Generalized Analytic Fourier-Feynman Transforms with Respect to Gaussian Processes on Function Space

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1 ilomat 3:6 6), DO.98/L6665C Published by aculty of Sciences and Mathematics, University of Niš, Serbia Available at: Generalized Analytic ourier-eynman Transforms with Respect to Gaussian Processes on unction Space Seung Jun Chang a, Hyun Soo Chung a, Ae Young Ko a, Jae Gil Choi a a Department of Mathematics, Dankook University, Cheonan 33-74, Korea Abstract. n this article, we introduce a generalized analytic ourier-eynman transform and a multiple generalized analytic ourier-eynman transform with respect to Gaussian processes on the function space C, T] induced by generalized Brownian motion process. We derive a rotation formula for our multiple generalized analytic ourier-eynman transform.. ntroduction Let C, T] denote one-parameter Wiener space; that is, the space of all real-valued continuous functions x on, T] with x) =. Let M denote the class of all Wiener measurable subsets of C, T] and let m w denote Wiener measure. Then C, T], M, m w ) is a complete measure space. The concept of the analytic eynman integral on the Wiener space C, T] was initiated by Cameron ]. The foundation of the definition of the analytic eynman integral also can be found in, 3]. There has been a tremendous amount of papers on the analytic eynman integral theory. urthermore, the concept of the analytic ourier-eynman transform on C, T] has been developed in the literature. or an elementary introduction of the analytic eynman integral and the analytic ourier-eynman transform, see 4] and the references cited therein. The concepts of the analytic Z h -Wiener integral the Wiener integral with respect to Gaussian paths Z h ) and the analytic Z h -eynman integral the analytic eynman integral with respect to Gaussian paths Z h ) on C, T] were introduced by Chung, Park and Skoug in ], and further developed in 4,, 3]. n 4,,, 3], the Z h -Wiener integral is defined by the Wiener integral C,T] Z hx, ))dm w x) where Z h x, ) is the Gaussian path given by the stochastic integral Z h x, t) = t hs)dxs) with h L, T]. On the other hand, in 5, 7 9], the authors studied a generalized analytic ourier-eynman transform and a generalized integral transform on the very general function space C, T]. The function space C, T], induced by generalized Brownian motion process, was introduced by J. Yeh 5, 6] and was used extensively in 5 9, ]. Mathematics Subject Classification. Primary 8C, 6J65; Secondary 46G, 6G5 Keywords. generalized Brownian motion process, Gaussian process, generalized analytic eynman integral, generalized analytic ourier-eynman transform, multiple generalized analytic ourier-eynman transform. Received: May 4; Accepted: 9 ebruary 5 Communicated by Svetlana Janković This research was supported by Basic Science Research Program through the National Research oundation of Korea NR) funded by the Ministry of Science, CT and uture Planning 5RCAA5497) and the Ministry of Education 5RDAA584). The corresponding author: Jae Gil Choi addresses: sejchang@dankook.ac.kr Seung Jun Chang), hschung@dankook.ac.kr Hyun Soo Chung), ayko@dankook.ac.kr Ae Young Ko), jgchoi@dankook.ac.kr Jae Gil Choi)

2 S.J. Chang et al. / ilomat 3:6 6), n this article, we introduce the generalized analytic Z h -eynman integral, the generalized analytic Z h -ourier-eynman transform, and the multiple generalized analytic ourier-eynman transform with respect to Gaussian paths on the function space C, T]. We also derive a rotation formula involving the two transforms.. Preliminaries n this section, we briefly list some of the preliminaries from 5, 7, 8, 5] that we will need to establish the results in this paper. Let at) be an absolutely continuous real-valued function on, T] with a) =, a t) L, T], and let bt) be a strictly increasing, continuously differentiable real-valued function with b) = and b t) > for each t, T]. The generalized Brownian motion process Y determined by at) and bt) is a Gaussian process with mean function at) and covariance function rs, t) = min{bs), bt)}. By Theorem 4. 