廣義布朗泛函之條件期望值

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(1)國立高雄大學應用數學系碩士論文. 廣義布朗泛函之條件期望值 Conditional Expectation of Generalized Brownian Functionals. 研究生：歐濟華撰指導教授：施信宏. 中華民國 102 年 7 月.

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(3) 誌謝在電機工程領域中已有近二十年的時間，每當想要詳細地探討工程應用的完整理論時，總是會遇到數學能力之不足。因此，一直有想要在數學領域上仔細研讀的願望。在經過高雄大學應用數學系的教導栽培之下，數學能力已逐漸成長，進而有很大的進步。甚至在電機工程領域上，因為有數學的基礎可以更能相得益彰。感激之意，環繞心中。首先，謹向指導教授施信宏博士致上最大敬意，感謝在他的全力指導之下，本論文得以順利完成。碩士班期間以來，他教導我正確的方法，使我能循序漸進，有條不紊地走入數學領域之中。在數學研究上，他宛如一盞明燈，帶領學生走向正確的研究之路。師恩浩蕩，永銘在心。另外也要感謝同實驗室林奕均同學和博士班學長鄭益新，他們總是適時地提供協助，正確地幫助我解決研究上的問題。也要感謝系辦公室的千惠姐和雅鳯姐，她們不厭其煩地協助我完成該做的行政事務，讓我碩士班的日子，一路走來平坦順利。最後，我要由衷地感謝我的家人：父母、嘉聆、承儒和承遠。您們的鼓勵和無微不至的照顧，一直是我最大的支柱，也是我心中快樂的來源。有了您們，我所有成就都有了存在的意義。如今，謹將這份喜悅及成果獻給您們。.

(4) Conditional Expectation of Generalized Brownian Functionals. by Chi-Hua Ou Advisor Hsin_Hung Shih. Department of Applied Mathematics, National University of Kaohsiung Kaohsiung, Taiwan 811, R.O.C. July 2013.

(5) Contents 中文摘要. ii. 英文摘要. iii. List of Notations. iv. 1. Introduction. 1. 2. Preliminaries. 2. 3. Classical Wiener Space with Time Interval (− ∞,+∞ ). 4. 4. Gel’fand Triple Centered Around Cameron-Martin Space. 17. 5. A Representation of Conditional Expectations of Generalized Brownian Functionals. 24. References. 33. i.

(6) 廣義布朗泛函之條件期望值指導教授：施信宏教授國立高雄大學應用數學系. 學生：歐濟華國立高雄大學應用數學系. 摘要. 根據重複對數法則，布朗運動具有漸近之上界包絡線。根據上述之事實，本論文定義一個函數空間，此函數空間的函數具有如下性質：函數除以 1 + t 2 將會趨近於 0，當 t 趨近無限大。藉由此函數空間，建構標準的可數希爾伯特空間之方法將被提出。最後，在本論文中，我們將針對可積分的布朗泛函推導一些重要結果，相應地，也將定義廣義布朗泛函之條件期望值。關鍵字：布朗運動、可數希爾伯特空間、廣義布朗泛函關鍵字. ii.

(7) Conditional Expectation of Generalized Brownian Functionals Advisor: Professor Hsin-Hung Shih Department of Applied Mathematics National University of Kaohsiung. Student: Chi-Hua Ou Department of Applied Mathematics National University of Kaohsiung. ABSTRACT. By the law of the iterated logarithm, Brownian motion has the asymptotic upper envelope almost surely. Based on the fact, the paper provides the space of the continuous functions which vanish at infinity by dividing. 1 + t 2 . By the space of functions, a method. to construct a standard countably Hilbert space is presented. Finally, in the paper, we propose some results to integrable Brownian functionals and, accordingly, give a definition of conditional expectations of generalized Brownian functionals. Keywords: Brownian motion, countably Hilbert space, generalized Brownian functionals. iii.

(8) List of Notations Notations. (1) For any bounded linear operator T on a normed linear space, ∥T ∥ means the operator norm of T . (2) For any metric space M , B(M ) is the Borel σ-field of M . (3) For any locally convex space X with its dual X ∗ , (x, F )X,X ∗ , x ∈ X and F ∈ X ∗ , means the X-X ∗ dual-pairing. (4) For a signed measure ν on an σ-field, |ν| means the total variation of ν. (5) For a real normed space X with the |·|X -norm, Xc = X +i X is the complexification of X with the | · |Xc -norm given by |x + i y|Xc = sup{∥eiθ (x + i y)∥Xc : θ ∈ [0, 2π]} for x, y ∈ X, where ∥x + i y∥2Xc = |x|2X + |y|2X . (6) For any function f : A → B, if E is a subspace of A, the restriction of f to E is denoted by f |E . (7) Let Σ be a real separable Hilbert space. Then HS n (Σ), n ∈ N, is the Hilbert space of all n-linear Hilbert-Schmidt operators from Σ × · · · × Σ to itself with the inner product ⟨⟨A, B⟩⟩HS n (Σ) and the corresponding Hilbert-Schmidt norm ∥ · ∥HS n (Σ) . (8) sgn(a) = 1 as a ≥ 0 and sgn(a) = −1 as a < 0.. iv.

(9) 1. Introduction. Brownian motion can be considered as the macroscopic picture emerging from a particle moving randomly in n-dimensional space. In the paper, we study one-dimensional Brownian motion. In most practical example, we often use random walk with some assumptions to create the Brownian motion. In fact, Brownian motion has some limitation, for examples, over a long time, Brownian motion can not grow arbitrarily fast. The low of the iterated logarithm presents the asymptotic upper envelope of the Brownian motion. By the law of the iterated logarithm, Brownian motion has the asymptotic upper envelope almost surely. Based on the fact, first of all, we define the space of the continuous √ functions which vanish at infinity by dividing 1 + t2 and denote it by C . We will develop a countably Hilbert space from C . And, by using the fact of Gel’fand triple, we then get a continuous inclusions which includes C . In the continuous inclusions, the new test function E and the new generalized functionals E ′ are obtained. The test function E and generalized functionals E ′ are similar as the Schwartz space S and its dual space S ′ , that is, the natural bilinear pairing of E and E ′ is equal to the natural bilinear pairing of S and S ′ . Note that (S ′ , µ) is called a white noise space, where µ is the standard Gaussian measure on S ′ . Based on E and E ′ , in the paper, an integral representation concerning conditional expectations of integrable random variables on a Banach space with respect to a Gaussian measure will be obtained. As an important example, we then apply this result to integrable Brownian functionals (BF’s, for short) ,and accordingly, give a definition of conditional expectations of generalized Brownian functionals (GBF’s for short) (resp. generalized white noise functionals (GWNF’s, for short)). These definitions will provide us a scheme for studying the generalized stochastic processes. The thesis is organized as follows. In Section2, we briefly introduce some definitions and theorems which will be used in the subsequent study. In Section 3, first of all, we define two spaces of the functions. And, an identification of the dual space of C is presented. In Section 4, by the two spaces of the functions, a continuous inclusions are obtained. Hence, we have the new test function E and the new generalized functionals E ′ . Finally, in Section 5, we show some important results to integrable Brownian functionals and give a definition of conditional expectations of generalized Brownian functionals. 1.

(10) 2. Preliminaries. In the section, we give a brief review of the deification of Brownian motion. Also the iterated logarithm of Brownian motion is described. Finally, Schwartz space S and its dual space S ′ are provided.. 2.1. Brownian Motion and the Law of the Iterated Logarithm. Let B = {B(t), t ∈ R} be a stochastic process. In the following, we give the definition of Brownian motion. Definition 2.1. [3] A stochastic process B = {B(t), t ∈ R} is called a Brownian motion if it satisfies the following three conditions. (1) B is a Gaussian system with E (B(t)) = 0 for every t. (2) B(0) = 0. [ ] (3) E (B(t) − B(s))2 = |t − s|. It is noted that X = {X(t) ≡ B(t); t ≥ 0} and Y = {Y (t) ≡ B(−t); t ≥ 0} are both one-dimensional standard Brownian motions starting from origin. Over a long time, Brownian motion can not grow arbitrarily fast. The law of the iterated logarithm provides an envelope, which determines the asymptotic growth of a Brownian motion. The following theorem shows the fact. Theorem 2.2. [8] Suppose {B(t) : t ≥ 0} is a standard one-dimensional Brownian motion. Then, almost surely, lim sup √ t→∞. 2.2. B(t, ω) =1 2t log log t. and lim inf √ t→∞. B(t, ω) = −1, 2t log log t. Schwartz Space and its Dual Space. We will give the definition of Schwartz space, which is consist of real-valued rapidly decreasing function on R.. 2.

(11) Definition 2.3. A function f on R is called rapidly decreasing if it is smooth and for all n, k ∈ N0 , |tn f (k) (t)| → 0. as |t| → ∞.. The Schwartz space S is defined as S = {f : f is real-valued rapidly decreasing function on R }. Furthermore, S ′ denotes the dual space of S. In fact, the probability (S ′ , µ) is called a white noise space, where the measure µ is the standard Gaussian measure on S ′ .. 3.

