巴拿赫空間上的高斯測度之拓樸擔台的研究

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(1)國立高雄大學應用數學系碩士論文. 巴拿赫空間上的高斯測度之拓樸擔台的研究 The study of the topological support of a Gaussian measure in a Banach space. 研究生：林奕均撰指導教授：施信宏. 中華民國 101 年 7 月.

(2) 致謝. 本論文的完成要感謝指導教授施信宏老師的教導，在修習實變函數論、隨機分析的課程時，老師總是將課程內容說明的很仔細，也看到老師在做研究時嚴謹的態度，討論問題時也總能不厭其煩地說明整個問題的來龍去脈以及問題的解決方向，使我心中的困惑能迎刃而解，也謝謝口試委員江鑑聲老師、陳正忠老師在百忙之中能撥空參與我的口試，在此向三位老師至上最誠摯的謝意。另外也要感謝系辦的千惠姐、雅鳳姐，每當有困難時，這二位大姐姐總能適時地提供協助，另外也要感謝博士班的鄭益新學長，在課業上有問題時，學長總會有耐心地和我討論，讓我省下了不少時間，謝謝高大應數的師長、學長姐、同學，言有盡而意無窮，對高大應數的感恩千言萬語也道不盡，在這裡的點點滴滴將永銘我心。最後要感謝我的父母，無時無刻地支持與鼓勵，是我在高大求學的最大後盾與動力，是你們的加油、鼓勵、支持，我才能夠無後顧之憂地完成學業，謝謝你們。. 林奕均書 2012.7.

(3) The study of the topological support of a Gaussian measure in a Banach space. by Yi-Chin Lin Advisor Hsin-Hung Shih. Department of Applied Mathematics, National University of Kaohsiung Kaohsiung, Taiwan 811, R.O.C. July 2012.

(4) Contents 中文摘要. ii. 英文摘要. iii. 1. Introduction. 1. 2. Preliminaries. 2. 3. The topological support of a probability measure on a Banach space. 7. 4. Gaussian measure on a Banach space. 10. 5. Application. 15. 6. References. 20. i.

(5) 巴拿赫空間上的高斯測度之拓樸擔台的研究指導教授：施信宏教授國立高雄大學應用數學系. 學生：林奕均國立高雄大學應用數學系. 摘要. 在參考文獻[4]中，伊藤證明了對於一個在實可分希爾伯特空間上的高斯測度，其拓樸擔台恰為最小的閉子空間且測度為 1。本學位論文主要研究如何將伊藤的結果推廣到定義在實可分巴拿赫空間上且期望值為零的高斯測度。最後，我們將提供一種由 ( B , µ ) 建構標準可數希爾伯特空間的方法作為一個簡單的應用。關鍵字：拓樸擔台、高斯測度、巴拿赫空間、可數希爾伯特空間關鍵字. ii.

(6) The study of the topological support of a Gaussian measure in a Banach space Advisor: Professor Hsin-Hung Shih Department of Applied Mathematics National University of Kaohsiung. Student: Yi-Chun Lin Department of Applied Mathematics National University of Kaohsiung. ABSTRACT. In [4] , K . Itô showed that for a given Gaussian measure on a real separable Hilbert space , it’s topological support coincides with the least closed subspace with the total measure. The purpose of this study will be devoted to extend Itô’s result to a Gaussian measure on a real separable Banach space. As a simple application , for any real separable Banach space B on which a Gaussian measure µ with zero mean is given , a method to construct a standard countably Hilbert space setting from ( B , µ ) will be presented . Keywords: topological support , Gaussian measure, Banach space , countably Hilbert space. iii.

(7) 1. Introduction. In [4], K. Itô showed that for a given Gaussian measure on a real separable Hilbert space, its topological support coincides with the least closed subspace with the whole measure. Based on this viewpoint, we always assume that the considered Gaussian measure is non-degenerate, whenever the underlying probability space is constructed in a real separable Hilbert space. By the Prohorov’s theorem (see Lemma 4.4), we see that the associated covariance operator to a given Gaussian measure on a real separable Hilbert space is a S-operator, which means a positive-definite, self-adjoint, and traceclass operator. As a result, the identity operator can not correspond to some Gaussian measure on a Hilbert space as a covariance operator. To overcome this gap, L. Gross in [2] introduced the theory of abstract Wiener space with the associated abstract Wiener measure, which is a Gaussian measure on a real separable Banach space with zero mean. Gross’s theory generalizes the notion of the classical Wiener measure on (C[0, 1], | · |∞ ). Moreover, we can show that every abstract Wiener measure is non-degenerate, the topological support of which is the whole space. Given a real separable Banach space on which a Gaussian measure with zero mean is defined, a problem immediately aries: What is the topological support of such a Gaussian measure? More precisely, we want to see whether Itô’s result holds in a Banach space or not. We organize this thesis as follows. In Section 2, we briefly introduce some notions which will be used in the subsequent study. In Section 3, we give the definition of topological support of a general probability measure on a real Banach space and then study its property. In Section 4, we start to study Gaussian measure on a Banach space. We will show that the Itô’s result still holds for Gaussian measure with zero mean on a real separable Banach space. Finally, as a simple application, we will provide a method to contruct a Gel’fand triple setting from (B, µ), where B is a real separable Banach space and µ is a Gaussian measure on B with zero mean.. 1.