6, p.87], the probability measure µ induced by Y, taking a separable version, is supported by C, T] which is equivalent to the Banach space of continuous functions x on, T] with x) = under the sup norm). Hence, C, T], BC, T]),µ) is the function space induced by Y where BC, T]) is the Borel σ-algebra of C, T]. We then complete this function space to obtain C, T], WC, T]), µ) where WC, T]) is the set of all Wiener measurable subsets of C, T]. We note that the coordinate process defined by e t x) = xt) on C, T], T] is also the generalized Brownian motion process determined by at) and bt). The function space C, T] reduces to the classical Wiener space C, T], considered in papers 4,,, 3] if and only if at) and bt) = t for all t, T]. or more detailed studies about this function space C, T], see 5 9,, 5]. A subset B of C, T] is said to be scale-invariant measurable provided ρb WC, T]) for all ρ >, and a scale-invariant measurable set N is said to be scale-invariant null provided µρn) = for all ρ >. A property that holds except on a scale-invariant null set is said to hold scale-invariant almost everywhere s-a.e.). A functional is said to be scale-invariant measurable provided is defined on a scale-invariant measurable set and ρ ) is WC, T])-measurable for every ρ >. f two functionals and G are equal s-a.e., we write G. Let L, T] be the space of functions on, T] which are Lebesgue measurable and square integrable with respect to the Lebesgue-Stieltjes measures on, T] induced by a ) and b ); i.e., L, T] := {v : T v s)dbs) < + and T v s)d a s) < + where a ) denotes the total variation function of the function a ). Then L, T] is a separable Hilbert space with inner product defined by u, v) := T ut)vt)dm a,b t) T ut)vt)dbt) + a t)], where m a,b denotes the Lebesgue-Stieltjes measure induced by a ) and b ). Note that u u, u) = if and only if ut) = a.e. on, T] and that all functions of bounded variation on, T] are elements of L, T]. Also note that if at) and bt) = t on, T], then L, T] = L, T]. n fact L, T], ) L,b, T],,b) = L, T], ) since the two norms,b and are equivalent. or each v L, T], the Paley-Wiener-Zygmund PWZ) stochastic integral v, x is given by the formula T v, x := lim n n v, φ j ) φ j t)dxt) j= }

3 S.J. Chang et al. / ilomat 3:6 6), for µ-a.e. x C, T] where {φ j } is a complete orthonormal set of real-valued functions of bounded j= variation on, T] such that φ j, φ k ) = δ jk, the Kronecker delta. f v is of bounded variation on, T], then the PWZ stochastic integral v, x equals the Riemann-Stieltjes integral T vt)dxt) for s-a.e. x C, T]. urthermore, for each v L, T], the PWZ stochastic integral v, : C, T] R is a Gaussian random variable with mean T vt)dat) = T vt)a t)dt and variance T v t)dbt) = T v t)b t)dt. or more details, see 5, 7, 8]. 3. Gaussian Processes or any h L, T] with h >, let Z h x, t) denote the PWZ stochastic integral Z h x, t) := hχ,t], x, 3.) let β h t) := t h u)dbu), and let γ h t) := t hu)dau). Then Z h : C, T], T] R is a Gaussian process with mean function t Z h x, t)dµx) = hu)dau) = γ h t) C,T] and covariance function Zh x, s) γ h s) ) Z h x, t) γ h t) ) min{s,t} dµx) = h u)dbu) = β h min{s, t}). C,T] n addition, by 6, Theorem.], Z h, t) is stochastically continuous in t on, T]. Of course if ht), then Z x, t) = xt). urthermore, if at) and bt) = t on, T], then the function space C, T] reduces to the classical Wiener space C, T] and the Gaussian process 3.) with ht) is an ordinary Wiener process. or h, h L, T] with h j >, j {, }, let Z h and Z h be the Gaussian processes given by 3.) with h replaced with h and h respectively. Then the process Z h,h given by : C, T] C, T], T] R Z h,h x, x, t) := Z h x, t) + Z h x, t) 3.) is also a Gaussian process with mean m h,h t) = γ h t) + γ h t) and variance v h,h t) = β h t) + β h t). More precisely, the covariance of the process Z h,h is given by C,T] Zh,h x, x, s) m h,h s) ) Z h,h x, x, t) m h,h t) ) dµ µ)x, x ) = β h min{s, t}) + β h min{s, t}) = v h,h min{s, t}). Let h and h be elements of L, T]. Then there exists a function s L, T] such that s t) = h t) + h t) 3.3) for m a,b -a.e. t, T]. Note that the function s satisfying 3.3) is not unique. We will use the symbol sh, h ) for the functions s that satisfy 3.3) above.