(12) 3. Classical Wiener Space with Time Interval (−∞, +∞). Let B = {B(t); t ∈ R} be an one-dimensional standard Brownian motion. It is noted that X = {X(t) ≡ B(t); t ≥ 0} and Y = {Y (t) ≡ B(−t); t ≥ 0} are both one-dimensional standard Brownian motions starting from origin. Then, by applying the law of iterated logarithm to X and Y , we have that, for almost all ω, B(t, ω) B(t, ω) = 1 and lim inf √ = −1, lim sup √ t→±∞ t→±∞ 2|t| log log |t| 2|t| log log |t| from which it follows that (2|t| log log |t|)−1/2 B(t, ω) is bounded by 2 as t is large enough, and hence. |B(t, ω)| lim sup √ ≤ 2 lim t→+∞ t→±∞ 1 + t2. √. 2t log log t = 0. 1 + t2. (3.1). Hence, if f is a sample path of B = {B(t); t ∈ R}, then f will satisfy the condition as shown in (3.1).. 3.1. Two Spaces of Real-Valued Functions. By the property as given in (3.1), we define the space C as |ϕ(t)| = 0}. C = {ϕ : R → R| ϕ is continuous f unction with ϕ(0) = 0 and lim √ t→±∞ 1 + t2 For each ϕ ∈ C , define. |ϕ(t)| |ϕ|C = sup √ . 1 + t2 t∈R. (3.2). It is obvious that | · |C is a norm on C . Then, we have the following Lemma. Lemma 3.1. (C , | · |C ) is a Banach space. Proof. First of all, we prove C is a linear space. Suppose α ∈ R and f, g ∈ C . |f (t)| lim αf = |α| lim √ = |α| · 0 = 0. t→±∞ t→±∞ 1 + t2 |f (t) + g(t)| |f (t)| + |g(t)| |f (t)| |g(t)| √ √ ≤ lim ≤ lim √ + lim √ = 0. t→±∞ t→±∞ t→±∞ 1 + t2 1 + t2 1 + t2 t→±∞ 1 + t2 Furthermore, in terms of f and g are continuous functions, f + g is continuous function. 0 ≤ lim. Hence, f + g ∈ C . That is, C is a linear space. 4.

(13) √ Let {f1 , f2 , . . .} be a Cauchy sequence in C . Suppose 0 < T < ∞, ∀ ε/(2 1 + T 2 ) > √ 0, ∃N ∈ N, ∋ |fm − fn |C < ε/(2 1 + T 2 ) > 0, ∀m, n ≥ N. Hence, ∀t ∈ [−T, T ], |fm − fn | √ |fm − fn | ≤ √ 1 + T2 2 1+t |fm − fn | √ ≤ sup √ 1 + T2 2 1+t t∈R √ < |fm − fn |C 1 + T 2 √ √ ≤ ε/(2 1 + T 2 ) 1 + T 2 ≤ ε. Let t0 ∈ [−T, T ], |fm (t0 ) − fn (t0 )| < ε, ∀m, n ≥ N . This leads that (f1 (t0 ), f2 (t0 ), . . .) is a Cauchy sequence of real number. In terms of R is complete, fn (t0 ) → f (t0 ) as n → ∞. Hence, we can get a function fT (t) and it has that fn (t) → fT (t), ∀t ∈ [−T, T ]. Let T2 > T1 , by using the same above steps, we can create the function fT2 such that fT2 (t) = fT1 (t) as t ∈ [−T1 , T1 ]. Let T2 → ∞, {fn (t)} uniformly converges to f (t) on (−∞, ∞). Since fn is continuous, f (t) is a continuous function. It is obvious that √ f (0) = limn→∞ f (0) = 0 and limt→±∞ |f (t)|/ 1 + t2 = 0. Since f (t) ∈ C , (C , | · |C ) is a Banach space. In the following, define the space H as H = {ϕ : R → R| ϕ is absolutely continuous f unction with ϕ(0) = 0 and ϕ˙ ∈ L2 (R)}. H is a Hilbert space with | · |H -norm induced by the inner product ⟨·, ·⟩H , where ∫ ∞ ˙ ψ(t) ˙ dt f or any ϕ, ψ ∈ H . ϕ(t) ⟨ϕ, ψ⟩H = −∞. By the properties of C and H , we have the following lemma. Lemma 3.2. H is a subspace of C . In particular, H is a dense subspace of C .. 5.

(14) Proof. Suppose f ∈ H , i.e., f˙ exists a.e. on R and. ∫∞ −∞. |f˙|2 dt = M < ∞.. ∫ t 1 |f (t)| lim √ ≤ lim √ |f˙(t)|dt t→∞ 1 + t2 t→∞ 1 + t2 ∫0 ∞ 1 = lim √ 1[0,t] (t)|f˙(t)|dt t→∞ 1 + t2 −∞ ) 21 (∫ ∞ ) 21 (∫ ∞ 1 ≤ lim √ 12 [0,t] (t)dt · |f˙(t)|2 dt t→∞ 1 + t2 −∞ −∞ √ tM ≤ lim √ t→∞ 1 + t2 = 0. Similarly, it is follows that limt→−∞. |f (t)| √ 1+t2. = 0. In the following, we prove that H is a. dense subspace of C . For 0 < T < ∞, define     f (−T ), t < −T C T = {g : R → R| g(t) = f (t), t ∈ [−T, T ] , where f ∈ C }    f (T ), t>T T Hence, if 0 < T1 < T2 , then C T1 ⊂ C T2 . It is trivial that C T ⊂ C and ∪∞ ↗ C. T =1 C. Define. AnT.   g(−T ),         q ∈ Q, = {g : R → R| g(t) = =0     p ∈ Q,      g(T ), ,. t < −T t = − 2kn T, k = 1, 2, . . . , n t=0 t=. k T, 2n. k = 1, 2, . . . , n. t>T and linear between these binary points}.. By the definition of A , if f ∈ AnT , then f (0) = 0, f˙ exists a.e. and f˙ ∈ L2 (R). It is T follows that AnT ⊂ H and ∪∞ n=1 An ⊂ H . T T We define the norm |f |T∞ = supt∈R in C T , Under | · |T∞ , ∪∞ n=1 An is a dense subset of C .. Since ∞. ∞. ∞. C = ∪ C T = ∪ ∪ AnT , T =1 n=1. T =1. T T ∞ hence, ∪∞ T =1 ∪n=1 An is a dense in C . Since H is a Hilbert space and An ⊂ H , hence. H is a dense subset of C . 6.

(15) Moreover, 1 | · |C ≤ √ | · |H 2 on H , since, for any ϕ ∈ H , |ϕ(t)| √ ≤ 1 + t2. (3.3).

(16) √ ∫ |t|

(17) ˙

(18) ds

(19) ϕ(s) |t| 0 √ ≤ |ϕ|H · , 2 1 + t2 1+t. ∀ t ∈ R.. Let iH ,C : H → C be the inclusion map. Then iH ,C is continuous by (3.3). Let C ∗ be the dual of C and i∗H ,C be the adjoint operator of iH ,C . Applying the Riesz representation theorem to identifying H with its dual, we have ⟨i∗H ,C (F ), ϕ⟩H = (ϕ, F )C ,C ∗ ,. 3.2. ∀ F ∈ C ∗, ϕ ∈ H .. A Characterization of i∗H ,C (C ∗ ). In this subsection, our aim is to give an explicit description of i∗H ,C (C ∗ ) as a subspace of H . First of all, for a locally compact Hausdorff space Ξ, we denote by C0 (Ξ) the Banach space consisting of all continuous function on Ξ vanishing at infinity with the uniform norm | · |∞ . For any ϕ ∈ C , it is associated a function ϕe ∈ C0 ((−∞, +∞)) given by { } √ 2 e e e ϕ(t) = ϕ(t)/ 1 + t . Let C = ϕ; ϕ ∈ C . Then Ce ⊂ C0 ((−∞, +∞)) ⊂ C . One notes

(20)

(21)

(22)

(23) that for any ϕ ∈ C , ϕe

(24) ∈ C0 ((0, +∞)); and similarly, ϕe

(25) ∈ C0 ((−∞, 0)). (0, +∞). (−∞, 0). ˙ y− be the function Conversely, for any y+ ∈ C0 ((0, +∞)) and y− ∈ C0 ((−∞, 0)), let y+ + defined by ˙ y− )(t) = (y+ +. Set ξ(t) =. √.    y+ (t), 0,    y− (t),. if t > 0, if t = 0, if t < 0.. ˙ y− ) ∈ C . Consider the mapping T from C into 1 + t2 , t ∈ R. Then ξ · (y+ +. C0 ((0, +∞)) × C0 ((−∞, 0)) by (

(26)

(27) T (ϕ) = ϕe

(28). (0, +∞).

(29)

(30) , ϕe

(31). ) , (−∞, 0). ϕ ∈ C.. ˙ y− ) Then it is a homeomorphism with the inverse T −1 given by T −1 (y+ , y− ) = ξ · (y+ + for any (y+ , y− ) ∈ C0 ((0, +∞)) × C0 ((−∞, 0)). If F ∈ C ∗ is given, there are two associated bounded linear functionals F+ , F− separately defined in C0 ((0, +∞)) and C0 ((−∞, 0)) by F+ (y+ ) = F (T −1 ((y+ , 0))) and F− (y− ) = F (T −1 ((0, y− ))), 7.

