GENERALIZED TRANSFORMS AND GENERALIZED CONVOLUTION PRODUCTS ASSOCIATED WITH GAUSSIAN PATHS ON FUNCTION SPACE

. In this paper we deﬁne a more general convolution product (asso-ciated with Gaussian processes) of functionals on the function space C a,b [0 ,T ]. The function space C a,b [0 ,T ] is induced by a generalized Brownian motion process. Thus the Gaussian processes used in this paper are non-centered processes. We then develop the fundamental relationships between the generalized Fourier–Feynman transform associated with the Gaussian process and the convolution product.


1.
Introduction. For f ∈ L 2 (R), let the Fourier transform of f be given by and for f, g ∈ L 2 (R), let the convolution of f and g be given by where dm n L (v) denotes the normalized Lebesgue measure (2π) −1/2 dv on R. The Fourier transform F satisfies Parseval's relation in the form Furthermore F acts like a homomorphism with convolution * and ordinary multiplication on L 2 (R). More precisely, one can see that for f, g ∈ L 2 (R), subsets of C 0 [0, T ] and let m denote the Wiener measure which is a Gaussian measure on C 0 [0, T ] with mean zero and covariance function r(s, t) = min{s, t}. Then, as it is well-known, (C 0 [0, T ], M, m) is a complete measure space. The concept of the analytic Fourier-Feynman transform (FFT) on the Wiener space C 0 [0, T ], initiated by Brue [1], has been developed in the literature. For instance see [2,16]. This transform and its properties are similar in many respects to the ordinary Fourier transform F. For an elementary introduction to the analytic FFT, see [21] and the references cited therein. We refer to [21] for the concepts, the precise definitions, and the notations of the scale-invariant measurability, the L p analytic FFT, and the convolution product (CP) on the classical Wiener space In [12], Huffman, Park and Skoug defined the CP for functionals on C 0 [0, T ], and they then obtained various results for the analytic FFT and the CP, see also [13,14,20]. In previous researches involving [12,13,14,20], the authors established a relationship between the analytic FFT and the corresponding CP of functionals F and G on C 0 [0, T ] of the form for scale-almost every y ∈ C 0 [0, T ], where T (p) q (F ) and (F * G) q denote the analytic FFT and the CP of functionals on C 0 [0, T ], respectively.
The concepts of the generalized Wiener integral (namely, the Wiener integral associated with Gaussian paths) and the generalized analytic Feynman integral (namely, the analytic Feynman integral associated with Gaussian paths) on C 0 [0, T ] were introduced by Chung, Park and Skoug [11] and further developed in [10,15,19]. In [10,11,15,19], the generalized Wiener integral was defined by the Wiener integral where h is a non-zero function in L 2 [0, T ] and t 0 h(s)dx(s) denotes the Paley-Wiener-Zygmund stochastic integral [18].
In [15], Huffman, Park and Skoug introduced a generalized FFT (GFFT) and a generalized CP (GCP) associated with the Gaussian process Z h given by (1.3) above, and they developed a relationship between the GFFT and the GCP for functionals in the Banach algebra S(L 2 [0, T ]) introduced by Cameron and Storvick in [3]. Also, in [15], the authors examined several relationships between the GFFT and the GCP. The basic relationship investigated is as follows: q,h (F ) and (F * G) q,h denote the analytic GFFT and the GCP on C 0 [0, T ], respectively. In [10], the authors presented a more general CP of functionals on C 0 [0, T ] and established equation (1.4).
On the other hand, in [5,6,8], the authors introduced the FFTs on the very general function space C a,b [0, T ] (rather than the Wiener space C 0 [0, T ]), and studied their properties and related topics. The function space C a,b [0, T ], induced by a generalized Brownian motion process (GBMP), was introduced by Yeh [22,23] and was used extensively in [4,5,6,7,8,9].
