A CAMERON–STORVICK THEOREM FOR THE ANALYTIC FEYNMAN INTEGRAL ASSOCIATED WITH GAUSSIAN PATHS ON A WIENER SPACE AND APPLICATIONS

. The purpose of this paper is to establish a Cameron–Storvick theorem for the analytic Feynman integral of functionals in non-stationary Gaussian processes on Wiener space. As interesting applications, we apply this theorem to evaluate the generalized analytic Feynman integral of certain polynomials in terms of Paley–Wiener–Zygmund stochastic integrals.


Introduction and preliminaries.
Let In [1] Cameron derived an integration by parts formula for the Wiener measure m. This is the first infinite dimensional integration by parts formula. In [11] Donsker also established this formula using a different method, and applied it to study Fréchet-Volterra differential equations. In [16,17] Kuo and Lee developed the parts formula to abstract Wiener spaces and applied their formula to evaluate some functional integrals. The integration by parts formula on C 0 [0, T ] introduced in [1] was improved in [7,21] to study the parts formulas involving the analytic Feynman integral and the analytic Fourier-Feynman transform. Since then the parts formula for the analytic Feynman integral is called the Cameron-Storvick theorem by many mathematicians.
The Wiener process used in [1,7,21] is a stationary process. The purpose of this paper is to establish a Cameron-Storvick theorem for the analytic Feynman integral of functionals in non-stationary Gaussian processes defined on the Wiener space C 0 [0, T ]. As interesting applications, we use our parts formula to evaluate the generalized analytic Feynman integral of certain polynomials in terms of Paley-Wiener-Zygmund stochastic integrals.

Preliminaries.
In this section we first present a brief background and some well-known results about the Wiener space C 0 [0, T ].
A subset B of C 0 [0, T ] is said to be scale-invariant measurable provided ρB ∈ M for all ρ > 0, and a scale-invariant measurable set N is said to be scale-invariant null provided m(ρN ) = 0 for all ρ > 0. A property that holds except on a scale-invariant null set is said to hold scale-invariant almost everywhere (s-a.e.). A functional F is said to be scale-invariant measurable provided F is defined on a scale-invariant measurable set and F (ρ · ) is Wiener-measurable for every ρ > 0.
The Paley-Wiener-Zygmund (PWZ) stochastic integral [18] plays a key role throughout this paper. Let {φ n } ∞ n=1 be a complete orthonormal set in L 2 [0, T ], each of whose elements is of bounded variation on [0, T ]. Then for each for all x ∈ C 0 [0, T ] for which the limit exists, where (·, ·) 2 denotes the L 2 -inner product. For each v ∈ L 2 [0, T ], the limit defining the PWZ stochastic integral v, x is essentially independent of the choice of the complete orthonormal set {φ n } ∞ n=1 and it exists for s-a.e.
For a more detailed study of the PWZ stochastic integral, see [14,19].
Throughout this paper we let In addition, by [23,Theorem 21 is a Wiener process (standard Brownian motion). We note that the coordinate process Z 1 is stationary in time, whereas the stochastic process Z h generally is not. For more detailed studies on the stochastic process Z h , see [9,10,20].
Let C, C + and C + denote the set of complex numbers, complex numbers with positive real part and non-zero complex numbers with nonnegative real part, respectively. For each λ ∈ C, λ 1/2 denotes the principal square root of λ; i.e., λ 1/2 is always chosen to have positive real part, so that λ −1/2 = (λ −1 ) 1/2 is in C + for all λ ∈ C + .
exists as a finite number for all λ > 0. If there exists a function J * (h; λ) analytic on C + such that J * (h; λ) = J(h; λ) for all λ > 0, then J * (h; λ) is defined to be the generalized analytic Wiener integral (associated with the Gaussian paths Z h (x, ·)) of F over C 0 [0, T ] with parameter λ, and for λ ∈ C + we write Let q = 0 be a real number and let F be a functional such that )m(dx) exists for all λ ∈ C + . If the following limit exists, we call it the generalized analytic Feynman integral (associated with the Gaussian paths Z h (x, ·)) of F with parameter q and we write anfq Next we give the definition of the first variation δF of a functional F . The following definition is due to by Chang, Cho, Kim, Song and Yoo [8].
Throughout this paper we shall always choose w to be an element of Let BV [0, T ] denote the space of the functions of bounded variation on [0, T ]. Then we obtain the following formula: for each ϕ ∈ BV [0, T ] and each h ∈ We finish this section by stating the following well-known translation theorem using the above notation [2,3,15]. (2.4)

