Jae Gil Choi and Young Seo Park

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1 Kangweon-Kyungki Math. Jour. 11 (23), No. 1, pp TRANSLATION THEOREM ON FUNCTION SPACE Jae Gil Choi and Young Seo Park Abstract. In this paper, we use a generalized Brownian motion process to define a translation theorem. First we establish the translation theorem for function space integrals. We then obtain the general translation theorem for functionals on function space. 1. Introduction. In [1], Cameron and Martin introduced the transformation of Wiener integrals under the translation. In [4], Chang, Skoug and Park studied translation theorems for Fourier-Feynman transforms and conditional Fourier-Feynman transforms. In this paper, we study a translation theorem for functionals on function space but with x in a very general function space C a,b [, T ] rather than in the Wiener space. The Wiener process used in [1,4] is free of drift and is stationary in time while the stochastic processes used in this paper is nonstationary in time and is subject to a drift a(t). In Section 2 of this paper, we give the basic concepts and notations. In Section 3, we study a translation theorem for function space integrals. Finally, in Section 4, we establish the general translation theorem for functionals on function space. 2. Definitions and preliminaries. Let D [, T ] and let (Ω, B, P ) be a probability measure space. A real valued stochastic process Y on (Ω, B, P ) and D is called a generalized Brownian motion process if Y (, ω) almost everywhere and Received January 3, Mathematics Subject Classification: 6J65,28C2. Key words and phrases: Generalized Brownian motion process, function space, function space integrals, translation theorem.

2 18 Jae Gil Choi and Young Seo Park for t < t 1 < < t n T, the n-dimensional random vector (Y (t 1, ω),, Y (t n, ω)) is normally distributed with density function (2.1) K( t, η) ( (2π) n exp 1 2 n () ) 1/2 ((η j a(t j )) (η j 1 a(t j 1 ))) 2 where η (η 1,, η n ), η, t (t 1,, t n ), a(t) is an absolutely continuous real-valued function on [, T ] with a(), a (t) L 2 [, T ], and b(t) is a strictly increasing, continuously differentiable real-valued function with b() and b (t) > for each t [, T ]. As explained in [8, p.18-2], Y induces a probability measure µ on the measurable space (R D, B D ) where R D is the space of all real valued functions x(t), t D, and B D is the smallest σ-algebra of subsets of R D with respect to which all the coordinate evaluation maps e t (x) x(t) defined on R D are measurable. The triple (R D, B D, µ) is a probability measure space. This measure space is called the function space induced by the generalized Brownian motion process Y determined by a( ) and b( ). We note that the generalized Brownian motion process Y determined by a( ) and b( ) is a Gaussian process with mean function a(t) and covariance function r(s, t) minb(s), b(t). By Theorem 14.2 [8, p.187], the probability measure µ induced by Y, taking a separable version, is supported by C a,b [, T ] (which is equivalent to the Banach space of continuous functions x on [, T ] with x() under the sup norm). Hence (C a,b [, T ], B(C a,b [, T ]), µ) is the function space induced by Y where B(C a,b [, T ]) is the Borel σ-algebra of C a,b [, T ]. Let L 2 a,b [, T ] be the Hilbert 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., (2.2) L 2 a,b[, T ] v : v 2 (s)db(s) < and v 2 (s)d a (s) < where a (t) denotes the total variation of the function a on the interval [, t].