4 S.J. Chang et al. / ilomat 3:6 6), We consider a stochastic process associated with the process Z sh,h ). Define a process R h,h : C, T], T] R by R h,h x, t) := Z sh,h )x, t) + t Then R h,h is a Gaussian process with mean R h,h x, t)dµx) C,T] = Z sh,h )x, t)dµx) + C,T] = γ h t) + γ h t) = m h,h t) h u) + h u) sh, h )u) ) dau). 3.4) t h u) + h u) sh, h )u) ) dau) and covariance Rh,h x, s) m h,h s) ) R h,h x, t) m h,h t) ) dµx) = C,T] min{s,t} s h, h )u)dbu) = = β h min{s, t}) + β h min{s, t}) = v h,h min{s, t}). min{s,t} h u) + h u)) dbu) Also, R sh,h ), t) is stochastically continuous in t on, T]. rom these facts, one can see that Z h,h and R h,h have the same distribution and that for any random variable on C, T], Z h,h x, x, ) ) dµ µ)x, x ) = R h,h x, ) ) dµx), 3.5) C,T] C,T] where by = we mean that if either side exists, both sides exist and equality holds. Remark 3.. n ], the authors investigated a rotation property of the function space measure µ. The result is summarized as follows: for a measurable functional and every nonzero real p and q, = px + qx )dµ µ)x, x ) C,T] p + q x + C,T] ) ) p + q p + q a dµx). But, by the observation presented above, we also obtain an alternative result such that = px + qx )dµ µ)x, x ) C,T] C,T] ) p + q x + p + q + p + q a ) dµx).

5 S.J. Chang et al. / ilomat 3:6 6), A Rotation Theorem for Generalized Analytic eynman ntegrals with Respect to Gaussian Processes Let C denote the set of complex numbers. Let C + := {λ C : Reλ) > } and let C + := {λ C : λ and Reλ) }. Let G be a stochastically continuous Gaussian process on C, T], T]. We define the G-function space integral the function space integral with respect to the Gaussian process G) for functionals on C, T] by the formula G ] G,x Gx, ))] := Gx, ) ) dµx) C,T] whenever the integral exists. Let be a C-valued scale-invariant measurable functional on C, T] such that J G; λ) := G,x λ / Gx, ))] exists and is finite for all λ >. f there exists a function J G; λ) analytic on C + such that J G; λ) = J G; λ) for all λ >, then J G; λ) is defined to be the analytic G-function space integral the analytic function space integral with respect to the process G) of over C, T] with parameter λ, and for λ C + we write anλ an λ G ] an λ G,x Gx, ))] Gx, ) ) dµx) := J G; λ). C,T] Let q be a nonzero real number and let Γ q be a connected neighborhood of iq in C + such that Γ q, + ) is an open interval. Let be a measurable functional whose analytic G-function space integral exists for all λ C +. f the following limit exists, we call it the generalized analytic G-eynman integral the generalized analytic eynman integral with respect to the process G) of with parameter q and we write anf q G ] anf q Gx, ))] := lim G,x λ iq an λ G,x Gx, ))], 4.) where λ approaches iq through values in Γ q. n the case of the generalized analytic Z h -eynman integral, if we choose h on, T], then the definition of the generalized analytic Z -eynman integral agrees with the previous definitions of the generalized analytic eynman integral 5, 8, 9]. Now we will establish a rotation formula of our generalized analytic eynman integral. Lemma 4.. Given h j L, T], j {, }, with h j >, let Z hj be the Gaussian processes given by 3.) with h replaced with h j, and let R h,h be the Gaussian process given by 3.4). Let be a scale-invariant measurable functional that the analytic function space integrals an λ ], an λ ] and an λ ] exist for every λ C Z h Z h R +. urthermore assume h,h that an λ an λ Z Z h,x h x, ) + Z h x, ))]] exists for every λ, λ ) C + C +. Then for each λ C +, an λ an λ Zh x, ) + Z h x, ) )]] = an λ R h,h,x Rh,h x, ) )]. 