White Noise Representation of Gaussian Random Fields Zachary Gelbaum Department of Mathematics Oregon State University Corvallis, Oregon 97331-4605, USA 2 1 0 2 Abstract n a We obtain a representation theorem for Banach space valued Gaussian random variables as J integrals againsta white noise. As a corollarywe obtain necessaryand sufficient conditions for the 6 existenceofawhitenoiserepresentationforaGaussianrandomfieldindexedbyacompactmeasure 2 space. As an application we show how existing theory for integration with respect to Gaussian ] processes indexed by [0,1]canbe extended to Gaussianfields indexed by compact measure spaces. R P Keywords: white noise representation, Gaussian random field, stochastic integral . h t a 1. Introduction m [ Muchofliterature regardingthe representationofGaussianprocessesasintegralsagainstwhite noise hasfocusedonprocessesindexedby R, inparticularcanonicalrepresentations(mostrecently 1 see [8] and references therein) and Volterra processes (e.g. [1, 3]). An example of the use of such v 5 integralrepresentationsistheconstructionofastochasticcalculusforGaussianprocessesadmitting 3 a white noise representation with a Volterra kernel (e.g. [1, 11]). 6 In this paper we study white noise representations for Gaussian random variables in Banach 5 spaces, focusing in particular on Gaussian randomfields indexed by a compact measure space. We . 1 show that the existence of a representation as an integral against a white noise on a Hilbert space 0 H is equivalent to the existence of a version of the field whose sample paths lie almost surely in 2 H. For example as a consequence of our results a centered Gaussian process Y indexed by [0,1] 1 t : admits a representation v 1 d i Y = h(t,z)dW(z) X t Z 0 ar forsomeh L2([0,1] [0,1],dν dν), ν ameasureon[0,1]andW the white noiseonL2([0,1],dν) if and only∈if there is×a version o×f Y whose sample paths belong almost surely to L2([0,1],dν). t The stochastic integral for Volterra processes developed in [11] depends on the existence of a white noise integral representation for the integrator. If there exists an integral representation for a given Gaussian field then the method in [11] can be extended to define a stochastic integral with respect to this field. We describe this extension for Gaussian random fields indexed by a compact measure space whose sample paths are almost surely square integrable. Email address: [email protected] (Zachary Gelbaum) Preprint submitted to Elsevier Section2containspreliminarieswewillneedfromMalliavinCalculusandthetheoryofGaussian measures over Banach spaces. In section 3, Theorem 1 gives our abstract representation theorem andCorollary2specializestoGaussianrandomfieldsindexedbyacompactmeasurespace. Section 4 contains the extension of results in [11]. 2. Preliminaries 2.1. Malliavin Calculus We collect here only those parts of the theory that we will explicitly use, see [15]. Definition 1. SupposewehaveaHilbertspaceH. Thenthereexistsacomplete probabilityspace (Ω, ,P) and a map W :H L2(Ω,P) satisfying the following: F → 1. W(h) is a centered Gaussian random variable with E[W(h)2]= h H k k 2. E[W(h1)W(h2)]= h1,h2 H h i This process is unique up to distribution andis calledthe Isonormal or White Noise Process onH. The classical example is H =L2[0,1] and W(h) is the Wiener-Ito integral of h L2. ∈ Let denote the set of random variables of the form S F =f(W(h1),...,W(hn)) forsomef C∞(Rn)suchthatf andallitsderivativeshaveatmostpolynomialgrowthatinfinity. ∈ For F we define the derivative as ∈S n DF = ∂jf(W(h1),...,W(hn))hj. X1 We denote by D the closure of with respect to the norm induced by the inner product S F,G =E[FG]+E[ DF,DG ]. D H h i h i (D is usually denoted D1,2.) We also define a directional derivative for h H as ∈ D F = DF,h . h H h i D is then a closed operator from L2(Ω) to L2(Ω,H) and dom(D) = D. Further, D is dense in L2(Ω). Thus we can speak of the adjoint of D as an operator from L2(Ω,H) to L2(Ω). This operator is called the divergence operator and denoted by δ. dom(δ) is the set of all u L2(Ω,H) such that there exists a constant c (depending on u) with ∈ E[ DF,u ] c F H | h i |≤ k k for all F D. For u