A GBMP on a probability space (Ω, Σ, P ) and a time interval [0, T ] is a Gaussian process Y ≡ {Y t } t∈[0,T ] such that Y 0 = c almost surely for some constant c ∈ R, and for any set of time moments 0 = t 0 < t 1 < · · · < t n ≤ T and any Borel set B ⊂ R n , the measure P (I t1,...,tn,B ) of the cylinder set I t1,...,tn,B of the form where η 0 = c, a(t) is a continuous real-valued function on [0, T ], and b(t) is a increasing continuous real-valued function on [0, T ]. Thus, the GBMP Y is determined by the continuous functions a(·) and b(·). For more details, see [22,23]. Note that when c = 0, a(t) ≡ 0 and b(t) = t on [0, T ], the GBMP is a standard Brownian motion (Wiener process). In this paper we set c = a(0) = b(0) = 0. Then the function space C a,b [0, T ] induced by the GBMP Y determined by the a(·) and b(·) can be considered as the space of continuous sample paths of Y . In view of equation (1.1), it is worth-while to study a fundamental relation between the GFFT and the GCP, such as (1.2) and (1.4), on the function space C a,b [0, T ]. However, by an effect of the drift a(t) of the GBMP Y , the analytic GFFT on C a,b [0, T ] has unusual behaviors. In particular, the relations such as (1.2) and (1.4) do not hold between the GFFT and the GCP on the function space C a,b [0, T ]. For some previous work on the GFFT and the GCP, see [7], and for a more detailed study of the effect of drift on the GBMP, see [4] and the references cited therein.
In this paper, we will establish fundamental relationships between the GFFT and the GCP in view of a concept of the rotation of Gaussian processes on the function space C a,b [0, T ]. This paper is organized as follows. In Section 2, we recall a brief background on the function space C a,b [0, T ]. In Sections 3 and 4, we expound the Gaussian processes which are defined as integral processes on C a,b [0, T ] and the concept of the GFFT associated with the Gaussian paths Z k (x, ·). In Section 5, we investigate the existence of the GFFT associated with the Gaussian paths of exponential-type functionals on C a,b [0, T ]. We also establish that the transforms used in this paper are onto transforms on the class E(C a,b [0, T ]), the space of exponential-type functionals. In Section 6, we define a more general CP of functionals on C a,b [0, T ]. It turns out, as noted in Remark 6.5 below, that the class E(C a,b [0, T ]) forms a noncommutative complex algebra with the convolution. We then proceed to derive a fundamental relationships between the generalized transform and the generalized convolution on the function space C a,b [0, T ].
The Wiener process used in [1,2,3,12,13,14,16,20] is stationary in time and is free of drift, while the Gaussian process used in [10,11,15,19] is non-stationary in time and is free of drift. However the stochastic processes used in this paper, as well as in [4,5,6,7,8,9], are non-stationary in time and are subject to a drift a(t), and can be used to explain the position of the Ornstein-Uhlenbeck process in an external force field [17]. But, by choosing a(t) ≡ 0 and b(t) = t on [0, T ], the function space C a,b [0, T ] reduces to the Wiener space C 0 [0, T ], and so the expected results on C 0 [0, T ] are immediate corollaries of our results in this paper.

2.
Preliminaries. In this section, we present the brief backgrounds which are needed in the following sections.
-measurable for all ρ > 0, and a scale-invariant measurable set N is said to be scale-invariant null provided µ(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 W(C a,b [0, T ])-measurable for every ρ > 0. If two functionals F and G defined on C a,b [0, T ] are equal s-a.e., we write F ≈ G. Note that the relation "≈" is an equivalence relation.