(8) 2. Preliminaries. In this section, we mainly review some notions and results which will be used in the subsequent study.. 2.1. Standard Measurable Spaces. Definition 2.1. A measurable space M is called a standard measurable space if it is Borel isomorphic to a Borel subset of R (viewed as a measurable space), that is, there is a Borel set E of R and a map f : M → E such that f is bijective and f and f −1 are both measurable. Theorem 2.2. [4] Let S and T be standard measurable spaces. If f : S → T is a measurable injection, then f (S) is Borel isomorphic to S (under f ) and f (S) is a Borel set of T . Corollary 2.3. let S and T be standard measurable spaces. If f : S → T is a measurable injection, then, for any Borel set E of S, f (E) is a Borel set of T . Proof. Since S and f (S) are Borel isomorphic to each other, f (E) is a Borel set of f (S), that is, there is a Borel set F of T such that f (E) = F ∩ f (S). From Definition 2.1, f (S) is a Borel set of T , and hence f (E) is certainly a Borel set of T . Theorem 2.4. [4] Every complete and separable metric space, viewed as a measurable space, is a standard measurable space.. 2.2. Abstract Wiener Space. In this subsection, we will briefly review the theory of abstract Wiener space, introduced by L. Gross (see [2]) Let H be a real separable Hilbert space with | · |-norm and the induced inner product h·, ·i, and let F be the collection of all finite-dimensional orthogonal projections of H. For any P1 , P2 ∈ F , P1 ≤ P2 if and only if P1 (H) ⊂ P2 (H). 2.

(9) A set A in H is called a cylinder set if there exists an P ∈ F such that . A = h ∈ H; P (h) ∈ ϕ−1 (E) , P. for any E ∈ B(Rn ), where n = dimP (H), and ϕP : P (H) → Rn is defined by ϕP (r1 e1 + · · · + rn en ) = (r1 , . . . , rn ), where {e1 , e2 , . . .} is an orthonormal basis of H. Let R be the collection of all cylinder sets of H. Then R is a field. A set function µ : R → [0, 1]. is called a Gauss cylinder set measure if for any A ∈ R of the form as above, n Z Z . 1 1 2 −1 µ h ∈ H; P (h) ∈ ϕP (E) = √ · · · e− 2 |~x| d~x. 2π E. One notes that µ is well-defined, that is, the definition of µ is independent of the choice of an orthonormal basis of H. In addition, a well-known fact is that a Gauss cylinder set measure µ is finitely additive in R, but not σ-additive. Definition 2.5. A seminorm k · k on H is called a measurable seminorm if for any ǫ > 0, there exists an Pǫ ∈ F such that for any P ∈ F with P ⊥ Pǫ , µ ({h ∈ H; kP (h)k > ǫ}) < ǫ.. Remark 2.6. (1) Let {e1 , e2 , . . .} be an orthonormal basis of H such that for any h ∈ H, P (h) =. n X j=1. hh, ej i · ej .. Then . {h ∈ H; kP (h)k > ǫ} = h ∈ H; (hh, e1 i, . . . , hh, en i) ∈ φ−1 ((ǫ, +∞)) ,. where φ(t1 , . . . , tn ) = k t1 e1 + · · · tn en k for any (t1 , . . . , tn ) ∈ Rn .. (2) The Hilbert norm | · | is not a measurable norm on H unless H is finite-dimensional. 3.