dom(δ) δ(u) is characterizedby ∈ ∈ E[Fδ(u)]=E[ DF,u ] H h i for all F D. ∈ For examples and descriptions of the domain of δ see [15], section 1.3.1. When we want to specify the isonormal process defining the divergence we write δW. We will also use the following notations interchangeably δW(u), udW. Z 2 2.2. Gaussian Measures on Banach Spaces Here we collect the necessary facts regarding Gaussian measures on Banach spaces and related notions that we willuse in what follows. Forproofs andfurther details see e.g. [4, 5, 6, 13,16]. All Banach spaces are assumed real and separable throughout. Definition 2. Let B be a Banachspace. A probability measure µ on the borel sigma field of B is called Gaussian if for every l B∗ the random variable l(x) : (B, ,µ) R is GaussianB. The ∈ B → mean of µ is defined as m(µ)= xdµ(x). Z B µ is calledcenteredif m(µ)=0. The (topological)support ofµ inB, denotedB0, is defined as the smallest closed subspace of B with µ-measure equal to 1. ThemeanofaGuassianmeasureisalwaysanelementofB,andthusitsufficestoconsideronly centered Gaussianmeasures as we canthen acquire any Gaussian measure via a simple translation of a centered one. For the remainder of the paper all measures considered are centered. Definition 3. The covarianceof a Gaussianmeasure is the bilinear form C :B∗ B∗ R given µ × → by C (k,l)=E[k(X)l(X)]= k(x)l(x)dµ(x). µ Z B Anygaussianmeasureiscompletely determinedby its covariance: if fortwoGaussianmeasures µ, ν on B we have C =C on B∗ B∗ then µ=ν. µ ν × If H is a Hilbert space then C (f,g)=E[ X,f X,g ]= x,f x,g dµ(x) µ h ih i Z h ih i B defines a continuous, positive, symmetric bilinear form on H H and thus determines a positive × symmetric operator K on H. K is of trace class and is injective if and only if µ(H) = 1. µ µ Conversely, any positive trace class operator on H uniquely determines a Guassian measure on H [6]. Whenever we consider a Gaussian measure µ over a Hilbert space H we can after restriction to a closed subspace assume µ(H)=1 and do so throughout. We will denote by H the Reproducing Kernel Hilbert Space (RKHS) associatedto a Gaussian µ measureµ onB . Therearevariousequivalentconstructionsofthe RKHS.We follow[16]andrefer the interested reader there for complete details. Foranyfixedl B∗, C (l, ) B (this isa nontrivialresultinthe theory). Considerthe linear µ ∈ · ∈ span of these functions, S =span C (l, ) : l B∗ . µ { · ∈ } Define an inner product on S as follows: if φ()= na C (l , ) and ψ()= mb C (k , ) then 1 i µ i 1 j µ j · · · · P P n m <φ,ψ > a b C (l ,k ). Hµ≡ i j µ i j X1 X1 H is defined to be the closure of S under the associatednorm . This norm is stronger than µ k·kHµ B, Hµ is a dense subset of B0 and Hµ has the reproducing property with reproducing kernel k·k C (l,k): µ φ(),C (l, ) =φ(l) l B∗, φ H . h · µ · iHµ ∀ ∈ ∈ µ 3 Remark 1. Often one begins with a collection of random variables indexed by some set, Yt t∈T. Forexamplesuppose(T,ν)isafinite measurespace. ThensettingK(s,t)=E[Y Y ]andsu{pp}osing s t that application of Fubini-Tonelli is justified we have for f,g L2(T) ∈ E[ Y,f Y,g ]= E[Y ,Y ]f(s)g(t)dνdν = K(s,t)(f),g h ih i Z Z s t h i T T where we denote K(s,t)f(s)dν(s) by K(s,t)(f). If one verifiesthatthis lastoperatoris positive T symmetric and trRaceclass then the abovecollection Yt t∈T determines a measureµ onL2(T)and { } the above construction goes through with C (f,g)= K(s,t)(f),g and the end result is the same µ h i with H a space of functions over T. µ Define HX to be the closed linear span of X(l) l∈B∗ in L2(Ω,P) with inner product X(l),X(l′) =C (l,l′)(againforsimplicity{assu}meX isnondegenerate). Fromthereproducing h iHX µ property we can define a mapping R from H to H given initially on S by X µ X n k R ( c C (l , ))= c X(l) X k µ k k · X1 X1 and extending to an isometry. This isometry defines the isonormal process on H . µ In the case that H is a Hilbert space and µ a Gaussianmeasure on H with covarianceoperator K it is known that H =√K(H) with inner product √K(x),√K(y) = x,y . µ h iHµ h iH It was shown in [12] that givena Banachspace B there exists a Hilbert space H suchthat B is continuously embedded as a dense subset of H. Any Gaussian measure µ on B uniquely extends to a Gaussian measure µ on H. The converse question of whether a given Gaussian measure on H H restrictsto a GaussianmeasureonB is far moredelicate. There aresomeknownconditionse.g. [7]. The particularcase whenX is a metric spaceand B =C(X)has been the subject of extensive research [14]. Let us note here however that either µ(B) = 0 or µ(B) = 1 (an extension of the classical zero-one law, see [4]). From now on we will not distinguish between a measure µ on B and its unique extension to H when it is clear which space we are considering. 