3 Translation theorem on function space 19 For convenience, let BV [, T ] be the space of bounded variation functions on [, T ]. We denote the function space integral of a B(C a,b [, T ])-measurable functional F by F (x)dµ(x) whenever the integral exists. C a,b [,T ] 3. Translation theorem for function space integrals. Let (C a,b [, T ], B(C a,b [, T ]), µ) be the function space induced by the generalized Brownian motion process defined in Section 1. In this section we will obtain a translation theorem for function space integrals over (C a,b [, T ], B(C a,b [, T ]), µ). For a partition τ t 1,, t n of [, T ] with t < t 1 < < t n T, define a function X τ : C a,b [, T ] R n by X τ (x) (x(t 1 ),, x(t n )). For x C a,b [, T ], define the function [X τ (x)] [x] n : [, T ] R by (3.1) [x] n (t) x(t j 1 ) + b(t) b(t j 1) (x(t j) x(t j 1 )) for each t [t j 1, t j ], j 1, 2,, n. In case, [x] n is called the polygonalized form of x. Similarly, for ξ (ξ 1, ξ 2,, ξ n ) R n, define the function [ ξ] n : [, T ] R by (3.2) [ ξ] n (t) ξ j 1 + b(t) b(t j 1) (ξ j ξ j 1 ) for each t [t j 1, t j ], j 1, 2,, n, and ξ. For any positive integer n, define the point t j by t j T n j where j 1, 2,, n.

4 2 Jae Gil Choi and Young Seo Park Lemma 3.1. Let [x] n be the polygonalized form of x as in (3.1) and let F be a bounded and continuous functional on C a,b [, T ]. Then (3.3) lim F ([x] n )dµ(x) F (x)dµ(x). n C a,b [,T ] C a,b [,T ] Proof. Let ε > be given. For any x C a,b [, T ], there exists an integer n n (ε) such that for all t t 1/n, we have (3.4) x(t ) x(t ) < ε 2. By using (3.1) and (3.4), we have for each n n and t j 1 t t j, (3.5) x(t) [x] n (t) x(t) x(t j 1) b(t) b(t j 1) (x(t j) x(t j 1 )) x(t) x(t j 1 ) + x(t j) x(t j 1 ) Hence ε 2 + ε 2 ε. (3.6) lim n [x] n(t) x(t) uniformly on [, T ]. Since F is continuous on C a,b [, T ], (3.7) lim n F ([x] n) F (x). Let F n (x) F ([x] n ). Then for all n N, F n F. So, by using the Bounded Convergence Theorem, we have the desired result. Lemma 3.2. Let ϕ(t) be of bounded variation function on [, T ] and let x (t) t ϕ(s)db(s). Then x C a,b [, T ] and we have (3.8) lim n (x (t j ) x (t j 1 )) 2 ϕ 2 (s)db(s)

5 and (3.9) lim n Translation theorem on function space 21 ((x(t j ) a(t j )) (x(t j 1 ) a(t j 1 )))(x (t j ) x (t j 1 )) ϕ(s)dx(s) ϕ(s)da(s). Proof. Since ϕ BV [, T ], x (t) t ϕ(s)db(s) is absolutely continuous on [, T ]. Observe that (x (t j ) x (t j 1 )) 2 ( ) 2 x (t j ) x (t j 1 ) (b(t j ) b(t j 1)) By using the Cauchy s Mean Value Theorem in the above equation, we have (3.1) (x (t j ) x (t j 1 )) 2 ϕ 2 (ξ j )() where ξ j [t j 1, t j ] for each j 1, 2,, n. Similarly, we have (3.11) ((x(t j ) a(t j )) (x(t j 1 ) a(t j 1 )))(x (t j ) x (t j 1 )) ( ) x (t j ) x (t j 1 ) (x(t j ) x(t j 1 )) b(t j ) b(t j 1 ) ( ) x (t j ) x (t j 1 ) (a(t j ) a(t j 1 )) ϕ(ξ j )(x(t j ) x(t j 1 )) ϕ(ξ j )(a(t j ) a(t j 1 )).