(10) (3) Let k · k be a measurable norm on H. Assume that H is not finitedimensional. Then it follows from (2) that (H, k · k) is not complete. Otherwise, by the Open Mapping Theorem, | · | is equivalent to k · k, leading to the result that | · | is measurable, a contradiction. Definition 2.7. Let k·k be a measurable norm defined on H. Then the triple (i, H, B) is called an abstract Wiener space (AWS, in short), where B is the completion of H with respect to k · k-norm and i is the canonical embedding of H into B. Remark 2.8. Let k · k be a measurable norm in H. Then k · k is always weaker than | · |-norm. We refer the reader to [6] for the proof. In what follows, all notations are the same as those mentioned above. As H is identified as a dense subspace of B, we identify B ∗ as a dense subspace of H ∗ under the adjoint operator i∗ of i by the following way: For any x ∈ H and η ∈ B ∗ , hx, i∗ (η)i = (i(x), η),. where (·, ·) is the dual pairing between B and B ∗ . Applying the Riesz representation theorem to identify H ∗ with H, we have the continuous inclusion maps B ∗ ⊂ H ⊂ B. A set of the form {x ∈ B; ((x, η1 ), . . . , (x, ηn )) ∈ E}. is called a cylinder set of B. Let RB be the collection of all cylinder sets of B. Define µ e {x ∈ B; ((x, η1 ), . . . , (x, ηn )) ∈ E} := µ {x ∈ H; (hx, η1 i, . . . , hx, ηn i) ∈ E} .. Theorem 2.9. (Gross [2]) µ e is σ-additive in the σ-field generated by RB . In general, we may replace µ by µt , where n Z Z . 1 1 2 −1 µt h ∈ H; P (h) ∈ ϕP (E) = √ · · · e− 2t |~x| d~x. 2πt E. µt is called the Gauss cylinder set measure in H of parameter t > 0. Define µ et on RB accordingly. Then µ et has a unique σ-additive extension to the Borel field generated by RB . 4.

(11) Notation. pt will denote the extension of µ et . It is called the abstract Wiener measure on B with variance parameter t. Then for any η ∈ B ∗ , Z t 2 ei(x, η) pt (dx) = e− 2 |η| . (2.1) B. Theorem 2.10. [6] The σ-field generated by RB is the Borel field of B. From (2.1), (·, η) is a random variable on (B, B(B), pt ) with mean zero and variance t|η|2 . For any h ∈ H, let {ηn } be a sequence in B ∗ such that |ηn −h| → 0 as n → ∞. Then {(·, ηn )} forms a Cauchy sequence in L2 (B, pt ), the L2 (B, pt )-limit of which is denoted by h·, hi. One notes that h·, hi is independent of the choice of {ηn } and distributed by the law of N(0, t|h|2 ). The following important theorem told us that the condition that k · k is measurable is also a necessary condition in order to make the Gauss cylinder set measure on B σ-additive. Theorem 2.11. (Dudley-Feldman-LeCam [1]) Let k · k be a weaker norm than |·| on H, and let B be the completion of H under k·k. Define the Gauss cylinder set measures µ and µ e on H and B respectively as before. Then µ e has σ-additive extension to the Borel field of B if and only if k · k is a measurable norm in H. Theorem 2.12. [6] Let (i, H, B) be an AWS. If h ∈ H, then pt (h, ·) is equivalent to pt and the Randon-Nikodym derivative is given by dpt (h, ·) 1 2 1 (x) = exp − |h| − hh, xi , x ∈ B, dpt 2t t where pt (h, E) = pt (E + h) for any E ∈ B(B).. 2.3. Reproducing Kernel Hilbert Space. Let (B, k · k) be a real separable Banach space. A probabnility measure µ on (B, B(B)) is called a Gaussian measure if the law of arbitrary (·, η), η ∈ B ∗ , considered as a random variable on (B, B(B), µ) is a Gaussian measure on (R, B(R)). Let µ be a Gaussian measure on B with zero mean, that is, (·, η) ∼ N(0, ση ), ση > 0. Then the following Fernique’s theorem is useful in estimating the integrability of some random variables of exponential type on (B, B(B), µ). 5.

(12) Theorem 2.13. [7, Theorem 2.6] There exists λ > 0 such that Z 2 eλkxk µ(dx) < +∞. B. A linear subspace H ⊂ B equipped with a norm | · | induced by an inner product h·, ·i is said to be a reproducing kernel Hilbert space (RKHS, in short) for µ if H is complete, continuously embedded in B and such that for arbitrary η ∈ B ∗ , the law of (·, η) is normally distributed by (·, η) ∼ N(0, |η|2), where |η| = sup{|(h, η)|; |h| ≤ 1, h ∈ H}. Theorem 2.14. [7] Let µ be a Gaussian measure on B with zero mean. Then there exists a unique reproducing kernel Hilbert space (H, | · |).. 6.

(13) 3. The topological support of a probability measure on a Banach space. In this section, we mainly study some general properties of the topological support of a probability measure on a Banach space. In what follows, the notation B(M) will always stands for the Borel σ-field of a metric space M, and Prob(M) the collection of all probability measures on the Borel measurable space (M, B(M)). Definition 3.1. Let (B, k·kB ) be a real Banach space. Then, for any µ ∈ Prob(B), the topological support of µ in B, denoted by supp (µ), is defined to be the smallest closed subset of B having µ-measure 1. Proposition 3.2. Let (B, k·kB ) be a real Banach space. Assume that B is separable. Then, for any µ ∈ Prob (B), supp (µ) exists. Proof. Let F be the family of all closed subsets of B having T µ-measure 1. It is obvious S that F is non-empty, since B ∈ F . Set A = F ∈F F . Then B \ A = F ∈F F c , so {F c : F ∈ F } forms an open covering of B \ A. Since B \ A is the second countable as a metric subspace of B, it has the so-called Lindelöf property. S Then we can choose countably many Fj , j ∈ Λ, from F such that B \ A = j∈Λ Fjc , where Λ ⊂ N is a indexing set. Consequently , µ (B \ A) = µ ≤. X j∈Λ. [. j∈Λ. Fjc. !. µ Fjc = 0,. implying µ (A) = 1, and thus A = supp (µ). Remark 3.3. In a metric space X, the following three statements are equivalent: (1) X is the second countable. (2) X is separable. (3) X has the Lindelöf property. 7.