3. White Noise Representation 3.1. The General Case The setting is the following: B is a Banach space densely embedded in some Hilbert space H (possibly with B =H), where H is identified with its dual, H =H∗. (A Hilbert space equal to its dual in this way is called a Pivot Space, see [2]). The classical definition of canonical representation has no immediate analogue for fields not indexed by R, but the notion of strong representation does. Let L : H H be unitary. µ Then WX(h) = RX(L∗(h)) defines an isonormal process on H and σ( WX(h) →h∈H) = σ(HX) = σ( X(l) l∈B∗) where the last inequality follows from the density of H i{n B∗. } { } We now state our representation theorem. Theorem 1. Let B be a Banach space, µ a Gaussian measure on B, and C the covariance of µ µ on B∗ B∗. Then µ is the distribution of a random variable in B given as a white noise integral × of the form X(l)= h(l)dW. (3.1) Z 4 for some h : B∗ H and a Hilbert space H, where h is a Hilbert-Schmidt operator on H. H → | Moreover, the representation is strong in the following sense: σ( WX(h) h∈H)=σ( X(l) l∈B∗). { } { } Proof. B H = H∗ as above. Let W be the isonormal process constructed above and C (l,k) X µ the covaria⊂nceofµ. LetL be aunitary mapfromH toH anddefine the function k (l):B∗ H µ L → by k (l) L(C (l, )). L µ ≡ · Consider the Gaussian random variable determined by Y(l) k (l)dW . ≡Z L X We have Cov(Y(l1),Y(l2))=hkL(l1),kL(l2)iH =hCµ(l1,·),Cµ(l2,·)iHµ =Cµ(l1,l2) so that µ is the distribution of Y(l) and d X(l)= k (l)dW . Z L X It is clear that kL is linear and if Cµ(h1,h2)= K(h1),h2 H, h1,h2 H, then from above h i ∈ k∗k =K. L L Because K is trace class this implies that k is Hilbert-Schmidt on H. L From the preceding discussion we have σ( WX(h) h∈H)=σ( X(l) l∈B∗). { } { } Remark 2. While the statement of the above theorem is more general than is needed for most applications, this generality serves to emphasize that having a “factorable”covarianceand thus an integral representation are basic properties of all Banach space valued Gaussian random variables. Remark 3. The kernel h(l) is unique up to unitary equivalence on H, that is if L′ =UL for some unitary U on H L as above, then d d Z hL′(l)dW =Z U(hL(l))dW =Z hL(l)dW. Remark 4. In the proof above, kL(l1),kL(l2) H =Cµ(l1,l2) (3.2) h i is essentially the “canonical factorization” of the covariance operator given in [17], although in a slightly different form. Remark 5. Inthelanguageofstochasticpartialdifferentialequations,whatwehaveshownisthat every Gaussian random variable in a Hilbert space H is the solution to the operator equation L(X)=W for some closed unbounded operator L on H with inverse given by a Hilbert-Schmidt operator on H. 5 3.2. Gaussian Random Fields The proof of Theorem 1 has the following corollary for Gaussian random fields: Corollary2. LetX beacompactHausdorffspace,ν apositiveRadonmeasureandH =L2(X,dν). If B isacollection ofcenteredGaussianrandomvariables indexedbyX,then B hasaversion x x { } { } with sample paths belonging almost surely H if and only if d B = h(x, )dW (3.3) x Z · for some h:X H such that the operator K(f) h(x,z)f(z)dν(z) is Hilbert-Schmidt. In this → ≡ X case (3.2) takes the form R E[B B ]= h(x,z)h(y,z)dν(z). x y Z X In other words, the field B determines a Gaussian measure on L2(X,dν) if and only if B x x admits an integral representation (3.3). 