6 22 Jae Gil Choi and Young Seo Park Hence equations (3.1) and (3.11) converge to the followings ϕ 2 (s)db(s) and ϕ(s)dx(s) ϕ(s)da(s) as n, respectively. Thus we have the desired results. Theorem 3.3. Let ϕ and x be given as in Lemma 3.2 and let F be a B(C a,b [, T ])-measurable functional. Then F (x+x ) is B(C a,b [, T ])- measurable and (3.12) F (y)dµ(y) F (x + x )J(x, x )dµ(x) C a,b [,T ] where (3.13) J(x, x ) exp 1 2 ϕ 2 (s)db(s) + C a,b [,T ] ϕ(s)da(s) ϕ(s)dx(s). Proof. It suffices to show the case in which the functional F is bounded on C a,b [, T ]. Let us first consider the case F is bounded, continuous, and F (y) for any y x C a,b [, T ] : x > M where M >. Then F (x + x ) is measurable and there exists a positive real number K such that F (x) K for all x C a,b [, T ]. Let τ : t < t 1 < < t n T be a partition of [, T ]. Define a function G on R n by G( ξ) F ([ ξ] n ) for each ξ R n. Then G is bounded and continuous. Hence we see that (3.14) F ([y] n ) G(y(t 1 ),, y(t n )). By using (3.14) and the Change of Variables Theorem, we have (3.15) F ([y] n )dµ(y) C a,b [,T ] C a,b [,T ] ( (2π) n R n G(y(t 1 ),, y(t n ))dµ(y) n () ) 1/2 G(v1,, v n ) exp 1 2 ((v j a(t j )) (v j 1 a(t j 1 ))) 2 d v.

7 Translation theorem on function space 23 Let β j x (t j ) and u j v j β j. Then we see that (3.16) F ([x + x ] n ) G(x(t 1 ) + x (t 1 ),, x(t n ) + x (t n )) G(x(t 1 ) + β 1,, x(t n ) + β n )) and (3.17) ((v j a(t j )) (v j 1 a(t j 1 ))) 2 ((u j a(t j )) (u j 1 a(t j 1 ))) 2 + (β j β j 1 ) 2 + 2(β j β j 1 )((u j a(t j )) (u j 1 a(t j 1 ))). By applying (3.16) and (3.17) above to the last equation of (3.15), we have (3.18) C a,b [,T ] F ([y] n )dµ(y) exp 1 (x (t j ) x (t j 1 )) 2 F ([x + x ] n ) 2 b(t j ) b(t j 1 ) C a,b [,T ] ((x(t j ) a(t j )) (x(t j 1 ) a(t j 1 ))) exp b(t j ) b(t j 1 ) (x (t j ) x (t j 1 )) dµ(x). Assume that x + x x C a,b [, T ] : y > M. Then we see that [x + x ] n x + x M and so we have (3.19) x(t j ) x (t j ) + x(t j ) + x (t j ) x + M

8 24 Jae Gil Choi and Young Seo Park for any x C a,b [, T ]. By using (3.11) and (3.19), we obtain that ((x(t j ) a(t j )) (x(t j 1 ) a(t j 1 )))(x (t j ) x (t j 1 )) b(t j ) b(t j 1 ) ϕ(ξ j )(x(t j ) x(t j 1 )) ϕ(ξ j )(a(t j ) a(t j 1 )) ϕ(ξ j )(x(t j ) x(t j 1 )) n + ϕ(ξ j )(a(t j ) a(t j 1 )) ϕ(ξ n)x(t n ) ϕ(ξ n ) x(t n ) + (ϕ(ξ j ) ϕ(ξ j 1 ))x(t j 1 ) + ϕ(ξ n)a(t n ) (ϕ(ξ j ) ϕ(ξ j 1 ))a(t j 1 ) ϕ(ξ j ) ϕ(ξ j 1 ) x(t j 1 ) + ϕ(ξ n ) a(t n ) + ( x + M + a ) ( ϕ + V T (ϕ)) ϕ(ξ j ) ϕ(ξ j 1 ) a(t j 1 ) where V T (ϕ) is the total variation of ϕ on [, T ]. So the integrand of (3.18) is bounded by the following K exp( x + M + a )( ϕ + V T (ϕ)). Thus, by using the Bounded Convergence Theorem, (3.8), and (3.9), the expression (3.17) converges to exp 1 ϕ 2 (s)db(s) 2 F (x + x ) exp C a,b [,T ] ϕ(s)dx(s) + ϕ(s)da(s) dµ(x). Hence by using Lemma 2.1, we obtain the equation (3.12) above.