4.) Proof. n view of the definition of the analytic function space integral with respect to the Gaussian process, we first note that the existences of the generalized analytic integrals an λ ], an λ ], an λ ], and an λ an λ Z h Z h R h,h Z h,x Z h,x Zh x, ) + Z h x, ) )]] guarantee that the function space integrals Zh,x λ / Z h x, ) )], Zh,x λ / Z h x, ) )], Rh,h,x λ / R h,h x, ) )], Zh,x Zh,x λ / Z h x, ) + λ / Z h x, ) )]], and Zh,x an ζ Zh x, ) + ζ / Z h x, ) )]]

6 S.J. Chang et al. / ilomat 3:6 6), all exist for any λ >, λ >, λ >, ζ C +, and ζ >. Next, the existence of the analytic function space integral J Z h, Z h ; λ, λ ) an λ an λ Zh x, ) + Z h x, ) )]], λ, λ ) C + C +, 4.3) also ensure that the analytic function space integral J Z h, Z h ; λ, λ) = an λ an λ Zh x, ) + Z h x, ) )]] is well-defined for all λ C +. n equation 4.3) above, J Z h, Z h ; λ, λ ) means the analytic function space integral, which is the analytic continuation of the function space integral Zh,x an λ Z h Zh x, ) + λ / Z h x, ) )]], λ, λ ) C +, + ). On the other hand, using the ubini theorem, 3.) and 3.5), it follows that for all λ >, Zh,x Zh,x λ / Z h x, ) + λ / Z h x, ) )]] = Zh,x Zh,x λ / Z h x, ) + Z h x, )] )]] = Zh,x Zh,x λ / Z h,h x, x, ) )]] = Rh,h,x λ / R h,h x, ) )]. We now use the analytic continuation to obtain our desired conclusion. Remark 4.. Let and J λ Z h ; λ ) := Zh,x an λ Zh x, ) + λ / Z h x, ) )]], λ, λ ) C +, + ), J λ Z h ; λ ) := Zh,x an λ λ / Z h x, ) + Z h x, ) )]] = an λ Zh,x λ / Z h x, ) + Z h x, ) )]], λ, λ ), + ) C +, JZ h, Z h ; λ, λ ) = Zh,x Zh,x λ / Z h x, ) + λ / Z h x, ) )]], λ, λ ), + ), + ). Also, let J λ Z h ; λ ), λ C +, denote the analytic continuation of J λ Z h ; λ ), let J λ Z h ; λ ), λ C +, denote the analytic continuation of J λ Z h ; λ ), and let J Z h, Z h ;, ) denote the analytic continuation on C + C + of the function JZ h, Z h ;, ). Clearly, J λ Z h ; λ ) = J Z h, Z h ; λ, λ ) where J Z h, Z h ;, ) is the analytic function on C + C + given in 4.3) above. rom the assumptions in Lemma 4., one can see that the three analytic function space integrals J λ Z h ; λ ), J λ Z h ; λ ), and J Z h, Z h ; λ, λ ) all exist, and J λ Z h ; λ ) = J λ Z h ; λ ) = J Z h, Z h ; λ, λ ) for all λ, λ ) C + C +. Theorem 4.3. Let Z h, Z h, R h,h, and be as in Lemma 4.. Then for a real q R \ {}, anf q anf q Zh x, ) + Z h x, ) )]] = anf q R h,h,x Rh,h x, ) )]. 4.4)

7 Proof. To obtain equation 4.4), one may establish that lim λ,λ iq λ,λ Γ S.J. Chang et al. / ilomat 3:6 6), an λ an λ Zh x, ) + Z h x, ) )]] = anf q R h,h,x Rh,h x, ) )]. But, as shown in the proof of Lemma 4., the assumption that the analytic function space integrals an λ ], Z h an λ ], and an λ ] exist for every λ C Z h R +, and the analytic function space integral an λ an λ Z h,h Z h,x h x, )+ Z h x, ))]] exists for every λ, λ ) C + C +, says the fact that an λ ] is analytic on C R +, as a function of λ, h,h and an λ an λ Z Z h,x h x, ) + Z h x, ))]] is analytic on C + C +, as a function of λ, λ ). Thus, to establish equation 4.4), it will suffice to show that lim λ iq λ Γ an λ an λ Zh x, ) + Z h x, ) )]] = anf q R h,h,x Rh,h x, ) )]. Using equation 4.) and the analytic continuation, we obtain the desired result. 