(14) Theorem 3.4. Let (B, k·kB ) be a real Banach space, and let µ ∈ Prob (B). Assume that supp (µ) exists. Then supp (µ) = {x ∈ B : ∀r > 0, µ (Nx (r)) > 0} ,. (3.1). where Nx (r) = {y ∈ B : kx − ykB < r} . For convenience, we denote by N(µ) the set in the right-hand side of (3.1). Proof. Let F ⊂ B be closed such that µ (F ) = 1. Claim that supp (µ) ⊂ N(µ) ⊂ F. (3.2). We divide the proof of (3.2) into the following two parts: (i) For x ∈ supp (µ), if there exists r > 0 such that µ (Nx (r)) = 0, then supp (µ) \ Nx (r) is a proper closed subset of supp (µ) with µ-measure 1, contradicting the fact that supp (µ) is the least closed set having µ-measure 1. Hence, for any r > 0, µ (Nx (r)) > 0, that is, x ∈ N(µ). (ii) Suppose that N(µ) \ F 6= ∅. For x ∈ N(µ) \ F , since F is closed, there exists r > 0 such that Nx (r) ∩ F = ∅. Since x ∈ N(µ), µ (Nx (r)) > 0. Then µ (F ) < µ (Nx (r) ∪ F ) ≤ µ(B) = 1, a contradiction.. One notes that supp(µ) is also a closed set with µ-measure 1. Then, by taking F = supp (µ) in (3.2), we immediately obtain that N(µ) = supp (µ).. Remark 3.5. In general, such a set N(µ) always exists (N(µ) may be an empty set). Moreover, N(µ) is a closed set. In fact, let x be a limit point of N(µ). Then, for any r > 0, there exists yr ∈ N(µ) such that yr ∈ Nx (r). Set r ′ = r − kx − yr kB . Then Nyr (r ′ ) ⊂ Nx (r) and µ (Nyr (r ′ )) > 0, since (yr ∈ N(µ)). Consequently, µ (Nx (r)) > 0, implying x ∈ N(µ). Counterexample. µ (N(µ)) need not be equal to µ (B). For a given p > 0, there exists a Gaussian measure µp in B = l∞ with the σ-field generated by ∗ l1 ⊆ l∞ such that µp (Nx (p)) = 0 for any x ∈ l∞ , and then µ(N(µp )) = 0, since N(µp ) = ∅. In fact, supp(µp ) does not exist in this example by the following Theorem 3.6, which does not contradict Proposition 3.2 because l∞ is not separable. 8.

(15) Theorem 3.6. Let (B, k·kB ) be a real Banach space. For µ ∈ Prob (B), µ (N(µ)) = 1 if and only if supp (µ) exists. Proof. (Sufficiency.) By Theorem 3.4, N(µ) = supp (µ). Hence µ (N(µ)) = µ (supp (µ)) = 1. (Necessity.) By the same argument in the proof of Theorem 3.4, N(µ) ⊆ F for any closed subset F of B with µ (F ) = 1. Then, by the fact that N(µ) is closed (see Remark 3.5) and the assumption µ (N(µ)) = 1, we immediately see from the definition of topological support that N(µ) is exactly the supp (µ).. 9.

(16) 4. Gaussian measure on a Banach space. Let B be a real separable Banach space with k·kB -norm, and B ∗ the dual space of B with k·kB∗ -norm. The symbol (x, η), x ∈ B and η ∈ B ∗ , means the dual pairing between B and B ∗ . Definition 4.1. (i) A Gaussian measure µ on (B, B (B)) is a Borel measure in B such that for any η ∈ B ∗ , the random variable x ∈ B 7→ (x, η) is normally distributed; that is, there exist mη , ση ∈ R with ση > 0 such that (·, η) ∼ N (mη , ση ). (ii) A Gaussian measure µ on a Banach space is non-degenerate if every non-empty open subset of B has positive µ-measure.. In this section, we will study the topological support of a Gaussian measure on a real separable Banach space. The main theorem will show that the topological support of such a Gaussian measure is exactly the least closed subspace of B with the total measure 1, which in fact extends the Itô’s result (see the following Lemma 4.6) about the topological support of a Gaussian measure on a real separable Hilbert space to Banach space. One notes that not all Gaussian measures on an Banach space are nondegenerate. The sufficient and necessary condition is given as follows, which indeed a general condition for any finite Borel measure on a Banach space. Lemma 4.2. Let (B, k · kB ) be a real separable Banach space, and let µ ∈ Prob(B). Then µ is non-degenerate if and only if the set N(µ) is equal to the whole space B. Proof. It is immediately obtained from the definition of N(µ) given in Theorem 3.4 below. In fact, there is a large class of non-degenerate Gaussian measures on a Banach space. More precisely, we have the following Proposition 4.3. Let the triple (i, H, B) be an AWS, and let pt be the abstract Wiener measure on B with variance parameter t > 0. Then pt is non-degenerate. 10.