(32) for any y+ ∈ C0 ((0, +∞)) and y− ∈ C0 ((−∞, 0)). Then, for any ϕ ∈ C , ) (

(33) (

(34)

(35)

(36) e F (ϕ) = F (ϕ · 1(0, +∞) ) + F (ϕ · 1(−∞, 0) ) = F+ ϕ

(37) + F− ϕe

(38) (0, +∞). ) . (−∞, 0). Applying the Riesz representation theorem to F+ and F− , there exist uniquely two regular signed measures λF+ and λF+ separately on B((0, +∞)) and B((−∞, 0)) such that ∫ ∫ e λF (dt), ∀ ϕ ∈ C . e (3.4) ϕ(t) F (ϕ) = ϕ(t) λF+ (dt) + − (−∞, 0). (0, +∞). Moreover, ∥F+ ∥ = |λF+ |((0, +∞)) and ∥F− ∥ = |λF− |((−∞, 0)). Remark 3.3. For F ∈ C ∗ , ∥F ∥ = CF := |λF+ |((0, +∞)) + |λF− |((−∞, 0)). In fact, it is obvious that ∥F ∥ ≤ CF . On the other hand, we take sequences {yn } of C0 ((0, +∞)) and {zn } of C0 ((−∞, 0)) such that |yn |∞ = 1, |zn |∞ = 1, |F+ (yn )| → ∥F+ ∥, and |F− (zn )| → ∥F− ∥. Set εn = sgn(F+ (yn )) and ϑn = sgn(F− (zn )). Let ˙ n zn ). ϕn = ξ · (εn yn +ϑ Then ϕn ∈ C with |ϕn |C = 1. Moreover, lim F (ϕn ) = lim εn · F+ (yn ) + lim ϑn · F− (zn ). n→+∞. n→+∞. n→+∞. = lim |F+ (yn )| + lim |F− (zn )| = ∥F+ ∥ + ∥F− ∥ = CF . n→+∞. n→+∞. For F ∈ C ∗ , it is associated a continuous mapping ΦF on R given by ΦF (0) = 0 and )  ∫ t (∫ 1   √ λF+ (du) ds, as t > 0,  2 1 + u 0 (s, +∞) ) ΦF (t) = ∫ 0 (∫ (3.5) 1   √  λF− (du) ds, as t < 0. 1 + u2 t (−∞, s] Proposition 3.4. Let F ∈ C ∗ and ΦF be given as in (3.5). Then ΦF admits the following properties: (i) ΦF is in H . (ii) The derivative Φ˙ F is right continuous with lim. t→±∞. 8. √. 1 + t2 |Φ˙ F (t)| = 0..

(39) (iii) Φ˙ F is a function of bounded variation on (−∞, +∞). (iv) Let VΦ˙ F be the function on (−∞, +∞) such that VΦ˙ F (t) is defined to be the total variation of Φ˙ F on (−∞, t] for any t ∈ R. Then ∫ ∞√ 1 + t2 d VΦ˙ F (t) < +∞. −∞. Proof. First of all, we note that for any a < b in R, ∫ b 1 √ |ΦF (b) − ΦF (a)| ≤ CF · ds, 1 + s2 a. (3.6). where CF is the same as one in Remark 3.3. From (3.5) it follows that ΦF is absolutely continuous in R. Applying the Cauchy-Schwartz inequality to the right-hand integral of (3.6), we see that |ΦF (b) − ΦF (a)| ≤. √. π CF ·. √. b − a,. by which it follows that √ |ΦF (t)| lim √ ≤ π CF · lim t→±∞ t→±∞ 1 + t2. √. |t| = 0. 1 + t2. On the other hand, applying the Lebesgue’s differentiation theorem to (3.5), ΦF is differentiable with derivative Φ˙ F (t), where ∫ 1   √ λF+ (du), as t ≥ 0,  2 1 + u (t, ∞) Φ˙ F (t) = ∫ −1   √ λF− (du), as t < 0,  1 + u2 (−∞, t]. (3.7). at almost every t ∈ R with respect to the Lebesgue measure. Identifying Φ˙ F with the right-hand term of (3.7), Φ˙ F is right continuous and |Φ˙ F (t)|2 ≤ So, ΦF ∈ H and |ΦF |H ≤. C2F , 1 + t2. √. ∀ t ∈ R.. π CF < +∞. In addition, it follows from (3.7) that   lim |λF+ |((t, +∞)) √ 2 ˙ =0 lim 1 + t |ΦF (t)| ≤ t→+∞ t→±∞  lim |λF |((−∞, t]) − t→−∞. 9.

(40) So far, we have completed the proof of (i) and (ii). To prove (iii), let {t0 < t1 < · · · < tj < 0 ≤ tj+1 < · · · < tn } be an arbitrarily selected sequence. Then n−1 ∑. j−1 ∑. |Φ˙ F (tj+1 ) − Φ˙ F (tj )| ≤. j= 0. |Φ˙ F (ti+1 ) − Φ˙ F (ti )| +. i= 0. n−1 ∑. |Φ˙ F (ti+1 ) − Φ˙ F (ti )|. i= j+1. + |Φ˙ F (tj+1 ) − Φ˙ F (0)| + |Φ˙ F (0)| + |Φ˙ F (tj )| ≤ 2 · CF , and thus Φ˙ F is a function of bounded variation on R. Finally, by the Radon-Nikodym theorem, there are two Borel measurable functions ζ+ and ζ− respectively on (0, +∞) and (−∞, 0) such that |ζ+ (t)| = |ζ− (u)| = 1 for any t, u and the Radon-Nikodym derivative [ ] − d λF± / d |λF± | is ζ± . Let µ+ ˙ F be the Lebesgue-Stieltjes measures corresponding ˙ F and µΦ Φ to Φ˙ F respectively on (0, +∞) and (−∞, 0). Then, for any 0 ≤ a < b and c < d < 0, ∫ µ+ ˙ F ((a, Φ. (a, b]. (a, b]. ∫

(41)

(42) 1 1

(43) −

(44) √ √ |λF+ |(dt) and

(45) µΦ˙

(46) ((c, d]) = |λF− |(dt). F 1 + t2 1 + t2 (c, d]. b]) = −. which implies that ∫

(47)

(48)

(49) +

(50)

(51) µΦ˙

(52) ((a, b]) = F. ∫. ζ (t) √+ |λF+ |(dt) and µ− ˙ F ((c, d]) = − Φ 1 + t2. (c, d]. ζ (t) √− |λF− |(dt), 1 + t2. For any 0 ≤ a < b, let Γ = {a = s0 < s1 < · · · < sm = b} be a arbitrary partition of [a, b]. By the right continuity of Φ˙ F , m ∑. VΦ˙ F (b) − VΦ˙ F (a) = sup Γ. |Φ˙ F (sj ) − Φ˙ F (sj−1 )|. j=1. = lim+ sup ε→0. { |Φ˙ F (s1 ) − Φ˙ F (a + ε)| +. Γ. m ∑ j=2.

(53)

(54)

(55)

(56)

(57) +

(58)

(59) +

(60) ≤ lim+

(61) µΦ˙

(62) ((a + ε, b]) =

(63) µΦ˙

(64) ((a, b]). F F ε→0. Similarly, for any c < d < 0,

(65)

(66)

(67) −

(68) VΦ˙ F (d) − VΦ˙ F (c) ≤

(69) µΦ˙

(70) ((c, d]). F 10. } |Φ˙ F (sj ) − Φ˙ F (sj−1 )|.

(71) Therefore, ∫ ∞√ −∞. ∫ 1 + t2 dVΦ˙ F (t) =. ∞. √. ∫. 0. ∫ 1 + t2 dVΦ˙ F (t) + √. ≤. (0, +∞) ∫ 0√. +. −1.

(72)

(73) 1 + t2

(74) µ+ ˙F Φ. 0. √. 1 + t2 dVΦ˙ F (t) ∫

(75) √

(76) 1 + t2

(77) (dt) + −∞. (−∞, −1].

(78)

(79)

(80) −

(81) µ

(82) Φ˙ F

(83) (dt). 1 + t2 dVΦ˙ F (t) ∫. = |λF+ |((0, +∞)) + |λF− |((−∞, −1]) +. 0. −1. √. 1 + t2 dVΦ˙ F (t). < + ∞. The proof is complete. We are ready to present a characterization of the dual C ∗ of C as a subspace of H . Theorem 3.5. (i) For any F ∈ C ∗ , i∗H ,C (F ) = ΦF . (ii) The image i∗H ,C (C ∗ ) coincides with the subspace of all functions ϕ ∈ H with the properties (ii) − (iv) stated in Proposition 3.4. (ii) Identifying C ∗ with i∗H ,C (C ∗ ), we have the duality relation: For any F ∈ C ∗ and ϕ ∈ C,. ∫ (ϕ, F )C ,C ∗ = −. ∞. ϕ(t) dF˙ (t).. −∞. (iv) For any F ∈ C ∗ , define |F |C ∗ to be the operator norm ∥F ∥. Then ∫ ∞√ 1 + t2 dVF˙ (t). |F |C ∗ = −∞. Proof. (i) For F ∈ C ∗ and ϕ ∈ H , since, by Proposition 3.4, ∫ ∞ ∫ ∞√ |ϕ(t)| dVΦ˙ F (t) ≤ |ϕ|C · 1 + t2 dVΦ˙ F (t) < +∞, −∞. the Riemann-Stieltjes integral. −∞. ∫∞ −∞. ϕ(t) dΦ˙ F (t) exists. Let ΓM = {tj }, M > 0, be a. partition of [0, M ], and let mj be the infimum of ϕ in [tj−1 , tj ]. Define the function. 11.