5. Multiple Generalized Analytic ourier-eynman Transform with Respect to Gaussian Processes We begin this section with the definitions of the generalized analytic ourier-eynman transform with respect to Gaussian process and the multiple generalized analytic ourier-eynman transform with respect to Gaussian processes of functionals on C, T]. Let be a scale-invariant measurable functional on C, T] and let G be a stochastically continuous Gaussian process on C, T], T]. or λ C + and y C, T], let )] T λ,g )y) := an λ G,x y + Gx, ) 5.) denote the analytic function space transform of. Let q be a nonzero real number and let Γ q be a connected neighborhood of iq in C + such that Γ q, + ) is an open interval. We define the L generalized analytic ourier-eynman transform with respect to the process G, T ) ) of, by the formula if it exists) q,g T ) )y) := lim q,g λ iq λ Γ T λ,g )y) 5.) for s-a.e. y C, T]. We note that T ) ) is defined only s-a.e.. We also note that if T) ) exists and if G, then T) q,g q,g exists and T ) G) T) ). rom equations 5.), 5.), and 4.), it follows that q,g q,g T ) q,g G) q,g )y) = anf q y + )] 5.3) G for s-a.e. y C, T]. Next, let G j, j {,..., n}, be stochastically continuous Gaussian processes on C, T], T]. or λ > and y C, T], define a transform M λ,g,...,g n ))y) as follows: M λ,g,...,g n ))y) n ) := y + λ / G j x j, ) dµ n x,..., x n ) C n,t] j= Gn,x n Gn,x n G,x y + λ / n j= )]] ]] G j x j, ).

8 S.J. Chang et al. / ilomat 3:6 6), Let M λ,g,...,g n ))y) also denote an analytic extension of M λ,g,...,g n ))y) as a function of λ C +. Given q R \ {} and a connected neighborhood Γ q of iq in C + such that Γ q, + ) is an open interval, we define the L multiple generalized analytic ourier-eynman transforms with respect to the Gaussian processes G,..., G n ), M ) q,g,...,g n ) of, by the formula if it exists) ) M ) q,g,...,g n )y) := lim ) λ iq λ Γ M λ,g,...,g n ))y) 5.4) for s-a.e. y C, T]. Clearly, we have that M λ,g) ) = T λ,g ) for all λ C +, and M ) ) = T) ) for any nonzero real q if q,g) q,g the transforms exist. Theorem 5.. Let Z h, Z h, and R h,h be as in Lemma 4.. Let be a scale-invariant measurable functional that the analytic transforms T λ,zh )y) an λ Z h y + )], T λ,zh )y) an λ Z h y + )], and T λ,rh,h )y) an λ R h,h y + )] exist for every λ C + and s-a.e. y C, T]. urthermore assume that the analytic function space integral an λ an λ y + Zh x, ) + Z h x, ) )]] exists for every λ, λ ) C + C + and s-a.e. y C, T]. Then for a real q R \ {}, M ) q,z h,z h ) )y) = T ) q,r h,h )y) for s-a.e. y C, T]. Proof. irst, proceedings as in the proofs of Lemma 4. and Theorem 4.3, we conclude that anf q anf q y + Zh x, ) + Z h x, ) )]] = anf q R h,h,x y + Rh,h x, ) )]. 5.5) for s-a.e. y C, T]. Next, in view of equation 5.4) and under the assumption, it follows that for s-a.e. y C, T], M ) )y) = lim q,z h,z h ) λ iq λ Γ = lim λ iq λ Γ = lim λ,λ iq λ,λ Γ M λ,zh,z h ))y) an λ an λ y + λ / Z h x, ) + Z h x, )] )]] an λ an λ y + λ / Z h x, ) + λ / Z h x, ) )]] = anf q anf q y + Zh x, ) + Z h x, ) )]]. 5.6) inally using equations 5.6), 5.5), and 5.3) with G replaced with R h,h, it follows that for s-a.e. y C, T], as desired. M ) q,z h,z h ) )y) = T ) q,r h,h )y), The following theorem follows by the use of mathematical induction.