(17) Proof. For any r > 0 and φ1 , φ2 ∈ B ∗ , we see by Theorem 2.12 that Z pt (Nφ1 (r)) = 1Nφ2 (r) (x + φ2 − φ1 ) pt (dx) ZB |φ1 − φ2 |2H 1 = exp − − (x, φ1 − φ2 ) pt (dx), 2t t Nφ2 (r) which implies that pt (Nφ1 (r)) = 0 if and only if pt (Nφ2 (r)) = 0,. (4.1). where | · |H is the Hilbert norm on H. Let {ηn } ⊂ B ∗ be a countable dense subset of B. Obviously, for an arbitrarily given r > 0, B = ∪n Nηn (r). From (4.1) it follows that pt (Nηn (r)) > 0 for any n ∈ N, and hence pt is non-degenerate. Consider a real separable Hilbert space H with the inner product h·, ·iH , the following is the well-known result by Yu. V. Prohorov. Lemma 4.4. [6] Let µ be a Gaussian measure on (H, h·, ·iH ). Then there is a unique self-adjoint, positive-definite, and trace-class operator Sµ from H into itself such that for any x, y ∈ H, Z hSµ x, yiH = hz, xiH hz, yiH µ (dz) . H. Remark 4.5. 1. A self adjoint, positive definite, and trace-class operator from H into itself is called an S-operator of H. 2. In the above Lemma 4.4, the operator Sµ is called the covariance operator of µ in H. So Prohorov’s theorem told us that the covariance operator of a Gaussian measure on a real separable Hilbert space exists, which is an S-operator of H. 3. Since the identity operator IH of H is not an S-operator of H, there does not exist a Borel measure µ in H such that for any x ∈ H, Z 1 1 eihz,xiH µ (dz) = e− 2 hIH x,xiH = e− 2 hx,xiH . H. 11.

(18) Next, we recall the well-known Itô theorem (see [4]), where Itô showed that there is a close relation between Sµ and supp (µ) for a Gaussian measure µ on H with mean 0 as follows: Lemma 4.6. (Itô’s Theorem [4]) Let µ be a Gaussian measure on H with mean 0, that is, Z 1 eihz,xiH µ (dz) = e− 2 hSµ x,xiH , ∀ x ∈ H. H. Then supp (µ) is the least closed subspace of H with µ-measure 1. In fact, supp (µ) is exactly the orthogonal complement of the null space of Sµ in H. In the rest of this thesis, we will show that Itô’s theorem is also valid for Gaussian measure µ with mean 0 on a real separable Banach space (B, k·kB ). First of all, let Bµ be the intersection of all closed subspace of B with µ-measure 1. Then Bµ is also a real separable Banach space with the norm inherited from B. Since B is separable, µ (Bµ ) = 1 by the same arguments as the proof of Proposition 3.2. So we still denote the restriction µ |Bµ of µ to (Bµ , k·kB ) by µ. Lemma 4.7. µ is a Gaussian measure on (Bµ , k · kB ) with mean 0. Proof. Let Bµ∗ be the dual space of Bµ . For any η ∈ Bµ∗ , it has an extension ηe ∈ B ∗ by the Hahn-Banach theorem with the same operator norm as η, and so Z Z i(x, η)Bµ ,Bµ ∗ e µ(dx) = ei(x, ηe) µ(dx) Bµ B Z µ = ei(x, ηe) µ(dx) B R − 12 B (x, ηe)2 µ(dx). =e. ,. where (·, ·)Bµ ,Bµ∗ is the Bµ -Bµ∗ dual pairing. Then this lemma immediately follows. Let L be the closure of the linear subspace {(·, η)Bµ,Bµ∗ ; η ∈ Bµ∗ } in R. L2 (Bµ , µ) and Hµ the linear subspace x ϕ(x) µ(dx); ϕ ∈ L of Bµ , Bµ where the integrals inside the brace exist as Bµ -valued Bochner integrals 12.

