White Noise Space Analysis and Multiplicative Change of Measures
aa r X i v : . [ m a t h - ph ] J a n WHITE NOISE SPACE ANALYSIS AND MULTIPLICATIVE CHANGEOF MEASURES
DANIEL ALPAY, PALLE JORGENSEN, AND MOTKE PORAT
Abstract.
In this paper we display a family of Gaussian processes, with explicit formulasand transforms. This is presented with the use of duality tools in such a way that thecorresponding path-space measures are mutually singular. We make use of a correspond-ing family of representations of the canonical commutation relations (CCR) in an infinitenumber of degrees of freedom.A key feature of our construction is explicit formulas for associated transforms; these areinfinite-dimensional analogues of Fourier transforms. Our framework is that of GaussianHilbert spaces, reproducing kernel Hilbert spaces, and Fock spaces. The latter forms thesetting for our CCR representations. We further show, with the use of representationtheory, and infinite-dimensional analysis, that our pairwise inequivalent probability spaces(for the Gaussian processes) correspond in an explicit manner to pairwise disjoint CCRrepresentations.
Contents
Introduction 11. Background and Tools 42. Conditional Negative Definite Functions 63. A One-Parameter Family of Gaussian Measures 103.1. Mutual singularity of the measures 144. Isometric Isomorphisms and Intertwining Operators 174.1. The symmetric Fock space 174.2. Generalized infinite Fourier transform 194.3. The canonical commutation relations 22References 25
Introduction
Our framework is an infinite-dimensional harmonic analysis and our main aims are four-fold. Starting with certain Gelfand triples, built over infinite-dimensional Hilbert space,we identify and construct, associated indexed families of infinite-dimensional probabilityspaces and corresponding stochastic processes. Second, we give conditions for when thesestochastic processes are Gaussian. Third, we show that different values of the index-variable
Date : January 6, 2021.2010
Mathematics Subject Classification.
Key words and phrases.
Gaussian processes, canonical commutation relations, infinite-dimensional har-monic analysis, Gelfand triples, reproducing kernel, unitary equivalent, Fock space, intertwining operators,measure preserving transformations, white noise space. yield mutually singular probability measures. Fourthly, we identify associated representa-tions of CCRs, and we show that pairwise different index values yield mutually inequivalentrepresentations.Our current focus lies at the crossroads of white noise analysis, Itˆo calculus, and algebraicquantum physics. While there is a very large prior literature in the area, we wish here toespecially call attention to the following papers by Albeverio et al. [1, 2, 3, 4, 5, 6].Some of the basic concepts that we study here can also be found, in special cases, and inone form or the other, with variants of our main themes, at [51, 52, 30, 39]. Our presentresults go beyond those of earlier papers. This includes (i) our making an explicit anddirect link between this wide family of stochastic processes, their harmonic analysis, andthe corresponding representations. Also, (ii) our infinite-dimensional harmonic analysis,and infinite-dimensional transform theory, are new. Also new are (iii) our results whichidentify which of the general stochastic processes are Gaussian.Our study of Gelfand triples in the framework of Schwartz tempered distributions ismotivated in part by Wightman’s framework for quantum fields, i.e., that of (unbounded)operator valued tempered distributions. For details, see [60].We now turn to the study of stochastic processes indexed by an Hilbert space H froma Gelfand triple S ( R ) ֒ → H ֒ → S ′ ( R ) , side to side with the question of when such processes are Gaussian. Here is a descriptionon how to obtain the Hilbert space H . Consider the real Schwartz space S := S ( R ) anda conditional negative definite (CND) function N : S → R that is continuous w.r.t theFr´echet topology and satisfies N (0) = 0. A general theory of Schoenberg then tells us that S can be isometrically embedded into an Hilbert space H , via a mapping s ϕ s such that N ( s ) = k ϕ s k H . This idea goes back to works by Schoenberg [46, 47, 48] and von Neumann [49], wherethe question ”when can a metric space be realized in a Hilbert space with a norm” wasconsidered. The Hilbert space H is taken to be the reproducing kernel Hilbert space (RKHS)with reproducing kernel (RK) of the form ϕ N ( s , s ) = (cid:0) N ( s ) + N ( s ) − N ( s − s ) (cid:1) / H from the CND function N , see Section 2. It further follows naturally from the Bochner–Minlos theorem, see e.g.,[61], applied for the positive definite functions Q λ ( s ) = exp {− λ N ( s ) / } with λ >
0, theexistence of a one-parameter family of Gaussian measures { P λ } λ> on S ′ , which satisfies E P λ (cid:2) exp { iX s } (cid:3) = exp {− λ N ( s ) / } , where X s : S ′ → R is given by the duality of the spaces S and S ′ , as X s ( ω ) = h ω, s i , and E P λ stands for the expectation w.r.t the measure P λ .The stochastic process given by { X s } s ∈S plays a main role in our analysis, as one of thepurposes in the paper is to explore its behavior w.r.t different measures (from a family ofmeasures described below, cf. equation (3.3)); in particular we are interested in the caseswhere it is Gaussian. We show that the Gaussian property of that process is equivalent to HITE NOISE SPACE ANALYSIS AND MULTIPLICATIVE CHANGE OF MEASURES 3 a scaling assumption on the CND function N , simply saying that N ( αs ) = α N ( s )for every s ∈ S and α ∈ R (cf. Proposition 2.3). As our settings are quite general, one canconsider many examples for the CND function N , which provide us with different examplesof Gaussian processes. In particular, one can choose the CND function N ( s ) = k s k L ( R ) ,so that we get the standard Brownian Motion, or