9 S.J. Chang et al. / ilomat 3:6 6), Theorem 5.. Given h j L, T], j {,..., n}, with h j >, let Z hj be the Gaussian processes given by 3.) with h replaced with h j, and let R h,...,h n : C, T], T] R be the Gaussian process given by R h,...,h n x, t) := Z sh,...,h n )x, t) + t n j= ] h j u) sh,..., h n )u) dau), where sh,..., h n ) is an element of L, T] which satisfies the condition s h,..., h n ) = n j= h j for m a,b -a.e. on, T]. Let be a scale-invariant measurable functional that the analytic function space transforms T λ,zhj )y) an λ y + )], j {,..., n}, and T )y) Z λ,rh hj,...,hn an λ ] exist for every λ C R + and s-a.e. h,...,hn y C, T]. urthermore assume that the analytic function space integral an λn an λ y + Z hn,x n n j= )]] ] Z hj x j, ) exists for every λ,..., λ n ) C n + and s-a.e. y C, T]. Then, for a real q R \ {} and s-a.e. y C, T], M ) q,z h,...,z hn ) )y) = T ) q,r h,...,hn )y). We note that the hypotheses and hence the conclusions) of Lemma 4., Theorems 4.3, 5., and 5. above are indeed satisfied by many large classes of functionals. These classes of functionals include: a) The Banach algebra SL, T]) defined by Chang and Skoug in 8]. b) Various spaces of functionals of the form x) = f α, x,..., α n, x ) for appropriate f as discussed in 5, 7, 9]. Acknowledgements The authors would like to express their gratitude to the editor and the referees for their valuable comments and suggestions which have improved the original paper. References ] M. D. Brue, A functional transform for eynman integrals similar to the ourier transform, Thesis, University of Minnesota, 97. ] R. H. Cameron, The LSTOW and eynman integrals, J. D Analyse Mathematique 96-63) ] R. H. Cameron, D.A. Storvick, An L analytic ourier-eynman transform, Michigan Math. J ) 3. 4] K. S. Chang, D. H. Cho, B. S. Kim, T. S. Song,. Yoo, Relationships involving generalized ourier-eynman transform, convolution and first variation, ntegral Transform Spec. unct. 6 5) ] S. J. Chang, J. G. Choi, D. Skoug, ntegration by parts formulas involving generalized ourier-eynman transforms on function space, Trans. Amer. Math. Soc ) ] S. J. Chang, D. M. Chung, Conditional function space integrals with applications, Rocky Mountain J. Math ) ] S. J. Chang, H. S. Chung, D. Skoug, ntegral transforms of functionals in L C, T]), J. ourier Anal. Appl. 5 9) ] S. J. Chang, D. Skoug, Generalized ourier-eynman transforms and a first variation on function space, ntegral Transforms Spec. unct. 4 3) ] J. G. Choi, S. J. Chang, Generalized ourier-eynman transform and sequential transforms on function space, J. Korean Math. Soc. 49 ) 65 8.

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