3.3. Some Consequences and Examples In principle, all properties of a field are determined by its integral kernel. Without making an exhaustive justification of this statement we give some examples: In Corollary 2 above, being the kernel of a Hilbert-Schmidt operator, h L2(X X,dν dν). ∈ × × This means that we can approximate h by smooth kernels (supposing these are available). If we assume h(x, ) is continuous as a map from X to H i.e. · lim h(x, ) h(y, ) =0 H x→yk · − · k for each y X and let h C∞(X), h L2 h it follows that h (x, ) h(x, ) 0 pointwise so n n n H ∈ ∈ → k · − · k → that if Bn = h (x, )dW x Z n · we have E[BnBn] E[B B ] x y → x y point-wise. This last condition is equivalent to Bn d B → and we can approximate in distribution any field over X with a continuous (as above) kernel by fields with smooth kernels. The kernel of a field over Rd describes its local structure [9]: The limit in distribution of X(t+c x) X(t) n lim − rcnn→→00 rn is h(t+c x) h(t) lim n − dW rcnn→→00Z rn 6 where h is the integral kernel of X, and this last limit is determined by the limit in H of h(t+c x) h(t) lim n − . rcnn→→00 rn TherepresentationtheoremyieldsasimpleproofoftheknownseriesexpansionusingtheRKHS. The setting is the same as in Theorem 1. Proposition3. LetY(l)beacenteredGaussian random variable in aHilbert space H with integral kernel h(l). Let e ∞ be a basis for H. Then there exist i.i.d. standard normal random variables k 1 { } ξ such that k { } ∞ Y(l)= ξ Φ (l) k k X1 where Φ (l)= h(l),e and the series converges in L2(Ω) and a.s. k k H h i Proof. For each l ∞ h(l)= Φ (l)e . k k X1 We have ∞ ∞ Y(l)= Φ (l)e dW = Φ (l)ξ Z k k k k X1 X1 where ξ = e dW are i.i.d. standard normal as dW is unitary from H to L2(Ω). As k k Φ (l){ }l2(N){tRhe serie}s converges a.s. by the martingalRe convergence theorem. k { }∈ 4. Stochastic Integration Combined with Theorem 1 above, [11] furnishes a theory of stochastic integration for Gaussian processes and fields, which we now describe for the case of a random field with square integrable sample paths as in Corollary 2. Denote by µ the distribution of B in H = L2(X,dν) and as above the RKHS of B by x x { } H H. Let µ ⊂ B = h(x, )dW x Z · andL∗(f)= h(x,y)f(y)dν(y). ThenL∗ :H H isanisometryandthemapv R (L∗(v)) µ B → 7→ ≡ W(v) : H RH (H is the closed linear span of B as defined in sec. 2) defines an isonormal B B x → { } process on H. Denote this particular process by W in what follows. First note that as H = L∗(H) and L is unitary, it follows immediately that D1,2 = L∗(D1,2) µ Hµ H where we use the notation in [15, 11] and the subscript indicates the underlying Hilbert space. 7 Thefollowingprooffrom[11]carriesoverdirectly: ForasmoothvariableF(h)=f(B(L∗(h1),...,B(L∗(hn)) we have n EhDB(F),uiHµ =Eh f′(B(L∗(h1),...,B(L∗(hn))L∗(hk),uiHµ X1 =E f′(B(L∗(h1),...,B(L∗(hn))hk,L(u) H h i X =E f′(W(h1),...,W(hn))hk,L(u) H h i X =E DW(F),L(u) H h i which establishes dom(δB)=L∗(dom(δW)) and udB = L(u)dW. Z Z The series approximationin [11] also extends directly to this setting. Theorem 4. If Φ is a basis of H then there exists i.i.d. standard normal ξ such that: k µ k { } { } 1. If f H and µ ∈ ∞ fdB = f,Φ ξ a.s. Z h kiHµ k X1 2. If u D then ∈ Hµ ∞ udB = ( u,Φ DB u,Φ ) a.s. Z X1 h kiHµ −h Φk kiHµ Proof. The proof of (1) and (2) follows that in [11]. Remark 6. For our purposes the method of approximation via series expansions above seems most appropriate. However in [1] a Riemann sum approximation is given under certain regularity hypotheses on the integral kernel of the process, and this could be extended in various situations as well. Remark 7. Theavailabilityofthe kernelabovesuggeststhe methodin[1]wherebyconditionsare imposed on the kernelin order to prove an Ito Formula as promising for extension to more general settings. 5. References References [1] Elisa Al`os, Olivier Mazet, and David Nualart. Stochastic calculus with respect to Gaussian processes. Ann. Probab., 29(2):766–801,2001. 8 [2] Jean-Pierre Aubin. Applied functional analysis. Pure and Applied Mathematics (New York). Wiley-Interscience, New York, second edition, 2000. With exercises by Bernard Cornet and Jean-Michel Lasry, Translated from the French by Carole Labrousse. [3] Fabrice Baudoin and David Nualart. Equivalence of Volterra processes. Stochastic Process. Appl., 107(2):327–350,2003. [4] Vladimir I. Bogachev. 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