_𝑌∣𝑌_𝑛+1𝜉_𝑛+1=𝛼_𝑗𝑠₂𝑏𝑠₁−𝑏𝑡_𝑗−1+𝑎𝑠₁−[𝑎]_𝑏𝑠₁+𝜉_𝑛+1𝑏𝑠₁×𝑎𝑠₂−[𝑎]_𝑏𝑠₂+𝜉_𝑛+1𝑏𝑠₂. (32)

Theorem 24. Let the assumptions be as given in Theorem 23. (1)If 𝑙≤𝑛, 𝑗≤𝑛 and 𝑙≠𝑗, then for a.e. 𝜉_𝑛∈ℝ^𝑛 (hence, 𝑃_𝑌𝑛 a.e. 𝜉_𝑛∈ℝ^𝑛) 𝐸𝐺_𝑌∣𝑌_𝑛𝜉_𝑛=𝑎𝑠₁−[𝑎]_𝑏𝑠₁+𝜉_𝑛𝑏𝑠₁×𝑎𝑠₂−[𝑎]_𝑏𝑠₂+𝜉_𝑛𝑏𝑠₂. (33)(2)If 𝑙≤𝑛, 𝑗≤𝑛 and 𝑙=𝑗, then for a.e. 𝜉_𝑛∈ℝ^𝑛 (hence 𝑃_𝑌𝑛 a.e. 𝜉_𝑛∈ℝ^𝑛) 𝐸𝐺_𝑌∣𝑌_𝑛𝜉_𝑛=𝛼_𝑗𝑠₂𝑏𝑠₁−𝑏𝑡_𝑗−1+𝑎𝑠₁−[𝑎]_𝑏𝑠₁+𝜉_𝑛𝑏𝑠₁×𝑎𝑠₂−[𝑎]_𝑏𝑠₂+𝜉_𝑛𝑏𝑠₂. (34)(3)If 𝑙=𝑛+1 and 𝑗≤𝑛, then for a.e. 𝜉_𝑛=(𝜉₁,…,𝜉_𝑛)∈ℝ^𝑛 (hence 𝑃_𝑌𝑛 a.e. 𝜉_𝑛∈ℝ^𝑛) 𝐸𝐺_𝑌∣𝑌_𝑛𝜉_𝑛=𝑎𝑠₁−[𝑎]_𝑏𝑠₁+𝜉_𝑛+𝛽_𝑛+1𝑠₁𝑎(𝑡)−𝑎𝑡_𝑛×𝑎𝑠₂−[𝑎]_𝑏𝑠₂+𝜉_𝑛𝑏𝑠₂. (35)(4)If 𝑙≤𝑛 and 𝑗=𝑛+1, then for a.e. 𝜉_𝑛=(𝜉₁,…,𝜉_𝑛)∈ℝ^𝑛 (hence 𝑃_𝑌𝑛 a.e. 𝜉_𝑛∈ℝ^𝑛) 𝐸𝐺_𝑌∣𝑌_𝑛𝜉_𝑛=𝑎𝑠₂−[𝑎]_𝑏𝑠₂+𝜉_𝑛+𝛽_𝑛+1𝑠₂𝑎(𝑡)−𝑎𝑡_𝑛×𝑎𝑠₁−[𝑎]_𝑏𝑠₁+𝜉_𝑛𝑏𝑠₁. (36)(5)If  𝑙=𝑛+1 and 𝑗=𝑛+1, then for a.e. 𝜉_𝑛=(𝜉₁,…,𝜉_𝑛)∈ℝ^𝑛 (hence 𝑃_𝑌𝑛 a.e. 𝜉_𝑛∈ℝ^𝑛) 𝐸𝐺_𝑌∣𝑌_𝑛𝜉_𝑛=𝜉²_𝑛+𝜉_𝑛𝑎(𝑡)−𝑎𝑡_𝑛×𝑎𝑠₁−[𝑎]_𝑏𝑠₁+𝑎𝑠₂−[𝑎]_𝑏𝑠₂+𝛽_𝑛+1𝑠₁+𝛽_𝑛+1𝑠₂+𝑎𝑠₁−[𝑎]_𝑏𝑠₁+𝛽_𝑛+1𝑠₁𝑎(𝑡)−𝑎𝑡_𝑛×𝑎𝑠₂−[𝑎]_𝑏𝑠₂+𝛽_𝑛+1𝑠₂𝑎(𝑡)−𝑎𝑡_𝑛+𝛽_𝑛+1𝑠₁𝛽_𝑛+1𝑠₂𝑏(𝑡)−𝑏𝑡_𝑛+𝛼_𝑛+1𝑠₂𝑏𝑠₁−𝑏𝑡_𝑛. (37)