A translation theorem.
It is well-known that there is no quasi-invariant measure on infinite dimensional linear spaces (see for instance [22] [2,3]. On the other hand, Cameron and Storvick [5,6] presented a translation theorem for the analytic Feynman integral of functionals on the Wiener space In our next theorem, we obtain a translation theorem for the generalized Wiener integral on C 0 [0, T ].
Proof. We first note that Using this we see that Also, using equation (2.2) it follows that Hence letting G(x) = F (Z h1 (x, ·)) and using equations (3.3), (2.4) with F and x 0 replaced with G and x 2 , (3.2), and (3.4), it follows that as desired.
Next we present a Cameron-Storvick theorem for the generalized analytic Feynman integral.
Next using equation (4.7) with ϕ and w ϕh1 replaced with g and w gh1 , respectively, and with F (x) = g, x , and using equation (5.2), we obtain the formula anfq Remark 5.2. Using (2.2), it follows that In this case, the two Gaussian random variables gh 1 , x and gh 2 , x have different Gaussian distributions. Thus in order to calculate the analytic Feynman integral (see Remark 2. Then in particular, setting w g (t) = t 0 g(s)ds and using (2.2) and (2.3), it follows that Letting m = 3 in equation (5.6) and applying equation (5.3) allows us to easily and completely evaluate the generalized analytic Feynman integral Then setting m = 4 in equation (5.6) allows us to completely evaluate the generalized analytic Feynman integral Using those calculations it follows I 3 = I 5 = 0 and with H m given by (5.7). In addition, for each j = 1, 2, . . . , k, let m j be a nonnegative integer, and let Ψ m1,...,m k (x) be the functional on C 0 [0, T ] defined by For example, we observe that We note that Ψ m1,...,m k (x) = Ψ m1,...,m k ,0,...,0 (x) for all positive integers k. The functionals given by (5.8) are called the Fourier-Hermite polynomials (or functionals). It was shown by Cameron and Martin [4] that the Fourier-Hermite polynomials form a complete orthonormal set in L 2 (C 0 [0, T ]). To prove the completeness of the class of the Fourier-Hermite polynomials, Cameron and Martin used the Wiener integration formula (5.4) above. The use is based on the followings: given a complete orthonormal set {α j } ∞ j=1 , each of whose elements is of bounded variation on [0, T ], let γ j (x) = α j , x for each j ∈ N. Then the class {γ j (x)} ∞ j=1 is a sequence of i.i.d standard Gaussian random variables. Thus, using equation (5.4), one can calculate the Wiener integrals, C0[0,T ] Ψ m1,...,m k (x)m(dx) of the Fourier-Hermite polynomials Ψ m1,...,m k .
However, we will pose the following question. When we evaluate the generalized analytic Feynman integral given by (5.9) we might not use the Wiener integration formula (5.4) because the set of Gaussian random variables { α j , Z hj (x, ·) } n j=1 = { α j h j , x } n j=1 is generally not independent. However, we will calculate the generalized analytic Feynman integral (5.9) via the following examples.
Example 5.6. Let h 1 and h 2 be elements of Supp 2 [0, T ] and set F (x) = x(T ) = 1, x . Then using the techniques as those used in (5.1) we observe that for w α1h1 (5.10) Next using equation (4.7) with ϕ ≡ 1, with h 2 and h 1 replaced with α 2 h 2 and α 1 h 1 , respectively, and with F (x) = 1, x , and using equation (5.10), we obtain the formula anfq   Using (5.13) inductively, and (5.11) we can obtain a positive answer of the Question 5.5 above. More precisely, using the recursive formula (5.13) and a tedious calculation, we conclude that