Let L 2 a,b [0, T ] be the space of functions on [0, T ] which are Lebesgue measurable and square integrable with respect to the Lebesgue-Stieltjes measures on [0, T ] induced by a(·) and b(·); i.e., where |a|(·) denotes the total variation function of a(·). Then L 2 a,b [0, T ] is a separable Hilbert space with inner product defined by

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where m |a|,b denotes the Lebesgue-Stieltjes measure induced by |a|(·) and b(·). In particular, note that ) is a separable Hilbert space. Throughout the rest of this paper, we consider the linear space is also a separable Hilbert space. Note that the two separable Hilbert spaces L 2 a,b [0, T ] and C a,b [0, T ] are topologically homeomorphic under the linear operator given by equation (2.1). The inverse operator of D is given by In this paper, in addition to the conditions put on a(t) above, we now add the condition 3. Gaussian processes. In order to present our results involving the analytic GFFT and the GCP, we follow the exposition of [5]. [5,9]. It is known that for each and it is a Gaussian random variable with mean (w, a) C a,b and variance w 2 and let Then the stochastic process Z k : and covariance function In addition, by [23, Theorem 21 , the sample paths of the coordinate process Y = {e t : 0 ≤ t ≤ T } referred in Section 2 above. Also, as mentioned in Section 2, if a(t) ≡ 0 and b(t) = t on [0, T ], the function space C a,b [0, T ] reduces to the classical Wiener space C 0 [0, T ] and the Gaussian process (3.1) with k(t) ≡ t is an ordinary Wiener process.
Let C * a,b [0, T ] be the set of functions k in C a,b [0, T ] such that Dk is continuous except for a finite number of finite jump discontinuities and is of bounded variation where DwDk denotes the pointwise multiplication of the functions Dw and Dk. In this case, (C * a,b [0, T ], ) forms a commutative algebra with the identity b. For more details, see [5]. Given Thus, throughout the rest of this paper, we require k to be in C * a,b [0, T ] for each process Z k . This will ensure that the Lebesgue-Stieltjes integrals Using equation (3.6) and the change of variable theorem, the function space integration formula , let Z k1 and Z k2 be the Gaussian processes given by (3.1), respectively. Then the process is also a continuous Gaussian process with mean function where γ k and β k are given by (3.2) and (3.3), respectively. In fact, the covariance function of X k1,k2 (x 1 , x 2 , ·) is given by = v k1,k2 (min{s, t}).

SEUNG JUN CHANG AND JAE GIL CHOI
Next we will consider a stochastic process associated with Z s(k1,k2) . Given nonzero functions k 1 and where Then R k1,k2 is a Gaussian process with mean function and covariance function = v k1,k2 (min{s, t}).
In view of the argument above and using an induction argument, it follows that for a finite sequence {k 1 , . . . , k m } of non-zero functions in C * a,b [0, T ] and any l ∈ {1, . . . , m}, the three processes whenever the integral exists. Throughout the rest of this paper, 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. Furthermore, 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 + . exists and is finite for all λ > 0. If there exists a function J * F (G; λ) analytic on C + such that J * F (G; λ) = J F (G; λ) for all λ > 0, then J * F (G; λ) is defined to be the analytic G-function space integral (namely, the analytic function space integral associated with the paths G(x, ·)) of F over C a,b [0, T ] with parameter λ, and for λ ∈ C + we write Let q be a non-zero real number and let F be a measurable functional whose analytic G-function space integral I an λ G [F ] exists for all λ ∈ C + . If the following limit exists, we call it the generalized analytic G-Feynman integral (namely, the generalized analytic Feynman integral associated with the paths G(x, ·)) of F with parameter q, and we write where λ approaches −iq through values in C + .
Next we state the definition of the GFFT associated with Gaussian paths on function space. G,x [F (y + G(x, ·))] exists. For p ∈ (1, 2], we define the L p analytic G-GFFT (namely, the GFFT associated with the Gaussian paths G(x, ·)), T (p) q,G (F ) of F , by the formula, if it exists; i.e., for each ρ > 0, for s-a.e. y ∈ C a,b [0, T ] whenever this limit exists.