9 Translation theorem on function space 25 Now, let F be a nonnegative, bounded and continuous functional on C a,b [, T ]. For each n N, define the function M n by 1 ( u n) M n (u) n + 1 u (n u n + 1) (n + 1 u). Then M n is a continuous real valued function. Let F n (x) F (x)m n ( x ). Then F n satisfies the hypothesis of the first case. So proceeding as in the proof above, we obtain C a,b [,T ] F n (y)dµ(y) exp 1 2 C a,b [,T ] ϕ 2 (s)db(s) + F n (x + x ) exp ϕ(s)da(s) ϕ(s)dx(s) dµ(x). Since F n is a monotone increasing sequence of functionals, F n F as n. Hence by using the Monotone Convergence Theorem, we have the desired result. Theorem 3.4. Let ϕ, x, and F be given as in Theorem 3.3. Then C a,b [,T ] F (x + x )dµ(x) exp 1 ϕ 2 (s)db(s) ϕ(s)da(s) 2 F (x) exp ϕ(s)dx(s) dµ(x). C a,b [,T ] Proof. Let G(x) F (x) exp ϕ(s)dx(s). Then by using equa-

10 26 Jae Gil Choi and Young Seo Park tion (2.12), we have C a,b [,T ] C a,b [,T ] C a,b [,T ] C a,b [,T ] F (x) exp ϕ(s)dx(s) dµ(x) G(x)dµ(x) G(x + x )J(x, x )dµ(x) F (x + x ) exp ϕ(s)d(x(s) + x (s)) exp 1 ϕ 2 (s)db(s) + ϕ(s)da(s) 2 1 T exp ϕ 2 (s)db(s) + ϕ(s)da(s) 2 C a,b [,T ] ϕ(s)dx(s) dµ(x) F (x + x )dµ(x). Hence we have the desired result. 4. The general translation theorem In this section we consider the general translation theorem for functionals on C a,b [, T ]. For u, v L 2 a,b [, T ], let (u, v) a,b u(t)v(t)d[b(t) + a (t)]. Then (, ) a,b is an inner product on L 2 a,b [, T ] and u a,b (u, u) a,b is a norm on L 2 a,b [, T ]. In particular note that u a,b if and only if u(t) a.e. on [, T ]. Furthermore (L 2 a,b [, T ], a,b) is a separable Hilbert space. Let e j be a complete orthogonal set of real-valued functions of bounded variation on [, T ] such that, j k (e j, e k ) a,b 1, j k,

11 and for each v L 2 a,b [, T ], let Translation theorem on function space 27 v n (t) (v, e j ) a,b e j (t) for n 1, 2,. Then for each v L 2 a,b [, T ], the Paley-Wiener- Zygmund(PWZ) stochastic integral v, x is defined by the formula v, x lim n v n (t)dx(t) for all x C a,b [, T ] for which the limit exists. Remark 4.1. Following are some facts about the PWZ stochastic integral (i) For each v L 2 a,b [, T ], the PWZ integral v, x exists for µ-a.e. x C a,b [, T ]. (ii) The PWZ integral v, x is essentially independent of the complete orthonormal set e j. (iii) If v BV [, T ], then PWZ integral v, x equals the Riemann -Stieltjes integral v(s)dx(s) for s-a.e. x C a,b[, T ]. (iv) The PWZ integral has the expected linearity properties. Lemma 4.1. Let ϕ n C a,b [, T ] BV [, T ] for each n N and let ϕ n converge in the space L 2 a,b [, T ] as n. Then for any real number λ, expλ ϕ n(t)dx(t) converges in the space L 2 (C a,b [, T ]) as n. Proof. The proof given in [3] with the current hypotheses on a(t) and b(t) also works here. Now, we obtain the general translation theorem of a functional on C a,b [, T ]. Theorem 4.2. Let ϕ(t) L 2 a,b [, T ] and let x (t) t ϕ(s)db(s). If F be a B(C a,b [, T ])-measurable functional on C a,b [, T ], then F (x+x ) is B(C a,b [, T ])-measurable and F (y)dµ(y) exp 1 2 (ϕ2, b ) + (ϕ, a ) (4.1) C a,b [,T ] C a,b [,T ] F (x + x ) exp ϕ, x dµ(x)