(84) ϕΓM on [0, M ] by setting ϕΓM (0) = 0 and ϕΓM = mj in (tj−1 , tj ]. Then, since ϕΓM → ϕ uniformly on [0, M ] as |ΓM | → 0, ∫ ∞ ∫ ϕ(t) dΦ˙ F (t) = lim. M. ϕ(t) dΦ˙ F (t) ∫ M = lim lim ϕΓM (t) dΦ˙ F (t) M →+∞ |ΓM |→0 0 ( ) ∑ mj · Φ˙ F (tj ) − Φ˙ F (tj−1 ) = lim lim M →+∞. 0. 0. M →+∞ |ΓM |→0. = − lim. j. lim. ∑. M →+∞ |ΓM |→0. ∫ = − lim. ∫ (tj−1 , tj ]. j. lim. M →+∞ |ΓM |→0. ∫. √. mj ·. (0, M ]. 1 λF+ (du) 1 + u2. ϕΓ (t) √M λF+ (dt) 1 + t2. ϕ(t) √ λF+ (dt) = − lim M →+∞ (0, M ] 1 + t2 ∫ ϕ(t) √ =− λF+ (dt). 1 + t2 (0, +∞). (3.8). Similarly, ∫. ∫. 0 −∞. ϕ(t) dΦ˙ F (t) =. ∫. 0−. −∞. ϕ(t) dΦ˙ F (t) = − (−∞, 0). ϕ(t) √ λF− (dt). 1 + t2. (3.9). One notes that, by (ii) in Proposition 3.4, lim|t|→+∞ ϕ(t) Φ˙ F (t) = 0. Then, combining (3.8)–(3.9) with (3.4) and applying the integration by parts formula, we get ∫ ∞ ∫ ∞ ∫ ∞ ∗ ˙ dt, ⟨iH ,C (F ), ϕ⟩H = − ϕ(t) dΦ˙ F (t) = Φ˙ F (t) dϕ(t) = Φ˙ F (t) ϕ(t) −∞. −∞. (3.10). −∞. where the last equality is obtained by Remark 3.6. Therefore, i∗H ,C (F ) = ΦF . Conversely, if ξ ∈ H satisfying the properties (ii)–(iv) stated in Proposition 3.4, it is easy to see that ∫∞ ˙ is in C ∗ , and i∗ (Fξ ) = ξ. the functional Fξ which carries ϕ ∈ C into − −∞ ϕ(t) dξ(t) H ,C So we obtain the assertion (ii). For (iii), we take a sequence {ϕn } of H such that ϕn → ϕ in C . Then, by (iv) in Proposition 3.4 and (3.10), ∫ ∞ ∫ ϕn (t) dF˙ (t) = (ϕ, F )C ,C ∗ = lim ⟨F, ϕn ⟩H = − lim n→+∞. To prove (iv), set k(t) =. n→+∞. √. −∞. ∞. ϕ(t) dF˙ (t).. −∞. 1 + t2 and let K be the space of all real-valued measurable. functions f on R satisfying f /k ∈ L∞ (R). If f ∈ K, we define |f |K to be the essential 12.

(85) supremum of f /k. Then (K, |·|K ) is a Banach space and C is a closed subspace of K. For F ∈ C ∗ , we apply the Hahn-Banach theorem to see that there is an Fe ∈ K ∗ , the dual of K, such that Fe|C = F and ∥Fe∥ = ∥F ∥. For a fixed ϕ ∈ C , let Γa,b = {a = t0 < t1 < · · · < ∑ tn = b} be an arbitrarily given partition of [a, b], and let ϕΓa,b = nj=1 ϕ(tj−1 )·1[tj−1 , tj ) . Set ε(t) = (1[t, +∞) , Fe)K,K ∗ , t ∈ R. Observe that if {tj−1 = tj−1,0 < tj−1,1 < · · · < tj−1,n = tj } j. be a partition of [tj−1 , tj ], then { nj } n ∑ ∑ k(tj−1 ) |ε(tj−1,i−1 ) − ε(tj−1,i )| j=1. =. i=1. ( n nj ∑∑. ). sgn (ε(tj−1,i−1 ) − ε(tj−1,i )) k(tj−1 ) · 1[tj−1,i−1 , tj−1,i ) , Fe. j=1 i=1. ≤ |F |C ∗ . K,K ∗. This implies that ε is a function of bounded variation on R, and ∫ ∞ k(t) dVε (t) ≤ |F |C ∗ ,. (3.11). −∞. where Vε (t) is the total variation of ε on (−∞, t] for any t ∈ R. On the other hand, as |Γa,b | → 0, ϕΓa,b → ϕ · 1[a, b] in K, and thus ( ) (ϕ · 1[a, b] , Fe)K,K ∗ = lim ϕΓa,b , Fe. K,K ∗. |Γa,b |→0. = lim. |Γa,b |→0. n ∑. ∫. b. ϕ(tj−1 ) (ε(tj−1 ) − ε(tj )) = −. ϕ(t) d ε(t). a. j=1. Letting a → −∞ and b → +∞, we see that (ϕ, Fe)K,K ∗ = − Therefore, ε = F˙ and, for any ϕ ∈ C ,.

(86) ∫

(87) |(ϕ, F )C ,C ∗ | = |(ϕ, Fe)K,K ∗ | =

(88)

(89). ∞. −∞. ∫. ∞. ϕ(t) d ε(t). −∞.

(90) ∫

(91) ϕ(t) dF˙ (t)

(92)

(93) ≤ |ϕ|C ·. ∞. −∞. k(t) dVF˙ (t).. Together with (3.11), we obtain the assertion (iv). The proof is complete. ∫∞ Remark 3.6. For ϕ ∈ H and F ∈ C ∗ , the Riemann-Stieltjes integral −∞ F˙ (t) dϕ(t) ∫∞ ˙ dt. In fact, for a < 0 < b, let Γ = {tj } coincides with the Lebesgue integral −∞ F˙ (t) ϕ(t) be a partition of [a, b], and define a function g by setting g = F˙ (tj ) in (tj−1 , tj ]. Then ∫ b ∫ b ∑ ˙ dt. F˙ (t) dϕ(t) = lim F˙ (tj ) (ϕ(tj ) − ϕ(tj−1 )) = lim g(t) ϕ(t) a. |Γ|→0. |Γ|→0. j. 13. a.

(94) Since F˙ is right continuous, g → F˙ pointwise a.e. in [a, b] as |Γ| → 0. Note that F˙ is bounded in [a, b]. Then, by the Lebesgue dominated convergence theorem, ∫ b ∫ b ˙ dt, ˙ F (t) dϕ(t) = F˙ (t) ϕ(t) a. a. and the equality still holds as a → +∞ and b → −∞.. 3.3. Classical Wiener Measure on (C , B(C )). Let ω+ be a measure on the Borel σ-field of C ([0, +∞)) of all continuous functions on [0, +∞) taking the value zero at 0, given by { }− 21∫ { } n (un −un−1 )2 (u2 −u1 )2 1 u2 ∏ − + +···+ tn −tn−1 ω+ (I) = (2π)n (tj − tj−1 ) e 2 t1 t2 −t1 du1 du2 · · · dun , j=1. D. where I is a cylinder set of the form {ϕ ∈ C ([0, +∞); (ϕ(t1 ), . . . , ϕ(tn )) ∈ D (∈ B(Rn ))}, {tj } being n time points with 0 = t0 < t1 < t2 < · · · < tn . A well-known fact is that a standard Brownian motion with time interval [0, +∞) can be represented as the process B + = {B + (t) = ϕ(t); ϕ ∈ C ([0, +∞)), t ≥ 0} on the probability space (C ([0, +∞)), B(C ([0, +∞))), ω+ ). One notes that C ([0, +∞)) is a Polish space under the metric ρ defined by [( ) ] ∞ ∑ 1 max |ϕ(t) − ψ(t)| ∧ 1 , ϕ, ψ ∈ C ([0, +∞)). ρ(ϕ, ψ) = n 0≤t≤n 2 n=1 It is easy to see that the inclusion map from C + into C ([0, +∞)) is continuous, where { } C + = ϕ|[0, +∞) ; ϕ ∈ C is viewed as a separable Banach space endowed with | · |C + -norm { }

(95)

(96) e

(97)

(98) given by ϕ|[0, +∞) + = sup |ϕ(t)|; t ≥ 0 . Since C + and C ([0, +∞)) are both standard C. measurable spaces, C + ∈ B(C ([0, +∞))) (see [5, Theorem 2.1.1]). By the law of iterated logarithm, ω+ (C + ) = 1. Analogously to the above argument, let ω− be a measure on the Borel σ-field of C ((−∞, 0]) of all continuous functions on (−∞, 0] starting from the origin, given by { }− 21∫ { } n (un −un−1 )2 (u2 −u1 )2 u2 1 ∏ + +···+ − tn−1 −tn du1 du2 · · · dun , ω− (J) = (2π)n (tj−1 − tj ) e 2 −t1 t1 −t2 j=1. D′. where J is a cylinder set of the form {ϕ ∈ C ((−∞, 0]); (ϕ(t1 ), . . . , ϕ(tn )) ∈ D′ (∈ B(Rn ))}, {tj } being n time points with 0 = t0 > t1 > t2 > · · · > tn . Then the 14.

(99) process B − = {B − (t) = ϕ(−t); ϕ ∈ C ([0, +∞)), t ≥ 0} is a standard Brownian motion on the probability space (C ((−∞, 0]), B(C ((−∞, 0]))), ω− ). In addition, C − = { } ϕ|(−∞, 0] ; ϕ ∈ C is a Borel subset of B(C ((−∞, 0])) with ω− (C − ) = 1, which is a { }

(100)

(101) e separable Banach space endowed with | · |C − -norm by

(102) ϕ|(−∞, 0]

(103) − = sup |ϕ(t)|; t≤0 . C. Denote by J be the homeomorphism from C onto the product space C + × C − by ( ) carrying ϕ ∈ C into ϕ|[0, +∞) , ϕ|(−∞, 0] . Set ω = (ω+ × ω− ) ◦J . Proposition 3.7. (i) The processes X = {X(t, ϕ) = ϕ(t); t ≥ 0, ϕ ∈ C } and Y = {Y (t, ϕ) = ϕ(−t); t ≥ 0, ϕ ∈ C } are two independent one-dimensional Brownian motions on (C , B(C ), ω). (ii) The process B = {B(t; ϕ) = ϕ(t); −∞ < t < ∞, ϕ ∈ C } is an one-dimensional standard Brownian motion on (C , B(C ), ω). (iii) For any F ∈ C ∗ ,. ∫ (ϕ, F )C ,C ∗ =. ∞. F˙ (t) dB(t; ϕ),. −∞. ∀ ϕ ∈ C,. which is defined pathwise, as a random variable on (C , B(C ), ω), has normal distribution with zero mean and variance |F |2H . (iv) The triple (iH ,C , H , C ) forms an abstract Wiener space. Proof. That X and Y are both standard Brownian motions on (C , B(C ), ω) follows from the above argument. And, for any U, V ∈ B(R) and s ≤ 0 ≤ t, ω ({ϕ ∈ C ; X(t, ϕ) ∈ U, Y (−s, ϕ) ∈ V }) = ω ({ϕ ∈ C ; ϕ(t) ∈ U, ϕ(s) ∈ V }) = ω+ ({ϕ ∈ C ([0, +∞)); ϕ(t) ∈ U }) × ω− ({ϕ ∈ C ((−∞, 0]); ϕ(s) ∈ V }) = ω ({ϕ ∈ C ; ϕ(t) ∈ U }) × ω ({ϕ ∈ C ; ϕ(s) ∈ V }) = ω ({ϕ ∈ C ; X(t, ϕ) ∈ U }) × ω ({ϕ ∈ C ; Y (−s, ϕ) ∈ V }) . Therefore, X and Y are independent to each other, and, for s < 0 ≤ t, ∫ ∫ ∫ 1 2 i r(ϕ(t)−ϕ(s)) i rX(t, ϕ) e ω(dϕ) = e ω(dϕ) · e−i rY (−s, ϕ) ω(dϕ) = e− 2 r (t−s) . C. C. C. 15.