(19) by using the Fernique theorem (see Theorem 2.13) and the Cauchy-Schwarz inequality. More precisely, for any ϕ ∈ L, Z Z k x ϕ(x)kB µ(dx) = |ϕ(x)| · kxkB µ(dx) Bµ. Bµ. ≤. (Z. Bµ. |ϕ(x)|2 µ(dx). )1/2 (Z. Bµ. kxk2B µ(dx). )1/2. < +∞.. Lemma 4.8. (i) Let ϕ ∈ L. Then Z x ϕ(x) µ(dx) = 0. if and only if. ϕ(x) = 0. Bµ. (ii) For η ∈ Bµ∗ , Z x (x, η)Bµ ,Bµ∗ µ(dx) = 0. if and only if. Bµ. µ − a.e. x ∈ Bµ .. (x, η)Bµ ,Bµ∗ = 0 ∀ x ∈ Bµ .. Proof. (i) The sufficiency is obvious. To show the necessity, assume that Z x ϕ(x) µ(dx) = 0. Bµ. Then, for any η ∈ Bµ∗ , Z. Bµ. (x, η)Bµ ,Bµ∗ · ϕ(x) µ(dx) =. Z. Bµ. x ϕ(x) µ(dx), η. !. = 0,. ∗ Bµ ,Bµ. R implying Bµ |ϕ(x)|2 µ(dx) = 0, whence ϕ(x) = 0, µ-a.e. x ∈ Bµ . R (ii) If Bµ x (x, η)Bµ ,Bµ∗ µ(dx) = 0, then we have by (i) that (x, η)Bµ ,Bµ∗ = 0, for µ-a.e. x ∈ Bµ . Let n o E = x ∈ Bµ ; (x, η)Bµ ,Bµ∗ = 0 .. It is easy to see that E is a closed subspace of B having µ-measure 1, which leads to Bµ ⊂ E. Thus E = Bµ . 13.

(20) *Z. Define an inner product on Hµ by + Z Z x ϕ(x) µ(dx), x ψ(x) µ(dx) :=. Bµ. Bµ. µ. Bµ. ϕ(x) ψ(x) ν(dx), ∀ ϕ, ψ ∈ L.. Such an inner product is meaningful by Lemma 4.8(i). Then (Hµ , h·, ·iµ) is a Hilbert space, and by Lemma 4.8(ii), Bµ∗ canRbe regarded as a dense subspace of Hµ by identifying arbitrary η ∈ Bµ∗ with Bµ x (x, η)Bµ ,Bµ∗ µ(dx). Denote by | · |µ the norm of Hµ . Observe that for any ϕ ∈ L, Z.

(21) (Z )12

(22) Z.

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(24).

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(26) x ϕ(x) µ(dx) ≤ kxk2B µ(dx) x ϕ(x) µ(dx)

(27) ,.

(28) Bµ.

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(30) Bµ Bµ B. and, for any η ∈. Bµ∗ ,. |η|µ ≤. µ. Z . Bµ. kxk2B µ(dx). 1/2. kηkBµ∗ .. Therefore, we have the following continuous inclusion maps: Bµ∗ ⊂ Hµ ⊂ Bµ .. Moreover, by Lemma 4.7 we see that for each η ∈ Bµ∗ , Z R − 21 Bµ (x, η)2 1 i(x, η)B ,B∗ 2 ∗ µ(dx) Bµ ,Bµ µ µ = e− 2 |η|µ . e µ(dx) = e Bµ. Consequently, Hµ is exactly the RKHS for µ by Theorem 2.14. In addition, µ is the σ-additive extension of the canonical Gaussian cylinder set measure µHµ to B(Bµ ), where µHµ is a finitely additive nonnegative set funtion on (Hµ , B(Hµ )) such that for any a ∈ R, Z a 1 u2 µHµ ({x ∈ Hµ ; hx, hiµ ≤ a}) = √ exp − du, ∀ h ∈ Hµ . 2 |h|2µ 2π |h|µ −∞. By using the result of Dudley, Feldman, and LeCam (see Theorem 2.11), (iHµ ,Bµ , Hµ , Bµ ) forms an abstract Wiener space, and µ is the associated abstract Wiener measure with variance parameter 1, where iHµ ,Bµ is the canonical embedding from Hµ into Bµ . By Proposition 4.3, µ is non-degenerate on Bµ . Then, by Lemma 4.2, Bµ is exactly the topological support of µ on B. In fact, we have finish the proof of the following Main Result. Theorem 4.9. Let B be a real separable Banach space and µ a Gaussian measure on B with mean 0. Then supp(µ) = Bµ .. 14.