Proof. For 𝜉_𝑛=(𝜉₁,…,𝜉_𝑛)∈ℝ^𝑛, let 𝜉_𝑛+1=(𝜉₁,…,𝜉_𝑛,𝜉_𝑛+1). (1) and (2) follow immediately from Theorem 23. If 𝑠∈[𝑡_𝑛,𝑡], then 12𝜋𝑏(𝑡)−𝑏𝑡_𝑛^1/2×_ℝ𝜉_𝑛+1𝑏(𝑠)−𝜉×exp^𝑛+1−𝜉_𝑛−𝑎(𝑡)−𝑎𝑡_𝑛²2𝑏(𝑡)−𝑏𝑡_𝑛𝑑𝜉_𝑛+1=12𝜋𝑏(𝑡)−𝑏𝑡_𝑛^1/2×_ℝ𝜉_𝑛+𝛽_𝑛+1(𝑠)𝜉_𝑛+1−𝜉_𝑛−𝜉×exp^𝑛+1−𝜉_𝑛−𝑎(𝑡)−𝑎𝑡_𝑛²2𝑏(𝑡)−𝑏𝑡_𝑛𝑑𝜉_𝑛+1=𝜉_𝑛+𝛽_𝑛+1(𝑠)𝑎(𝑡)−𝑎𝑡_𝑛 (38) by which (3) and (4) follow. If 𝑠₁,𝑠₂∈[𝑡_𝑛,𝑡], then 12𝜋𝑏(𝑡)−𝑏𝑡_𝑛^1/2×_ℝ𝜉_𝑛+1𝑏𝑠₁𝜉_𝑛+1𝑏𝑠₂−𝜉×exp^𝑛+1−𝜉_𝑛−𝑎(𝑡)−𝑎𝑡_𝑛²2𝑏(𝑡)−𝑏𝑡_𝑛𝑑𝜉_𝑛+1=12𝜋𝑏(𝑡)−𝑏𝑡_𝑛^1/2×_ℝ𝜉_𝑛+𝛽_𝑛+1𝑠₁𝜉_𝑛+1−𝜉_𝑛×𝜉_𝑛+𝛽_𝑛+1𝑠₂𝜉_𝑛+1−𝜉_𝑛−𝜉×exp^𝑛+1−𝜉_𝑛−𝑎(𝑡)−𝑎𝑡_𝑛²2𝑏(𝑡)−𝑏𝑡_𝑛𝑑𝜉_𝑛+1=𝜉²_𝑛+𝜉_𝑛𝑎(𝑡)−𝑎𝑡_𝑛𝛽_𝑛+1𝑠₁+𝛽_𝑛+1𝑠₂+𝛽_𝑛+1𝑠₁𝛽_𝑛+1𝑠₂𝑎(𝑡)−𝑎𝑡_𝑛²+𝑏(𝑡)−𝑏𝑡_𝑛 (39) by which we have (5).

For 𝑗=1,…,𝑛+1, let ℎ_𝑗=𝜒_{(𝑡𝑗−1,𝑡𝑗]}ℎ and 𝑔_𝑗=ℎ_𝑗/𝑏(𝑡_𝑗)−𝑏(𝑡_𝑗−1). Let 𝑉 be the subspace of 𝐿₂[0,𝑡] generated by {𝑔₁,…,𝑔_𝑛+1}, 𝑉^⟂ the orthogonal complement of 𝑉, and 𝒫^⟂∶𝐿₂[0,𝑡]→𝑉^⟂ the orthogonal projection. Let ℳ(𝐿₂[0,𝑡]) be the class of all ℂ-valued Borel measures of bounded variation over 𝐿₂[0,𝑡], and let 𝒮_𝑤𝜑𝛼 be the space of all functions 𝐹 which for 𝜎∈ℳ(𝐿₂[0,𝑡]) have the form 𝐹(𝑥)=_{𝐿2[0,𝑡]}exp{𝑖(𝑣,𝑥)}𝑑𝜎(𝑣) (40) for 𝑤_𝜑𝛼 a.e. 𝑥∈𝐶[0,𝑡]. Note that 𝒮_𝑤𝜑𝛼 is a Banach algebra [4].

Now, we have the following theorems which are our final results.

Theorem 25. Let ℎ be of bounded variation and 𝑎 absolutely continuous on [0,𝑡]. Let 𝐹∈𝒮_𝑤𝜑𝛼 and 𝜎∈ℳ(𝐿₂[0,𝑡]) be related by (40). Then, 𝐸𝐹_𝑌=_{𝐿2[0,𝑡]}−1exp2‖𝑣ℎ‖²𝑑𝜎_𝑎(𝑣), (41) where (𝑑𝜎_𝑎/𝑑𝜎)(𝑣)=exp{𝑖(𝑣,𝑎)} for 𝑣∈𝐿₂[0,𝑡]. Moreover for a.e. 𝜉_𝑛+1∈ℝ^𝑛+1 (hence for 𝑃_𝑌𝑛+1 a.e. 𝜉_𝑛+1∈ℝ^𝑛+1) 𝐸[𝐹_𝑌∣𝑌_𝑛+1](𝜉_𝑛+1) is given by 𝐸𝐹_𝑌∣𝑌_𝑛+1𝜉_𝑛+1=_{𝐿2[0,𝑡]}exp𝑖𝑣,𝑎−[𝑎]_𝑏+𝜉_𝑛+1𝑏−12𝒫^⟂(𝑣ℎ)²𝑑𝜎(𝑣). (42)