(104) The remaining proof of (i) and (ii) is just a routine verification. The statement (iii) immediately follows from (ii) and Theorem 3.5. In addition, the conclusion of (iii) implies that ω is a σ-additive extension of Gauss cylinder set measure in H . Then, by the Dudley-Feldman-LeCam’s theorem (see [1]), | · |C is a measurable norm in H , and thus (iH ,C , H , C ) is an abstract Wiener space. Afterwards, we will work with the classical Wiener space (C , B(C ), ω) as the underlying probability space. In view of Proposition 3.7(iii), if ϕ ∈ H and if {ϕn } is a sequence in C ∗ such that |ϕn − ϕ|C ∗ → 0 as n → +∞, we define ⟨·, ϕ⟩ as the L2 (C , ω)-limit of (·, ϕn )C ,C ∗ . Then. ∫ ⟨·, ϕ⟩ =. ∞. ˙ dB(t), ϕ(t). −∞. [ω]-a.e. in C ,. where the right-hand integral is a Wiener integral. Set H1 = {⟨·, ϕ⟩; ϕ ∈ H }. Then H1 forms a Gaussian system with mean vector 0 and covariance matrix is ∫ ∞ ˙ ψ(t) ˙ dt, ϕ, ψ ∈ H . Vϕ,ψ = E[⟨·, ϕ⟩⟨·, ψ⟩] = ϕ(t) −∞. Example 3.8. As a simple application of Proposition 3.7(iii), there is a dual representation of Brownian motion as follows: For each t ∈ R, define ∫ s   1[0, t] (u) du, if t ≥ 0,  max{0, min{s, t}} = 0 ∫ αt (s) = s   max{0, min{−s, −t}} = − 1[t, 0] (u) du, if t < 0,. s ∈ R.. 0. It is easy to see that αt ∈ C ∗ . Then the Brownian motion {B(t); t ∈ R} can be represented by B(t; ϕ) = ϕ(t) = (ϕ, αt )C ,C ∗ ,. 16. ∀ ϕ ∈ C..

(105) 4. Gel’fand Triple Centered Around Cameron-Martin Space. Let Hn (t) = (−1)n et (d/dt)n e−t be the Hermite polynomial of the degree n and let 1 2 hn (t) = √√ Hn (t) e−t /2 π2n n! 2. 2. be the corresponding Hermite function. As is well-known, the set {hn ; n ∈ N0 ≡ N∪{0})} is a complete orthonormal basis (CONS for abbreviation) for L2 (R, m) (dm means the Lebesgue measure on R). Moreover, these Hermite functions are eigenfunctions of the self-adjoint operator A: Ahn = (2n + 2)hn for any n ∈ N0 , where A = −d2 /dt2 + (1 + t2 ). ∫t For any continuous function f on R, we define I(f )(t) = 0 f (u) du, t ∈ R. It is clear that the set {I(hn )} forms a CONS for H . For any p ∈ R, let Ep be the completion of H with the | · |p -norm, where (∞ )1/2 ∑ |ϕ|p ≡ (2n + 2)2p |⟨ϕ, I(hn )⟩H | ,. ϕ∈H.. n=0. Then (Ep , | · |p ) is a real separable Hilbert space with the dual space Ep′ which is unitarily ∑ equivalent to E−p by carrying F ∈ Ep′ into ∞ n=0 (F, I(hn ))Ep′ ,Ep I(hn ). If p ≥ 0, Ep = {ϕ ∈ H ; |ϕ|p < +∞}. Let E be the projective limit of {Ep ; p ≥ 0}. Then E is a nuclear space with the dual E ′ which is the inductive limit of Ep′ , p > 0. One notes that all of the | · |p -norms, p ∈ R, are pairwise comparable and compatible in the sense of Gel’fand and Vilenkin [2]. By applying the Riesz representation theorem to identifying H with ∑ its dual and identifying F ∈ Ep′ , p ∈ R, with ∞ n=0 (F, I(hn ))Ep′ ,Ep I(hn ), we have the following continuous inclusions: E ⊂ Eq ⊂ Ep ⊂ H = E0 ⊂ E−p ⊂ E−q ⊂ E ′ ,. 0 < p < q < +∞.. Remark 4.1. (1) From the above construction, every ϕ ∈ H is regarded as an elememt [ϕ] of E ′ by ([ϕ], ψ)E ′ ,E = ⟨ψ, ϕ⟩H . Then, for any F ∈ E ′ , it can be expressed as F =. ∞ ∑. (F, I(hn ))E ′ ,E · [I(hn )],. n=0. where the sum converges with respect to the topology of E ′ . Moreover, { } ∞ ∑ Ep = F ∈ E ′ ; (2n + 2)2p |(F, I(hn ))E ′ ,E |2 < +∞ , ∀ p ∈ R. n=0. 17.

(106) (2) Let S be the Schwartz space of real-valued rapidly decreasing functions on R. A well-known fact is that f ∈ S if and only if Ap f ∈ L2 (R, m) for any p > 0. For ϕ ∈ H , since ∞ ∑. 2p. (2n + 2). |⟨ϕ, I(hn )⟩H | = 2. n=0. ∞ ∑. 2p. (2n + 2). n=0.

(107) ∫

(108)

(109)

(110). ∞. −∞.

(111) 2

(112) ˙ hn (t) dt

(113) , ϕ(t)

(114). ∀ p ∈ R,. we see that ϕ ∈ E if and only if ϕ = I(η) for some η ∈ S. That is, E = {I(η); η ∈ S}. In order to combine these two Gel’fand triples: C ∗ ⊂ H ⊂ C and E ⊂ H ⊂ E ′ , we need the following inequality. Lemma 4.2. There is a constant K > 0 such that, for any n ∈ N0 and ϕ ∈ C , ∫ 7 |ϕ(t)| |h˙ n (t)| dt ≤ K · (2n + 2) 4 · |ϕ|C . C. Proof. By a standard estimate on Hermite functions (see [4]), we have √ √ n n+1 t hn (t) = hn−1 (t) + hn+1 (t), 2 2 √ √ ∫ ∞ n n + 1 hn−1 (t)− hn+1 (t), and h˙ n (t) = |hn (u)| du = O(n1/4 ). 2 2 −∞ Then ∫. ∞. |ϕ(t)| |h˙ n (t)| dt ∫ ∞ ≤ |ϕ|C · (1 + t2 ) |h˙ n (t)| dt −∞

(115)

(116) √ √ ∫ ∞

(117) n

(118) n+1

(119) 2

(120) (1 + t )

(121) hn−1 (t) − hn+1 (t)

(122) dt = |ϕ|C ·

(123)

(124) 2 2 −∞ { √ √ ∫ ∞ n(n − 1)(n − 2) n n ≤ |ϕ|C · |hn−3 (t)| + |hn−1 (t)| 8 2 2 −∞ } √ √ (n + 1)(n + 2)(n + 3) n+5 n+1 |hn+1 (t)| + |hn+3 (t)| dt + 2 2 8. −∞. 7. ≤ C1 (2n + 2) 4 · |ϕ|C ,. ∀ n ≥ 3,. 18.

(125) where the constant C1 is independent of the choice of ϕ. Thus we get that, for any n ∈ N0 , ∫ ∞ |ϕ(t)| |h˙ n (t)| dt −∞ { } ∫ ∞√ 7 − 47 2 ˙ ≤ max C1 , (2k + 2) 1 + t |hk (t)| dt; k = 0, 1, 2 · (2n + 2) 4 · |ϕ|C , −∞. which is the desired inequality by setting K = max{· · · }. By applying Lemma 4.2, we can compare the | · |C and | · |p ’s norms as follows. Theorem 4.3. The norms | · |C and | · |p are comparable and compatible in E, provided that p < − 94 . Proof. Let ϕ ∈ E. Then, by Lemma 4.2, {∞ }

(126) ∫ ∞

(127) 2 ∞ ∑ ∑

(128)

(129) 7 |ϕ|2p = (2n + 2)2p

(130)

(131) ϕ(t) h˙ n (t) dt

(132)

(133) ≤ K 2 · (2n + 2) 2 +2p · |ϕ|2C , −∞. n=0. (4.1). n=0. where the sum in the last term is finite provided that p < − 49 . Thus the norms | · |C and | · |p , p < − 94 , are comparable. Next, to prove the compatibility, let that {ϕk } in E be a Cauchy sequence with respect to both | · |C - and | · |p - norms (p < − 49 ). If |ϕk |C → 0, then, by the above inequality, |ϕk |p → 0. Conversely, if |ϕk |p → 0, set ϕ ∈ C such that |ϕk − ϕ|C → 0. We have to show that ϕ ≡ 0. Observe that, for any η ∈ S,

(134) ∫ ∞

(135)

(136) ∫ ∞

(137)

(138)

(139)

(140)

(141)

(142)

(143)

(144)

(145) ϕ(t) η(t) ˙ dt = lim ϕ (t) η(t) ˙ dt k

(146)

(147)

(148) k→+∞

(149) −∞. −∞. =. lim |⟨ϕk , I(η)⟩H | ≤ |I(η)|−p · lim |ϕk |p = 0.. k→+∞. k→+∞. In particular, for an arbitrarily given infinitely differentiable function ζ with compact support, since the mapping ∫ t ∈ R 7→ Tζ (t) ≡. (∫. t. −∞. ζ(u) du −. ∞. )∫ ζ(u) du. −∞. t. γ(u) du,. −∞. /∫ 1 − 1 − 1 where γ(t) = kγ e 1−t2 1(−1, 1) (t) with kγ = 1 −1 e 1−s2 ds, is in the space S, we have [∫ ∞ ]) ∫ ∞( ∫ ∞ ϕ(t) − ϕ(u) γ(u) du ζ(t) dt = ϕ(t) T˙ζ (t) dt = 0. −∞. −∞. Then ϕ is the constant function. ∫∞ −∞. −∞. ϕ(u) γ(u) du. Since ϕ(0) = 0, so ϕ ≡ 0. The proof is. complete. 19.

