A Modified Analytic Function Space Feynman Integral and Its Applications

Let C 0 [0, T] denote the one-parameter Wiener space, that is, the space of continuous real-valued functionsx on [0, T]with x(0) = 0, and letm denoteWienermeasure. Since the concept of the Feynman integral was introduced by Feynman and Kac, many mathematicians studied the “analytic” Feynman integral of functionals in several classes of functionals [1– 7]. Recently the authors have introduced an approach to the solutions of the diffusion equation and the Schrödinger equation via the Fourier-type functionals on Wiener space [6]. The function space C a,b [0, T], induced by a generalized Brownian motion, was introduced by Yeh in [8] and studied extensively in [9–11]. In [11] the authors have studied the generalized analytic Feynman integral for functionals in a very general function space C a,b [0, T]. In this paper, we present an analysis of the generalized analytic Feynman integral on function space. We define a modified generalized analytic function space Feynman integral (AFSFI) and then explain the physical circumstances with respect to an anharmonic oscillator using the concept of the modified generalized analytic Feynman integral on function space. The Wiener process used in [1–7] is stationary in time and is free of drift while the stochastic process used in this paper, as well as in [9–12], is nonstationary in time, is subject to a drift a(t), and can be used to explain the position of the Ornstein-Uhlenbeck process in an external force field [13].


Introduction
Let  0 [0, ] denote the one-parameter Wiener space, that is, the space of continuous real-valued functions  on [0, ] with (0) = 0, and let  denote Wiener measure.Since the concept of the Feynman integral was introduced by Feynman and Kac, many mathematicians studied the "analytic" Feynman integral of functionals in several classes of functionals [1][2][3][4][5][6][7].Recently the authors have introduced an approach to the solutions of the diffusion equation and the Schrödinger equation via the Fourier-type functionals on Wiener space [6].
In this paper, we present an analysis of the generalized analytic Feynman integral on function space.We define a modified generalized analytic function space Feynman integral (AFSFI) and then explain the physical circumstances with respect to an anharmonic oscillator using the concept of the modified generalized analytic Feynman integral on function space.
The Wiener process used in [1][2][3][4][5][6][7] is stationary in time and is free of drift while the stochastic process used in this paper, as well as in [9][10][11][12], is nonstationary in time, is subject to a drift (), and can be used to explain the position of the Ornstein-Uhlenbeck process in an external force field [13].
A subset  of  , [0, ] is said to be scale-invariant measurable provided  ∈ W( , [0, ]) for all  > 0, and a scale-invariant measurable set  is said to be a scale-invariant 2 Journal of Function Spaces null set provided () = 0 for all  > 0. A property that holds except on a scale-invariant null set is said to hold scaleinvariant almost everywhere(s-a.e.) [15].
Let  2 , [0, ] be the Hilbert space of functions on [0, ] which are Lebesgue measurable and square integrable with respect to the Lebesgue Stieltjes measures on [0, ] induced by (⋅) and (⋅); that is, where ||() denotes the total variation of the function  on the interval [0, ].
(1) The PWZ stochastic integral ⟨V, ⟩ is essentially independent of the complete orthonormal set for each  > 0.
We finish this section by stating the notion of generalized analytic function space Feynman integral, cf.[10,11].Definition 1.Let C denote the complex numbers, let C + = { ∈ C : Re() > 0}, and let C+ = { ∈ C :  ̸ = 0 and Re() ≥ 0}.Let  :  , [0, ] → C be a measurable functional such that, for each  > 0, the function space integral exists.If there exists a function  * () analytic in C + such that  * () = () for all  > 0, then  * () is defined to be the analytic function space integral of  over  , [0, ] with parameter , and for  ∈ C + we write Let  ̸ = 0 be a real number and let  be a functional such that  * () exists for all  ∈ C + .If the following limit exists, we call it the generalized AFSFI of  with parameter  and we write where  → − through values in C + .