(31) 5. Application. In this section, we will present a simple application of the Main Theorem 4.9. Given a Gaussian measure µ with mean 0 on a real separable Banach space B, we will show that a Gel’fand triple settig can be constructed from B. As a consequence, we may apply Hida’s white noise theory to study the B-valued stochastic analysis, stochastic differential equation, etc.. Let Bµ be the smallest closed subspace of B with µ-measure 1. By Theorem 4.9, Bµ is the topological support of µ. Pick up a countable dense subset {xn } of Bµ . By the Hahn-Banach theorem, there exists yn ∈ B ∗ such that kyn kBµ∗ = 1 and (xn , yn )B. = kxn kB ,. ∗ µ ,Bµ. for any n ∈ N, where kyn kBµ∗ = sup. 06=x∈Bµ.

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(34) (x, yn )B.

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(36) ∗

(37) µ ,B. kxkB. µ. .. For any x, y ∈ Bµ , define [x, y] =. ∞ X. 2−n (x, yn )B. ∗ µ ,Bµ. n=1. (y, yn)B. .. ∗ µ ,Bµ. Lemma 5.1. (i) [·, ·] is an inner product in Bµ . (ii) For any x ∈ Bµ , [x, x]1/2 ≤ kxkB . Proof. = 0 for any n ∈ N. For any ǫ > 0, there. (i) If [x, x] = 0 then (x, yn )B. ∗ µ ,Bµ. is an kǫ ∈ N such that xkǫ ∈ Nx (ǫ), implying that 0 = (x, ykǫ )B. ∗ µ ,Bµ. = (x − xkǫ , ykǫ )B. ∗ µ ,Bµ. + (xkǫ , ykǫ )B. ≥ kxkǫ kB − kx − xkǫ kB · kykǫ kBµ∗ > kxkǫ kB − ǫ. 15. ∗ µ ,Bµ.

(38) . So kxkǫ kB < ǫ. Clearly, lim xk1/n = x in Bµ , whence xk1/n → kxkB n→∞ B as n → ∞, and then x = 0. Other rules of inner product are obviously satisfied by [·, ·]. P −n (ii) [x, x] ≤ ∞ kxk2B · kyn k2Bµ∗ = kxk2B , for any x ∈ Bµ n=1 2 Let K be the completion of Bµ under |·|−1 -norm, where |x|−1 := [x, x]1/2 , x ∈ Bµ .. By Lemma 5.1(ii), the inclusion map iBµ ,K from (Bµ , k·kB ) into K, |·|−1 is continuous, so B(K) ∩ Bµ ⊂ B(Bµ ). On the other hand, since (Bµ , k·kB ) and K, |·|−1 are both standard measurable spaces by Theorem 2.4, we see that iBµ ,K (E) ∈ B (K) for any E ∈ B (Bµ ), that is, B(Bµ ) ⊂ B(K).. Hence B (Bµ ) = B (K) ∩ Bµ . Now, we define µK to be a measure on (K, B(K)) by µK (E) = µ(E ∩ Bµ ),. ∀ E ∈ B(K).. One notes that for any k ∈ K, the mapping x ∈ Bµ 7→ [x, k] is in Bµ∗ by Lemma 5.1(ii). By the Hahn-Banach theorem, there is a unique φk ∈ B ∗ such that [x, k] = (x, φk ), for any x ∈ Bµ . Then. Z. ir[x, k]. e. K. µ (dx) =. K. Z. eir[x, k] µK (dx). Bµ. =. Z. eir[x, k] µ (dx). Bµ. Z. eir(x, φk ) µ (dx) B 2Z r 2 = exp − (x, φk ) µ(dx) , 2 B. =. 16. ∀ r ∈ R,.

(39) R whence [·, k] ∼ N 0, B (x, φk )2 µ (dx) , k ∈ K. In other words, µK is a Gaussian measure on K with measure 0. By Lemma 4.4, the covariance operator SµK of µK is an S-operator of K. Moreover, by Lemma 4.6, supp(µK ) is equal to the orthogonal complement of the null space of SµK . One notes that µK (supp(µK )) = µ(supp(µK ) ∩ Bµ ) = 1. From the continuity of iBµ ,K , we see that supp(µK ) ∩ Bµ is also a closed subspace of Bµ , which is contained in Bµ and has µ-measure 1. Then we conclude that Bµ = supp(µK ) ∩ Bµ , implying that Bµ ⊂ supp µK . Thus. the closure of Bµ in K ⊂ the closure of supp µK in K = supp µK ⊂ K. By the denseness of Bµ in K,. K = supp µK ,. from which it follows that the null space of SµK = {0}, that is, SµK is oneto-one. Note that SµK has the spectral decomposition: SµK (x) = P. ∞ X. λn [x, en ] en ,. n=1. x ∈ K,. where λn > 0 for any n, λn < +∞, and {en } is a complete orthonormal basis (CONS, in short) of K. Let ( ) ∞ 2 X [x, e ] n 1/2 < +∞ . H0 = SµK (K) = x ∈ K : λn n=1 1/2. 1/2. For any x, y ∈ H0 , say x = SµK (k), y = SµK (h), k, h ∈ K, ! X [x, en ] · [y, en ] hx, yi0 := [k, h] = , λn n 1/2 and |x|0 := 0 . Then (H0 , h·, ·i0 ) is a real separable Hilbert space with n hx, xio 1/2 a CONB λn en .. 17.