(150) From (4.1) and Theorem 4.3, there is a chain of continuous inclusions: H ⊂ C ⊂ E−p , For p >. 9 4. 9 p> . 4. (4.2). ′ and F ∈ E−p , F |C ∈ C ∗ . Since, for any n ∈ N0 , ′ , ⟨i∗H ,C (F |C ), I(hn )⟩H = (I(hn ), F |C )C ,C ∗ = (I(hn ), F )E−p ,E−p. we have i∗H ,C (F |C ). =. ∞ ∑. ′ · I(hn ) = F, (I(hn ), F )E−p ,E−p. n=0 ′ which implies that E−p ⊂ C ∗ ⊂ H . Combining with (4.2), there is a chain of continuous. inclusions: 9 q>p> . 4. E ⊂ Eq ⊂ Ep ⊂ C ∗ ⊂ H ⊂ C ⊂ E−p ⊂ E−q ⊂ E ′ ,. Since the inclusion map iH ,E−p : H → E−p , p > 94 , is a Hilbert-Schmidt operator, the triple (iH ,E−p , H , E−p ), p > 94 , is also an abstract Wiener space. Remark 4.4. (1) Let ϕ ∈ C and regard it as an element of E ′ . If we take a sequence {ϕk } in E such that ϕk → ϕ in C , then ϕk → ϕ in E ′ . Hence, by Proposition 3.4, Remark 3.6, Remark 4.1(2), and applying the integration by parts formula, we see that (ϕ, ψ)E ′ ,E = lim (ϕk , ψ)E ′ ,E k→+∞ ∫ ∞ ¨ dt = − lim ϕk (t) ψ(t) k→+∞ −∞ ∫ ∞ ¨ dt =− ϕ(t) ψ(t) ∫−∞ ∞ ˙ =− ϕ(t) dψ(t) −∞. = (ϕ, ψ)C ,C ∗ ,. ∀ ψ ∈ E,. (4.3). where ψ¨ means the second order derivative of ψ. Combining (5.3) with Remark 4.1 yields that ϕ=−. ∞ {∫ ∑ n=0. provided that p >. ∞. −∞. } ˙ ϕ(t) hn (t) dt · [I(hn )]. 9 . 4. 20. in E−p ,. (4.4).

(151) (2) If f is a polynomially bounded function on R, that is, there are r, C > 0 such that |f (t)| ≤ C(1 + t2 )r , then f can be regarded as an element of E ′ by the following rule:. ∫ (f, ψ)E ′, E = −. ∞. ¨ dt, f (t) ψ(t). ∀ ψ ∈ E.. −∞. (4.5). One notes that every ϕ ∈ C is polynomially bounded.. • A connection between the Wiener space and the white noise space Let S ′ be the space of tempered distributions on R. It is noted that if ηk → η in S, then I(ηk ) → I(η) in E. Accordingly, we define a linear mapping J : E′ → S′. by J (F ) = xF , F ∈ E ′ ,. where xF ∈ S ′ is given by (xF , η)S ′, S = (F, I(η))E ′, E ,. η ∈ S.. We remark that, since every ϕ ∈ C is locally integrable, it can be also regarded as an element of S ′ by. ∫ (ϕ, η)S ′, S =. ∞. ϕ(t) η(t) dt, −∞. η ∈ S.. Proposition 4.5. The mapping J enjoys the following properties: (i) For any ϕ ∈ C , J (ϕ) = Dϕ, where Dϕ is the distributional derivative in S ′ . (ii) J is a homeomorphism with the inverse J −1 given by (J −1 (x), I(η))E ′, E = (x, η)S ′, S ,. x ∈ S ′ , η ∈ S.. (iii) For any p ∈ R, J |Ep is an unitary operator from Ep onto Sp , where E0 ≡ H , and Sp is the completion of S with the || · ||p -norm given by || η ||2p. =. ∞ ∑ n=0.

(152) ∫

(153) (2n + 2)

(154)

(155). ∞. 2p. −∞.

(156) 2

(157) η(t) hn (t) dt

(158)

(159) ,. η ∈ S.. Proof. The statement (i) follows from (5.3). Since, for any η ∈ S, |I(η)|p = || η ||p , the statements (ii) and (iii) immediately follow. 21.

(160) In what follows, we always denote J −1 (x) by Jx for any x ∈ S ′ . Let CS = J (C ) endowed with the | · |CS -norm given by |x|C = |Jx |C , x ∈ CS . Then CS is a real separable S. Banach space. Denote its dual by CS∗ . For f ∈ CS∗ , it corresponds to an element ϕf ∈ C ∗ given by (ϕ, ϕf )C ,C ∗ := (Dϕ, f )CS ,CS∗ ,. ϕ ∈ C.. Then, by the Riesz presentation theorem, there is a unique Tf ∈ L2 (R, m) such that for any ϕ ∈ H ,. ∫ (Dϕ, f )CS ,CS∗ =. ∞. −∞. ˙ Tf (t) dt. ϕ(t). Consequently, by Theorem 3.5, Remark 3.6, and applying the integration by parts formula, ∫ ∞ ∫ ∞ ˙ ˙ ˙ Tf (t) dt, ϕ ∈ H , ϕ(t) ϕf (t) dt = ϕ(t) −∞. −∞. which implies that Tf = ϕ˙ f . Identifying f with Tf , we now summarize the above discussion to obtain the following result. Proposition 4.6. (i) CS∗ = J (C ∗ ). In fact, for f ∈ CS∗ and x ∈ CS , ∫ ∞ (x, f )CS ,CS∗ = − Jx (t) df (t). −∞. (ii) There is a chain of continuous inclusions: S ⊂ Sq ⊂ Sp ⊂ CS∗ ⊂ L2 (R, m) ⊂ CS ⊂ S−p ⊂ S−q ⊂ S ′ ,. 9 q>p> . 4. (iii) For any η ∈ S and x ∈ CS , (x, η)S ′ ,S = (Jx , Jη )E ′ ,E = (x, η)CS ,CS∗ . For any U ∈ B(S ′ ), define µ(U ) = ω(J −1 (U ) ∩ C ). It is obvious that µ is a probability measure on (S ′ , B(S ′ )) with µ(CS ) = 1. Moreover, for any B(S ′ )-integrable function φ, ∫ ∫ φ(x) µ(dx) = φ(J (ϕ)) ω(dϕ). S′. C. 22.

(161) Together with Proposition 3.75, we see that, for any η ∈ S, ∫ ∫ i (x, η)S ′, S e µ(dx) = ei (J (ϕ), η)S ′, S ω(dϕ) ′ S ∫C = ei (ϕ, I(η))E′,E ω(dϕ) ∫C ∫ 1 ∞ 2 = ei (ϕ, I(η))C ,C ∗ ω(dϕ) = e− 2 −∞ |η(t)| dt . C. Therefore, µ is the white noise measure on (S ′ , B(S ′ )); (S ′ , B(S ′ ), µ) is the white noise space. The Brownian motion B = {B(t); t ∈ R} on (S ′ , B(S ′ ), µ) can be represented by ( ) B(t, x) = (x, J (αt ))CS ,CS∗ · 1CS (x) = Jx (t) · 1CS (x) , where αt ∈ C ∗ is given as one in Example 3.8. Then, for any η ∈ S and x ∈ CS , it follows from Proposition 4.6 that ∫ (x, η)S ′ ,S =. ∞. η(t) dB(t, x), −∞. where the right-hand integral is a Riemann-Stieltjes integral.. 23.

(162) 5. A Representation of Conditional Expectations of Generalized Brownian Functionals. In the section, first of all, an integral representation concerning conditional expectations of integrable random variables will be provided. By this integral representation, we will give a definition of conditional expectations of generalized Brownian functionals.. 5.1. Integral Representation of Conditional Expectations. In this section, an integral representation concerning conditional expectations of integrable random variables on a Banach space with respect to a Gaussian measure will be obtained. As an important example, we then apply this result to integrable Brownian functionals (BF’s, for short) on (C , B(C ), ω) (resp. white noise functionals (WNF’s, for short) on (S ′ , B(S ′ ), µ)), and accordingly, give a definition of conditional expectations of generalized Brownian functionals (GBF’s for short) (resp. generalized white noise functionals (GWNF’s, for short)). These definitions will provide us a scheme for studying the generalized stochastic processes. Let (B, ∥ · ∥) be a real Banach space and P : B → B be a continuous projection, that is, P is a bounded linear operator such that P 2 = P . Then Q = IB − P (IB is the identity map of B) is also a continuous projection on B and B = P (B) ⊕ Q(B) (the symbol ⊕ means the direct sum). For any operator T on B, we denote by kerT the kernel of T . Then P (B) = kerQ and Q(B) = kerP , whence P (B) and Q(B) are both closed in B. Let ρ : B → P (B) × Q(B) be defined by ρ(x) = (P (x), Q(x)), x ∈ B, where P (B) × Q(B) is regarded as a product space of P (B) and Q(B) with the ∥ · ∥1 -norm, that is ∥(x, y)∥1 = ∥x∥ + ∥y∥. Then, by the open mapping theorem, ρ is an isomorphism. Let Λ be a non-degenerate probability measure on (B, B(B)) (not necessarily Gaussian). For any U ∈ B(P (B)), define ΛP (U ) = Λ(ρ−1 (U × Q(B)) (= Λ(U ⊕ Q(B))) ; and similarly, for any V ∈ B(Q(B)), define ΛQ (E) = Λ(ρ−1 (P (B) × V ) (= Λ(P (B) ⊕ V )) . Then we have the following general result, which plays a key role in deriving an integral representation of conditional expectations. 24.