Proof. By Corollary 5 and the Fubini’s theorem 𝐸𝐹_𝑌=_{𝐿2[0,𝑡]𝐶[0,𝑡]}exp{𝑖(𝑣,𝑌(𝑥,⋅))}𝑑𝑤_𝜑𝛼(𝑥)(𝑣)=𝑑𝜎_{𝐿2[0,𝑡]}{𝑖(−1exp𝑣,𝑎)}exp2‖𝑣ℎ‖²𝑑𝜎(𝑣), (43) where the last equality follows from the following integral formula: _ℝexp−𝑎𝑢²𝜋+𝑖𝑏𝑢𝑑𝑢=𝑎^1/2−𝑏exp²4𝑎 (44) for 𝑎∈ℂ₊ and any real 𝑏. Furthermore, for a.e. 𝜉_𝑛+1∈ℝ^𝑛+1, we have by Theorems 3.4 and 3.6 in [8] 𝐸𝐹_𝑌∣𝑌_𝑛+1𝜉_𝑛+1=_{𝐿2[0,𝑡]}exp𝑖𝑣,𝑎−[𝑎]_𝑏+𝜉_𝑛+1𝑏×_𝐶[0,𝑡]exp𝑖𝑣,𝑍(𝑥,⋅)−[𝑍(𝑥,⋅)]_𝑏𝑑𝑤_𝜑𝛼(𝑥)(𝑣)=𝑑𝜎_{𝐿2[0,𝑡]}exp𝑖𝑣,𝑎−[𝑎]_𝑏+𝜉_𝑛+1𝑏−1×exp2𝒫^⟂(𝑣ℎ)²𝑑𝜎(𝑣) (45) which completes the proof.

Theorem 26. Let the assumptions be as given in Theorem 25 and for 𝜉_𝑛∈ℝ^𝑛 let 𝐴𝑣,𝜉_𝑛=exp𝑖𝑣,𝑎−[𝑎]_𝑏+𝜉_𝑛𝑏−12𝒫^⟂(𝑣ℎ)². (46) Then, for a.e. 𝜉_𝑛∈ℝ^𝑛 (hence for 𝑃_𝑌𝑛 a.e. 𝜉_𝑛∈ℝ^𝑛) 𝐸[𝐹_𝑌∣𝑌_𝑛](𝜉_𝑛) is given by 𝐸𝐹_𝑌∣𝑌_𝑛𝜉_𝑛=_{𝐿2[0,𝑡]}𝐴𝑣,𝜉_𝑛exp𝑖𝑣ℎ,𝑔_𝑛+1𝑎(𝑡)−𝑎𝑡_𝑛𝑏(𝑡)−𝑏𝑡_𝑛−12𝑣ℎ,𝑔_𝑛+1²𝑑𝜎(𝑣), (47) where ⟨⋅,⋅⟩ denotes the inner product on 𝐿₂[0,𝑡].

Proof. Let 𝜉_𝑛=(𝜉₁,…,𝜉_𝑛)∈ℝ^𝑛 and let 𝜉_𝑛+1=(𝜉₁,…,𝜉_𝑛,𝜉_𝑛+1). By the definition of the Paley-Wiener-Zygmund integral it is not difficult to show 𝑣,𝜉_𝑛+1𝑏=𝑣,𝜉_𝑛𝑏+𝑣ℎ,𝑔_𝑛+1𝑏(𝑡)−𝑏𝑡_𝑛𝜉_𝑛+1−𝜉_𝑛 (48) so that we have for a.e. 𝜉_𝑛=(𝜉₁,…,𝜉_𝑛)∈ℝ^𝑛𝐸𝐹_𝑌∣𝑌_𝑛𝜉_𝑛=12𝜋𝑏(𝑡)−𝑏𝑡_𝑛^1/2×_{𝐿2[0,𝑡]}𝐴𝑣,𝜉_𝑛×_ℝexp𝑖𝑣ℎ,𝑔_𝑛+1𝑏(𝑡)−𝑏𝑡_𝑛𝜉_𝑛+1−𝜉_𝑛−𝜉_𝑛+1−𝜉_𝑛−𝑎(𝑡)−𝑎𝑡_𝑛²2𝑏(𝑡)−𝑏𝑡_𝑛𝑑𝜉_𝑛+1(𝑣)=𝑑𝜎_{𝐿2[0,𝑡]}𝐴𝑣,𝜉_𝑛exp𝑖𝑣ℎ,𝑔_𝑛+1𝑎(𝑡)−𝑎𝑡_𝑛𝑏(𝑡)−𝑏𝑡_𝑛−12𝑣ℎ,𝑔_𝑛+1²𝑑𝜎(𝑣) (49) by Theorem 25 and the change of variable theorem.

Acknowledgment

This research was supported by Basic Science Research Program through the National Research Foundation (NRF) of Korea funded by the Ministry of Education, Science and Technology (2012-0002477).