Analogue of the Generalized AFSFI
The differential equation is called the diffusion equation with initial condition (, 0) = (), where Δ is the Laplacian and  is an appropriate potential function.Many mathematicians have considered the Wiener integral of functionals of the form where  is a real number.It is a well-known fact that the Wiener integral of the functional having the form forms the solution of the diffusion equation ( 9) by the Feynman-Kac formula.If time is replaced by an imaginary time, this diffusion equation becomes the Schrödinger equation with the initial condition (, 0) = ().Hence the solution to the Schrödinger equation ( 12) can be obtained via the analytic Feynman integral.An approach to finding the solution to the diffusion equation ( 9) and the Schrödinger equation ( 12) involves the harmonic oscillator () = (/2) 2 ; for a more detailed study, see [6].However, it can be difficult to obtain the solution for the diffusion equation ( 9) and the Schrödinger equation ( 12) with respect to anharmonic oscillators.
In this paper, we consider the following functional: where () is a real number with respect to  and ℎ() is a realvalued function on [0, ].When ℎ() =  for all  ∈ [0, ] and () is independent of the value , the functional in ( 13) reduces the functional in (11).That is to say, our functional ( 13) is more generalized compared with the functional in (11).Hence, all results and formulas for the functional in (11) are special cases of our results and formulas.
We will now explain the importance of the functionals given by (13).For a positive real number , when the potential function is () = (/2) 2 , the diffusion equation ( 9) is called the diffusion equation for a harmonic oscillator with .For  ∈ R, is just the translation of ; thus, it is called the diffusion equation for a harmonic oscillator with  1 .However, for an appropriate function ℎ() on [0, ], may be an anharmonic oscillator.For example, consider the following. ( In this case, the diffusion equation ( 9) is called the diffusion equation for anharmonic oscillator with  3 because it contains the " 3 -term." This means that the status of the harmonic oscillator can be exchanged for the status of the anharmonic oscillator under certain physical circumstances.We can explain this phenomenon by considering the Wiener integral of the functional in (13). ( In this case, the diffusion equation ( 9) is called the diffusion equation for double-well potential with  4 .
As such, it is a harmonic oscillator.
We are now ready to state the definition of the modified generalized AFSFI.Definition 2. Let ℎ ∈  , [0, ] be given.Let  :  , [0, ] → C be such that, for each  > 0, the function space integral exists for all  > 0 where () is a nonnegative real number which depends on .If there exists a function  * () analytic in C + such that  * () = () for all  > 0, then  * () is defined to be the modified analytic function space integral of  over  , [0, ] with parameter , and for  ∈ C + we write Let  ̸ = 0 be a real number and let  be a functional such that ∫ ()  ,ℎ  , [0,] () () exists for all  ∈ C + .If the following limit exists, we call it the modified generalized AFSFI of  with parameter  and we write where  approaches − through values in C + .
Remark 3. We have the following assertions with respect to the modified generalized AFSFI.
We conclude this section by listing several integration formulas for simple functionals to compare with the generalized AFSFI and the modified generalized AFSFI.For all nonzero real number , we have Tables 1 and 2.

Some Properties for the Modified Generalized AFSFI
In this section we establish a Fubini theorem for the modified analytic function space integrals and the modified generalized AFSFIs for functionals on  , [0, ].We also use these Fubini theorems to establish various modified generalized analytic Feynman integration formulas.First, we define a function to simply express many results and formulas in this paper.For  ≥ 2, define a function   : C + → C+ by where ∑  =1  −1/2  ̸ = 0 and ∑  =1  −1  ̸ = 0. Note that   is a symmetric function for all  = 2, 3, . ... In this paper we will assume that, for all ( 1 , . . .,   ) ∈ C + and (∑  =1  −1  ) where ≐ means that if either side exists, both sides exist and equality holds.
Proof.First, using the symmetric property, for all ,  > 0, This can be analytically continued in  and  for (, ) and so we have, for all (, ) ∈ Next, let  be a subset of C+ × C+ containing the point (− 1 , − 2 ) and it is such that (, ) ∈  implies that + ̸ = 0. Note that the function is continuous on  and is uniformly continuous on  provided  is compact.Then by the continuity of H and (27), we can establish (25) as desired.
The following theorem was established in [12,17].Formula (29) is called the Fubini theorem with respect to the function space integrals.Theorem 5. Let  be as in Theorem 4 above.Then To establish Theorem 7, we need the following lemma.Lemma 6.Let  be as in Theorem 4 above.Then for all (, ) ∈ where  = /( + ) and () =  2 (, ).
Proof.Using (29), it follows that for  > 0 and  > 0 This last expression is defined for  > 0 and  > 0. For  > 0, it can be analytically continued in  ∈ C + .Also for  > 0, it can be analytically continued in  ∈ C + .Therefore since  ∈ C + and  ∈ C + implies that /( + ) ∈ C + , we conclude that the last expression in proof of Lemma 6 can be analytically continued into C + to equal the analytic function space integral which completes the proof of Lemma 6 as desired.
The following theorem is the main result with respect to the modified generalized AFSFI.

Theorem 7. Let 𝐹 be as in Lemma 6 above. Then for all 𝑞
where Proof.First note that, for all  1 , where  and () are as in Lemma 6. Hence we complete the proof as desired.

Examples
In this section, we provide several brief examples in which we apply our formulas and results.) (V,   )}  (V) ,

Corollary 8 .
Let  be as in Theorem 7 above.Then one has the following assertions.