(163) Lemma 5.1. For any D ∈ B(B), ∫ ∫ ∫ 1D (x) (ΛP × ΛQ ) ◦ ρ(dx) = 1D (P (x) + Q(y)) Λ(dx)Λ(dy). B. B. B. Proof. Foe any U ∈ B(P (B)) and V ∈ B(Q(B)), (ΛP × ΛQ ) ◦ ρ(U + V ) = ΛP (U ) · ΛQ (V ) = Λ(ρ−1 (U × Q(B)) · Λ(ρ−1 (P (B) × V ) = Λ × Λ ({(x, y) ∈ B × B; P (x) ∈ U, Q(y) ∈ V }) , that is, ∫. ∫ ∫ 1U +V (x) (ΛP × ΛQ ) ◦ ρ(dx) =. B. 1U +V (P (x) + Q(y)) Λ(dx)Λ(dy). B. B. Let { D=. ∫ D ∈ B(B);. ∫ ∫ 1D (x) (ΛP × ΛQ ) ◦ ρ(dx) =. B. B. } 1D (P (x) + Q(y)) Λ(dx)Λ(dy) .. B. It is obvious that D is a λ-system containing the π-system B(P (B)) + B(Q(B)). Then, by Dynkin’s π-λ theorem, D contains the σ-field generated by B(P (B)) + B(Q(B)). Hence D = B(B), and the proof is complete. Hereafter, we assume further that B is separable and Λ is a Gaussian measure on (B, B(B)) with zero mean, that is, the law of arbitrary η ∈ B ∗ , the dual of B, considered as a random variable on (B, B(B), Λ) has a normal distribution in R with zero mean. Applying the Kuelbs theorem (see [6, 7]), there exists a unique reproducing kernel subspace (H, | · |H ) of B such that the triple (i, H, B) forms an abstract Wiener space, where i is the inclusion map from H into B. For any h ∈ H, let {ηn } be a sequence in B ∗ such that ηn → h in H. Then {(·, ηn )B,B∗ } forms a Cauchy sequence in L2 (B, Λ), the L2 (B, Λ)-limit of which is denoted by ⟨·, h⟩. One notes that ⟨·, h⟩ is independent of the choice of {ηn } and distributed by the law of N (0, |h|2H ). In the case of (i, H, B), Lemma 5.1 can be improved as follows. Theorem 5.2. If P |H is an orthogonal projection on H (and so is Q|H ), then Λ = (ΛP × ΛQ ) ◦ ρ. 25.

(164) Moreover, for any complex-valued measurable f on B, we have ∫ ∫ ∫ f (x) Λ(dx) = f (P (x) + Q(y)) Λ(dx)Λ(dy), B. B. B. provided that both integrals exist. Proof. Let P ∗ , Q∗ , and i∗ be the adjoint operators of P , Q, and i respectively. For any ζ ∈ B ∗ and h ∈ H, ⟨i∗ (P ∗ (ζ)), h⟩H = (h, P ∗ (ζ))B,B ∗ = (P (h), ζ)B,B ∗ = ⟨P (h), i∗ (ζ)⟩H. (since P (H) ⊂ H). = ⟨P (i∗ (ζ)), h⟩H. (since P |H is self-adjoint),. where ⟨·, ·⟩H is the inner product induced by the | · |H -norm. Hence i∗ (P ∗ (ζ)) = P (i∗ (ζ));. (5.1). i∗ (Q∗ (ζ)) = Q(i∗ (ζ)).. (5.2). and similarly,. That is, P (B ∗ ) ⊂ B ∗ and Q(B ∗ ) ⊂ B ∗ . Then, for any ζ ∈ B ∗ , it follows from (5.1), (5.2), and Lemma 5.1 that ∫ ei (x, ζ)B,B∗ (ΛP × ΛQ ) ◦ ρ(dx) B ∫ ∫ = ei (P (x)+Q(y), ζ)B,B∗ Λ(dx)Λ(dy) ∫ ∫B B i (P (x), ζ)B,B ∗ Λ(dx) · ei (Q(y), ζ)B,B∗ Λ(dy) = e ∫B ∫B ∗ ∗ ei (y, Q (ζ))B,B∗ Λ(dy) = ei (x, P (ζ))B,B∗ Λ(dx) · B. B − 21 |i∗ (P ∗ (ζ))|2H. = e. = e− 2 |P (i 1. ∗. ∗ (ζ))|2 H. − 12 |i∗ (Q∗ (ζ))|2H. ·e. · e− 2 |Q(i 1. ∗ (ζ))|2 H. = e− 2 |i (ζ)|H ∫ = ei (x, ζ)B,B∗ Λ(dx). 1. 2. B. Thus Λ = (ΛP × ΛQ ) ◦ ρ. Finally, applying the monotone convergence theorem together with Lemma 5.1, the remaining part of this theorem is easy to be obtained. 26.

(165) In the sequel, for a sub-σ-field G of B(B), we always assume that it has a generating set K in B ∗ , that is, G is generated by random variables (·, ζ)B,B ∗ , ζ ∈ K. One notes that the closed subspace K. H. in H generated by K is uniquely determined by G . In. other words, if K ′ is another generating set of G in B ∗ , then K ′ H. suppose that K ′ \ K. H. H. H. = K . Otherwise,. ̸= ∅. Then there exists a non-zero element β ∈ K ′. H. such that it. H. is orthogonal to K , implying that the random variable ⟨·, β⟩ is independent to G . On the other hand, ⟨·, β⟩ is G -measurable. As a result, ⟨·, β⟩ = 0, [Λ]-a.e. in B, leading to H. H. β = 0, a contradiction. Similarly, the case that K \ K ′ ̸= ∅ is unlikely to occur. In view of the above argument, the closed subspace K. B. in B generated by K is also. uniquely determined by G . We denote it by KG . In addition, we denote by PG the continuous projection on B with the range KG , if such an operator exists. The following is the result concerning the integral representation of conditional expectations. Theorem 5.3. Let G be a sub-σ-field of B(B). Assume that the associated continuous projection PG exists, and set QG = IB − PG . If PG |H is an orthogonal projection on H, then, for any f ∈ L1c (B, Λ), the conditional expectation E[ f | G ] of f relative to G has the following integral representation: ∫ E[ f | G ](x) = f (PG (x) + QG (y)) Λ(dy),. [Λ]-a.e. x ∈ B.. B. Proof. Let K ⊂ B ∗ be a generating set of G . Consider a cylinder set Z defined by ζ1 , . . . , ζn ∈ K, that is, Z = {x ∈ B; ((x, ζ1 )B,B ∗ , . . . , (x, ζn )B,B ∗ ) ∈ U (∈ B(Rn ))}. Observe that, for any ζ ∈ B ∗ and x, y ∈ B, we have by (4.1) and (4.2) that (PG (x) + QG (y), ζ)B,B ∗ = (x, PG (ζ))B,B ∗ + (y, QG (ζ))B,B ∗ ,. 27. ∀ x, y ∈ B.. (5.3).

(166) Therefore, by (5.3) and Theorem 5.2, ∫ ∫ E[ f | G ](x) Λ(dx) = f (x) Λ(dx) Z ∫Z = 1Z (x) f (x) Λ(dx) ∫B ∫ = 1Z (PG (x) + QG (y)) f (PG (x) + QG (y)) Λ(dx)Λ(dy) B B ∫ ∫ = 1Z (x) f (PG (x) + QG (y)) Λ(dx)Λ(dy) B B } ∫ {∫ f (PG (x) + QG (y)) Λ(dy) Λ(dx). (5.4) = Z. B. Finally, we show that the function. ∫. x ∈ B 7→ B. f (PG (x) + QG (y)) Λ(dy). is G -measurable. Since the set {ei (·, ζ)B,B∗ ; ζ ∈ B ∗ } is a total subset in L1c (B, Λ), there ∑ is a sequence {φn }, where φn = j aj,n ei (·, ζj,n )B,B∗ , ζj,n ∈ B ∗ and aj,n ∈ C, such that φn → f in L1c (B, Λ). By Theorem 5.2 and the Fubini theorem, ∫ ∫ φn (PG (·) + QG (y)) Λ(dy) → f (PG (·) + QG (y)) Λ(dy) in L1c (B, Λ). B. B. Since, by (5.1), ∫ B. φn (PG (·) + QG (y)) Λ(dy) } {∫ ∑ i (QG (y), ζj,n ) i P (·), ζj,n ) ∗ B,B B,B ∗ = aj,n e Λ(dy) · e ( G j. =. ∑. {∫. B i (QG (y), ζj,n ). aj,n. j. it is G -measurable, and so is. e. B,B ∗. } i ·, P (i∗ (ζj,n ))) B,B ∗ , Λ(dy) · e ( G. B. ∫ B. f (PG (·) + QG (y)) Λ(dy). Combining with (5.4), the proof. is complete. Come back to our study with the underlying probability space (C , B(C ), ω). Two typical sub-σ-fields of B(C ) are considered: (1) BK , the σ-field with a generating set K, where K = {ϕ1 , . . . , ϕn } ⊂ C ∗ is an orthonormal set in H ; (2) Bt (t > 0), the σ-field with the generating set {αs ; −∞ < s ≤ t}, where αs ’s are given as those in Example 3.8. Then BK is the σ-field generated by random variables (·, ϕi )C ,C ∗ , i = 1, 2, . . . , n, while Bt is the σ-field generated by B(s) for −∞ < s ≤ t. 28.