We note that for 1 ≤ p ≤ 2, T  A functional given by (5.1) is called an exponential functional. Also, given q ∈ R \ {0}, τ ∈ C a,b [0, T ], and k ∈ C * a,b [0, T ], let E q,τ,k be the class of all functionals having the form Ψ q,τ,k for s-a.e. x ∈ C a,b [0, T ], where Ψ w is given by equation (5.1) and K a q,τ,k is a complex number given by The functionals given by equation (5.2) and linear combinations (with complex coefficients) of the Ψ q,τ,k w 's are called the (partially) exponential-type functionals on C a,b [0, T ]. The functionals given by (5.1) are also partially exponential-type functionals because Ψ q,τ,0 For notational convenience, we let Then, using the fact that T ]) of partially exponential-type functionals is a commutative (complex) algebra under the pointwise multiplication and with identity Ψ 0 ≡ 1 because Note that every exponential-type functional is scale-invariant measurable. Since we shall identify functionals which coincide s-a.e. on C a,b [0, T ], E(C a,b [0, T ]) can be regarded as the space of all s-equivalence classes of exponential-type functionals.
Theorem 5.2. Let Ψ w ∈ E be given by equation (5.1). Then for all p ∈ [1, 2], any non-zero real q, and each non-zero function k in C * a,b [0, T ], the L p analytic Z k -GFFT of Ψ w , T (p) q,Z k (Ψ w ) exists and is given by the formula where Ψ q,·,k (w) is given by equation (5.4). Thus, T Since E(C a,b [0, T ]) = SpanE, for each exponential-type functional F in E(C a,b [0, T ]), it can be written as where (K a q,w,k ) −1 denotes the reciprocal number of K a q,w,k . Using this and the linearity of the analytic GFFT T (p) q,Z k , again, one can see that for every functional F ∈ E(C a,b [0, T ]) given by equation (5.6), In view of these observation and Theorem 5.2, we obtain the following theorem.
T ]) be given by equation (5.6). Then for all p ∈ [1, 2], any non-zero real q, and each non-zero function q,Z k (F ) exists and is given by the formula where Ψ q,·,k (wj ) is given by equation (5.4) with w replaced with w j for each j ∈ {1, . . . , n}.
exists and is an element of E(C a,b [0, T ]). In particular, given an exponential functional Ψ w in E, for s-a.e. y ∈ C a,b [0, T ], where K a q l ,w,k l is given by equation (5.3) with (q, τ, k) replaced with (q l , w, k l ) for each l ∈ {1, . . . , m}. Thus for each functional F ∈ E(C a,b [0, T ]) given by equation (5.6), it follows that In our next theorem we show that the composition of GFFTs associated with different Gaussian processes can be reduced to a single GFFT.
Theorem 5.5. Given non-zero functions k 1 and k 2 in C * a,b [0, T ], let s(k 1 , k 2 ) be a function in C * a,b [0, T ] which satisfies the relation (3.8), and let F be an exponentialtype functional in E(C a,b [0, T ]). Then, for all p ∈ [1, 2] and any non-zero real q, the L p analytic R k1,k2 -GFFT, T Proof. In view of Theorem 5.3, it will suffice to show that for each exponential functional Ψ w ∈ E, Using equations (5.8) and (5.3), it first follows that On the other hand, using (3.9) together with (3.10), (5.1), the Fubini theorem, and (3.7), it follows that for all λ > 0 and s-a.e. y ∈ C a,b [0, T ], (5.12) But, using (3.8), (3.5), and (3.10), it follows that Thus, using equation (5.12) together with (5.13) and (5.14), it follows that for s-a.e. y ∈ C a,b [0, T ]. Now, by an analytic continuation, it follows that for s-a.e. y ∈ C a,b [0, T ], T (p) q,R k 1 ,k 2 (Ψ w )(y) = K a q,w,k1 K a q,w,k2 Ψ w (y). Using an induction argument, we obtain the following corollary.