(40) Proposition 5.2. (i) (iH0 ,K , H0 , K) forms an abstract Wiener space, where iH0 ,K : H0 → K is the inclusion map (ii) H0 ⊂ Bµ ⊂ K (iii) µK is the abstract Wiener measure of (iH0 ,K , H0 , K).. Proof. (i) Let {en } and {λn } be given as above. Then ∞ X . iH0 ,K (λn1/2 en ),. iH0 ,K (λ1/2 n en ). n=1. . =. ∞ X. λn < +∞,. n=1. implying that the operator iH0 ,K is Hilbert-Schmidt. Hence the | · |−1 norm is measurable on H0 and the triple (iH0 ,K , H0 , K) is an AWS. (iii) Let i∗H0 ,K be the adjoint operator of iH0 ,K . For any f ∈ K ∗ , there is a unique kf ∈ K such that p p (x, f )K,K∗ = hx, kf i−1 = h SµK (x), SµK (kf )i0 = hx, SµK (kf )i0 for any x ∈ H0 , from which it follows that i∗H0 ,K (f ) = SµK (kf ) and Z 1 ei(x, f )K,K∗ µK (dx) = e− 2 hSµK (kf ), kf i−1 K. 1. 2. = e− 2 |SµK (kf )|0 1. 2. = e− 2 |iH0 ,K (f )|0 . ∗. (5.1). From the equality (5.1) it follows that µK is the abstract Wiener measure of (iH0 ,K , H0 , K). (ii) Suppose on the contrary that there is an element x ∈ H0 \ Bµ . Note that µK and µ(· + x) are equivalent to each other (see [6]). Then µK (Bµ ) = µK (Bµ + x) = 1. Since Bµ ∩ (Bµ + x) = emptyset, µK (K) ≥ µK (Bµ ) + µK (Bµ + x) = 2, a contradiction. 18.

(41) From (5.1) and Theorem 2.14 it follows that H0 is the unique RKHS for µ . On the other hand, for any f ∈ K ∗ , Z Z i(x, f |Bµ )Bµ ,Bµ ∗ i(x, f )K,K ∗ K e µ (dx) = e µ(dx) K Bµ ( ) Z 1 = exp − (x, f |Bµ )2Bµ ,Bµ∗ µ(dx) 2 Bµ K. 1. 2. = e− 2 |f |Bµ |µ .. (5.2). Obviously, Hµ is also continuously embedded in K. Then, from (5.2), it follows that Hµ is also a RKHS for µK . Consequently, by uniqueness (see Theorem 2.14) H0 = Hµ . Therefore, we have the continuous inclusion maps: p K ∗ ⊂ Bµ∗ ⊂ Hµ = H0 = SµK (K) ⊂ Bµ ⊂ K,. where K ∗ is regarded as a dense subspace of Bs∗ by identifying arbitrary f ∈ K ∗ with f |Bµ . Moreover, for each η ∈ Bµ∗ , Z R 1 2 − 21 Bµ (x, η)2B ,B∗ µ(dx) i(x, η)Bµ ,Bµ ∗ µ µ e µ(dx) = e = e− 2 |η|µ . Bµ. 19.

(42) 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] L. Gross, Abstract Wiener spaces, in: Proc. Fifth Berkeley Sympos. Math. Statist. and Probab. vol. II: Constribution to Probability Theory, Part I (1967), 31-42, University of California Press, Berkeley. [3] K. Itô, The topological support of Gauss measure on Hilbert space, Nagoya Math. J. 38 (1970), 181-183. [4] K. Itô, Foundations of Stochastic Differential Equations in Infinite Dimensional Spaces, CBMS-NSF Regional Conference Series in Applied Mathematics, vol. 47, SIAM, Philadelphia, 1984. [5] J. Kuelbs, Gaussian measures on Banach spaces, J. Funct. Anal. 5 (1970), 354-367. [6] H.-H. Kuo, Gaussian Measures in Banach Spaces, Lect. Notes in Math., vol. 463, Springer-Verlag, Berlin/New York, 1975. [7] G. Da Prato, J. Zabczyk, Stochastic Equations in Infinite Dimensions, Encyclopedia Math. Appl., vol. 44, Cambridge University Press, Cambridge, 1992. [8] Y. Yamasaki, Measures on Infinite Dimensional Spaces, Series in Pure Mathematics, vol. V, World Scientific, Singapore, 1985.. 20.

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