(167) Proposition 5.4.. {. (i) KBK is the closed subspace. n ∑. } aj ϕj ; aj ′ s ∈ R. of C , and the associated continu-. j=1. ous projection PBK is given by PBK =. n ∑. (·, ϕi )C ,C ∗ · ϕi .. i=1. (ii) KBt is the closed subspace {ϕ ∈ C ; ϕ(u) = ϕ(t), ∀ u ≥ t} of C , and the associated continuous projection PBt is given by PBt (ϕ)(u) = ϕ(min{u, t}),. ∀ u ∈ R, ϕ ∈ C .. Proof. The assertion (i) is a trivial result. To verify (ii), one notes that, since αs |[t, +∞) is a constant function for any s ≤ t, so is ϕ|[t, +∞) for any ϕ ∈ KBt . Conversely, if ϕ ∈ C and ϕ|[t, +∞) is a constant function, let n  (i − 2n ), if i = 0, 1, . . . , 2n ,   n 2 tn,i =     t (i − 2n ), if i = 2n + 1, 2n + 2, . . . , 2n+1 − 1, 2n for n ∈ N. Set αt − αtn,i ϕn,i = √ n,i+1 , for i = 0, 1, . . . , 2n+1 . tn,i+1 − tn,i Then {ϕn,i } is an orthonormal set of KBt . Let ψn =. 2n+1 ∑−1. (ϕ, ϕn,i )C ,C ∗ · ϕn,i ,. i=0. where.   ϕ(−n), if s ≤ −n,          (s − t )ϕ(t n,i n,i+1 ) + (tn,i+1 − s)ϕ(tn,i ) , if tn,i ≤ s ≤ tn,i+1 , ψn (s) =  tn,i+1 − tn,i     for some i = 0, 1, . . . , 2n+1 − 1,      ϕ(t), if s ≥ t.. Then ψn ’s ∈ KBt , and ψn → ϕ in C as n → +∞. Thus ϕ ∈ KBt . The proof is complete. 29.

(168) It is evident that PBK |H and PBt |H are both orthogonal projections on H . Applying Theorem 5.3 yields the following results: Corollary 5.5. For any f ∈ L1c (C , ω), we have ( n ) ∫ n ∑ ∑ E[ f | BK ] = f (·, ϕi )C ,C ∗ · ϕi + ϕ − (ϕ, ϕi )C ,C ∗ · ϕi ω(dϕ); C. ∫ E[ f | Bt ] =. C. i=1. i=1. f (PBt (·) + QBt (ϕ)) ω(dϕ).. The above results can be established in white noise analysis. Corollary 5.6. Let φ ∈ L1c (S ′ , µ) and {e1 , e2 , . . . , en } ⊂ S be an orthonormal set. Then E[ φ| (·, ei )S ′,S ; i = 1, 2, . . . , n ] ( n ) ∫ n ∑ ∑ = φ (·, ei )S ′,S · ei + y − (y, ei )S ′,S · ei µ(dy). S′. i=1. i=1. In addition, for µ-a.e. x ∈ CS , E[ φ| B(s); −∞ < s ≤ t ](x) ∫ = φ(D(PBt (Jx )) + y − D(PBt (Jy ))) µ(dy), CS. where D(·) means the distributional derivative in S ′ . Remark 5.7. If Jx ∈ H , that is, x ∈ L2 (R, m), then, for any η ∈ S, ∫ ∞ (D(PBt (Jx )), η)S ′,S = − PBt (Jx )(s) η(s) ˙ ds −∞ ∫ t =− PBt (Jx )(s) η(s) ˙ ds + PBt (Jx )(t) η(t) −∞ ∫ t = η(s) dPBt (Jx )(s) −∞ ∫ t = η(s) J˙x (s) ds (by Remark 3.6) ∫−∞ ∞ = 1(−∞, t] (s) x(s) η(s) ds, −∞. from which it follows that D(PBt (Jx )) = 1(−∞, t] · x. 30.

(169) 5.2. A Definition of Conditional Expectations of GBF’s. We start with the construction of test Brownian functionals on the classical Wiener space (C , B(C ), ω). For a fixed Banach space B which is either C or E−p with p > 0, let E(B) be the class of those functions φ defined on C so that (1) φ has an analytic extension φ(z) e to Bc ; (2) φ e satisfies the exponential growth condition: ′. ∃ c, c′ > 0 ∋ |φ(z)| e ≤ c ec |z|Bc ,. ∀ z ∈ Bc .. By the Fernique theorem (see [7]), E(B) ⊂ L2c (C , ω). For m ∈ N and φ ∈ E(B), define ∥φ∥Em (B) = sup{|φ(z)| e e−m |z|Bc ; z ∈ Bc }. Let Em (B) = {φ ∈ E(B); ∥φ∥Em (B) < +∞}. Then {(Em (B), ∥ · ∥Em (B) )} is an increasing sequence of Banach spaces and E(B) =. ∪. Em (B).. m∈N. Endow E(B) with the inductive limit topology induced by the family {Em (B)}. Then E(B) becomes a locally convex topological algebra. For convenience, we use the shorthands Em,p and Ep for Em (E−p ) and E(E−p ), respectively. Let E∞ =. ∩. Ep. p>0. topologied as the projective limit of {Ep }. Denote the dual spaces of E(C ), Ep , and E∞ ∗ by E(C )∗ , Ep∗ , and E∞ respectively, which are topolozied by the weak∗ -topology. Then we. have the following chain of continuous inclusions: ∗ E∞ ⊂ Ep ⊂ Eq ⊂ E(C ) ⊂ L2c (C , ω) ⊂ E(C )∗ ⊂ Eq∗ ⊂ Ep∗ ⊂ E∞ ,. as p ≥ q > 49 ,. where L2c (C , ω) is identified with its dual by the Riesz representation theorem. E∞ will ∗ will be referred as the generalserve as the space of test functions, while members of E∞. ized Brownian functionals (GBF’s’s for short) in our investigation. In addition, let ⟨⟨·, ·⟩⟩0 , 31.

(170) ∗ ⟨⟨·, ·⟩⟩p , and ⟨⟨·, ·⟩⟩∞ stand for the dual pairing of E(C )∗ -E(C ), Ep∗ -Ep , and E∞ -E∞ , respec-. tively. This completes the construction of test and generalized functions in our study. We ∗ remark that E∞ , Ep∗ , and E(C )∗ are all sequentially complete by the Banach-Steinhaus. theorem. Let BK and Bt be the σ-fields given as those in Corollary 5.5, where we suppose further that K is a subset of E. Then, for any φ ∈ E∞ , E[φ|BK ] ∈ E∞ . In fact, if p >. 9 4. and φ ∈ Em, p for some m ∈ N, then we have by Corollary 5.5 and the Fernique theorem that ∥E[φ|BK ]∥Em′, p ≤ Const.∥φ∥Em,p , where m′ = m · {. ∑n i=1. |ϕi |p |ϕi |−p }. Thus the mapping carrying φ into E[φ|BK ] is a. continuous operator from E∞ into itself. It leads to the following definition. ∗ Definition 5.8. Let F ∈ E∞ and K ⊂ E. Then the conditional expectation E[F |BK ] of ∗ F relative to BK is a member of E∞ defined by. ⟨⟨ E[F |BK ], φ⟩⟩∞ = ⟨⟨ F, E[φ|BK ]⟩⟩∞ ,. φ ∈ E∞ .. ∗ Next, we would like to define E[F |Bt ], t > 0, for F ∈ E∞ . In general, we can only. conclude that E[φ|Bt ] ∈ E(C ) even if φ ∈ E∞ . Indeed, it follows from Corollary 5.5 that, as φ ∈ Em (C ), E[φ|Bt ] ∈ Em (C ) and it satisfies ∥E[φ|Bt ]∥Em (C ) ≤ Const.∥φ∥Em (C ) . It seems to be natural to define, for F ∈ E(C )∗ , ⟨⟨ E[F |Bt ], φ⟩⟩0 = ⟨⟨ F, E[φ|Bt ]⟩⟩0 ,. φ ∈ E(C ).. However, up to now, we can not find a characterization concerning elements of E(C )∗ . ˙ We do not even know if the white noise B(t) is in E(C )∗ or not.. 32.

(171) References [1] R. M. Dudley, J. Feldman, L. LeCam, On semi-norms and probabilities, and abstract Wiener spaces, Ann. Math. 93 (1971), 390-408. [2] I. M. Gel’fand, N. Y. Vilenkin, Generalized Functions, vol. 4, Academic Press, 1964. [3] T.Hida and Si Si, Lectures on White Noise Functionals, World Scientific, 2008. [4] E. Hille, R. S. Phillips, “Functional Analysis and Semigroups”, Amer. Math. Soc. Colloquium Publ. 31, Providence, R. I., 1957. [5] K. Itô, Foundations of Stochastic Differential Equations in Infinite Dimensional Spaces, CBMS-NSF Regional Conference Series in Applied Mathematics, vol. 47, SIAM, Philadelphia, 1984. [6] J. Kuelbs, Gaussian measures on a Banach space, J. Funct. Anal. 5 (1970), 354-367. [7] H.-H. Kuo, Gaussian Measures in Banach Spaces, Lectures Notes in Math., vol. 463, 1975. [8] P. Morters and Y. Peres, Brownian Motion, Cambridge University Press, 2010.. 33.

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