is an onto transform. 6. Generalized convolution product with respect to Gaussian process. In this section we define a GCP of functionals on C a,b [0, T ], and investigate the fundamental relationships between the GFFT and the GCP. We first give the definition of the GCP with respect to Gaussian process on the function space C a,b [0, T ]. When λ = −iq, we denote (F * G) λ,G by (F * G) q,G . Remark 6.2. When a(t) ≡ 0, then GCP with respect to the Gaussian process Z k given by (3.1) is commutative. That is to say, (F * G) q,Z k = (G * F ) q,Z k . However, from (6.1) and (3.4), one can see that It generally does not hold that (F * G) q,Z k = (F * G) q,Z −k , because for almost every functional F on C a,b [0, T ].
Our first theorem gives an expression for the GCP (Ψ w1 * Ψ w2 ) q,Z k of exponential functionals Ψ w1 and Ψ w2 in E. Theorem 6.3. Let Ψ w1 and Ψ w2 be exponential functionals in E. Then the GCP of Ψ w1 and Ψ w2 , (Ψ w1 * Ψ w2 ) q,Z k exists for all non-zero real numbers q and each non-zero function k in C * a,b [0, T ], and is given by the formula for s-a.e. y ∈ C a,b [0, T ]. From the third expression of (6.2), we assert that the GCP with respect to Z k , (Ψ w1 * Ψ w2 ) q,Z k is an element of the class E(C a,b [0, T ]) for each non-zero function k in C * a,b [0, T ]. Proof. Using (3.7) we first observe that for all λ > 0 and s-a.e. y ∈ C a,b [0, T ], . But the last expression is an analytic function of λ throughout C + . Hence, in view of Definition 6.1, (Ψ w1 * Ψ w2 ) q,Z k exists and is given by the second expression of equation (6.2) for all q ∈ R \ {0}. The third expression of equation (6.2) follows from the conventions (5.3) and (5.2). Next, by the fact that E(C a,b [0, T ]) forms a complex algebra under the pointwise multiplication (see Remark 5.1 above), one can see that is an element of the complex algebra E(C a,b [0, T ]).
Throughout the rest of this paper, for any functionals F and G in E(C a,b [0, T ]), we will always express F by (5.6), and G by Based on the expressions (5.6) and (6.3), we obtain the following theorem.
Theorem 6.4. Let F and G be exponential-type functionals in E(C a,b [0, T ]). Then for any non-zero real q and each non-zero function k in C * a,b [0, T ], the GCP (F * G) q,Z k of F and G is given by the formula (F * G) q,Z k ≈ n j=1 m l=1 c j e l (Ψ wj * Ψ v l ) q,Z k .
Thus, we assert that (F * G) q,Z k exists and is an element of the class E(C a,b [0, T ]). Remark 6.5. Given a non-zero real q and a non-zero function k in C * a,b [0, T ], define an operation * q,k on E(C a,b [0, T ]) as follows: for any functionals F and G in E(C a,b [0, T ]), let * q,k (F, G) = (F * G) q,Z k . Then, by Theorem 6.4, the operation * q,k is well-defined and thus the linear space E(C a,b [0, T ]) is a noncommutative algebra with the operation * q,k .
Using the next two theorems, we establish interesting relationships between the GFFT and the GCP defined on the function space C a,b [0, T ]. These results are improvements of (1.2) and (1.4) holding on the Wiener space C 0 [0, T ], to the function space C a,b [0, T ]. Theorem 6.6. Let Ψ w1 and Ψ w2 be exponential functionals in E. Then for all p ∈ [1, 2], any non-zero real q, and each non-zero function k in C * a,b [0, T ], T (p) q,Z k (Ψ w1 * Ψ w2 ) q,Z k (y) and T (p) q,Z −k (Ψ w1 * Ψ w2 ) q,Z k (y) for s-a.e. y ∈ C a,b [0, T ], respectively.