12 28 Jae Gil Choi and Young Seo Park where and (ϕ 2, b ) (ϕ, a ) ϕ 2 (t)b (t)dt ϕ(t)a (t)dt ϕ(t)db(t) ϕ(t)da(t). Proof. It suffices to show the case in which the functional F is bounded and continuous on C a,b [, T ]. Let e j be complete orthonormal set in L 2 a,b [, T ] with e j C a,b [, T ] BV [, T ] for each j N. For each n N, let (4.2) ϕ n (t) (ϕ, e j ) a,b e j (t). Then ϕ n C a,b [, T ] BV [, T ] and (4.3) ϕ n ϕ a,b as n. For each n N, define (4.4) x,n (t) t ϕ n (s)db(s). Then, by using equation (3.12), we see that (4.5) C a,b [,T ] exp 1 2 F (y)dµ(y) ϕ 2 n(s)db(s) + C a,b [,T ] ϕ n (s)da(s) F (x + x,n ) exp ϕ n (s)dx(s) dµ(x).

13 Translation theorem on function space 29 By using Cauchy-Schwarz inequality, we have t x (t) x,n (t) (ϕ(s) ϕ n (s))db(s) (4.6) (ϕ(s) ϕ n (s))χ [,t] (s) d[b(s) + a (s)] ϕ ϕ n a,b b(t) + a (t) ϕ ϕ n a,b b(t ) + a (T ). Hence by using (4.6) and (4.3), we obtain that (4.7) x x,n. Thus for all x C a,b [, T ] (4.8) F (x + x,n ) F (x + x ). Since F is bounded, by applying the Bounded Convergence Theorem, we have (4.9) F (x + x,n ) F (x + x ) 2 dµ(x). C a,b [,T ] Note that ϕ n (t)dx(t) ϕ, x µ a.e.x C a,b [, T ]. So by using (4.3) and Lemma 4.1 with λ 1 exp ϕ n (t)dx(t) exp ϕ, x in the space L 2 (C a,b [, T ]). Further, by equation (4.9) above, we obtain (4.1) C a,b [,T ] F (x + x,n ) exp C a,b [,T ] ϕ n (t)dx(t) dµ(x) F (x + x ) exp ϕ, x dµ(x). Hence by using equations (4.5) and (4.1) we have the desired equation (4.1). We next use Theorem 4.2 to evaluate a translation theorem of functionals on C a,b [, T ].

14 3 Jae Gil Choi and Young Seo Park Theorem 4.3. Let ϕ, x, and F be given as in Theorem 4.2. Then F (x + x )dµ(x) C a,b [,T ] exp 1 2 (ϕ2, b ) (ϕ, a) C a,b [,T ] F (x) exp ϕ, x dµ(x). Proof. Let G(x) F (x) exp ϕ(s)dx(s). Then, proceeding as in the proof of Theorem 3.4, we have the desired result. References [1] R.H. Cameron and W.T. Martin, Translations of Wiener integrals under translations, Annals of Math. 45(2), [2] R.H. Cameron and R.E. Graves, Additive functionals on a space of continuous functions I, Trans. Amer. Math. Soc. 7, [3] S.J. Chang and D.M. Chung, Conditional function space integrals with applications, Rocky Mountain J. Math , [4] S.J. Chang, C. Park and D. Skoug, Translation theorems for Fourier-Feynman transforms and conditional Fourier-Feynman transforms, Rocky Mountain J. of Math. 3(2), [5] G.W. Johnson and D. Skoug, Notes on the Feynman integral I, Pacific J. Math. 93, [6], Notes on the Feynman integral II, J. Func. Anal. 41, [7], Notes on the Feynman integral III : The Schrödinger equation, Pacific J. Math. 15, [8] J. Yeh, Stochastic Processes and the Wiener Integral, Marcel Dekker, Inc., New York, Department of Mathematics Dankook University Cheonan , Korea jgchoi@dankook.ac.kr Department of Mathematics Dankook University Cheonan , Korea yseo@dankook.ac.kr

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