another function to get the fractionalBrownian Motion (cf. Examples 2.4-2.7)There are many dichotomy results regarding Gaussian measures, such as Kakutani’s di-chotomy theorem on infinite product measures [40], which states that any two Gelfand triplemeasures on S ′ are either equivalent or mutual singular. Also in [30, Chapter 5] it is proventhat any two measures, from a given one-parameter family of measures, are mutually sin-gular; this corresponds to the analysis in this paper with the choice of N ( s ) = k s k L ( R ) .Another interesting result in that spirit is presented in [39], where explicit conditions fortwo measures being equivalent or mutually singular are given and the theory of RK Hilbertspaces is heavily used. By using the theory in [39] we obtain the first main result in thepaper (cf. Theorem 3.9), that is establishing the crucial fact of the measures { P λ } λ> beingmutually singular. To do so, we explore the covariance function Γ λ ( s , s ) of the Gaussianprocess { X s } s ∈S w.r.t the measure P λ , that isΓ λ ( s , s ) = λ (cid:0) N ( s + s ) − N ( s − s ) (cid:1) / , and compare between such covariance functions for distinct values of λ . Next, we makeanother important connection between the Hilbert space H and the family of Gaussianmeasures { P λ } λ> . For that we let H λ be the scaled Hilbert space, that is the space H withthe scaled norm k · k H λ = λ k · k H , and then present an explicit transform (cf. equation (4.1)) W λ : Γ sym ( H λ ) → L ( S ′ , P λ )that is an isometric isomorphism from the symmetric Fock space of H λ onto L ( S ′ , P λ ) (cf.Theorem 4.2). By using this transform we will be able to move from the notion of equivalenceof the Gaussian spaces (or more precisely the Gaussian measures) to the equivalence of therepresentations of the CCR algebra on the symmetric Fock spaces; see the proof of Corollary4.19. Another explicit formula which fits in naturally in the analysis presented in this paperis a generalized (infinite dimensional) Fourier transform (cf. equation (4.5)) , which takes L ( S ′ , P λ ) onto the RKHS that is given by the RK Q λ ( s , s ) = exp {− λ N ( s − s ) / } . The last transform is a generalization of results from [30] to a wider family of conditionalnegative definite functions, other than N ( s ) = k s k L ( R ) .Finally, we present the second main result of the paper (cf. Corollary 4.19) which isan explicit example of unitarily inequivalent representations of the CCR algebra of an in-finite dimensional Hilbert space H ; for details on the CCRs and representations, see e.g.,[51, 52, 54]. This example is built in a natural way from our analysis, while due to theStone–von Neumann theorem it is impossible to do so in the finite dimensional case. Inthe proof we use the mutual singularity of the measures from the family { P λ } λ> and theintertwining operator W λ , to deduce that any two representations are disjoint, as the cor-responding measures are mutually singular. D. ALPAY, P. JORGENSEN, AND M. PORAT
The outline of the paper is as follows: Section 1 is devoted to present some preliminariesresults and definitions, which include positive definite kernels, conditional negative definitefunctions, the Bochner–Minlos theorem and (symmetric) Fock spaces.In Section 2 we lay down our basic setting of the paper; we present in details some ofthe results by Schoenberg, connect those with the way we build our associated white noisespace from the CND function N we started with, and establish the condition on N that isequivalent for the studied stochastic process to be Gaussian (Proposition 2.3). Towards theend of the section we present some examples which fit into our setting.In Section 3 we then build a family of L path spaces which correspond to the family ofGaussian measures { P λ } λ> obtained from the previous section after the simple change ofmultiplication by a scalar λ >
0. Then in Subsection 3.1 we prove the mutual singularityof the measures (Theorem 3.9) and give another example of a one-parameter family ofmutually singular measures which involves the Fourier transform (Example 3.12).In Section 4 we further study the intertwining operators between the symmetric Fockspace and the L space from the previous section (Theorem 4.2), while also define aninfinite dimensional generalized Fourier transform and explore its properties (Lemma 4.7and Theorem 4.8) . Using these intertwining operators and the mutual singularity of themeasures from previous sections, we establish the connection to representations of the CCRalgebra of the (infinite dimensional Hilbert space) H (Corollary 4.19).1. Background and Tools
We begin with presenting some preliminary background and definitions. Let X be anon-empty set. A function ϕ : X × X → R is called positive definite (PD) kernel, if ϕ issymmetric (i.e., ϕ ( x, y ) = ϕ ( y, x ) for all x, y ∈ X ) and n X j =1 n X k =1 c j c k ϕ ( x j , x k ) ≥ n ∈ N , x , . . . , x n ∈ X and c , . . . , c n ∈ C . A function f : X → R is called positivedefinite (PD) , if the kernel k f ( x, y ) = f ( x − y ) is a positive definite kernel, i.e., if f is oddand n X j =1 n X k =1 c j c k f ( x j − x k ) ≥ n ∈ N , x , . . . , x n ∈ X and c , . . . , c n ∈ C . A function ϕ : X × X → R is called conditional negative definite (CND) kernel, if n X j,k =1 c j c k ϕ ( x j , x k ) ≤ n ∈ N , x , . . . , x n ∈ X and c , . . . , c n ∈ C with P nj =1 c j = 0. Similarly to the positivedefinite definition, a function f : X → R is called conditional negative definite (CND) , if ϕ f ( x, y ) := f ( x − y ) is a CND kernel. Our conventions regarding CND functions follow [20].The class S := S ( R ) ⊂ L ( R ) is the Schwartz class of functions on R which are rapidlydecreasing, smooth and C ∞ . The dual space of S is the space S ′ := S ′ ( R ) of tempereddistributions ,while any ω ∈ S ′ defines a linear functional h ω, ·i : S 7→ R , also called the HITE NOISE SPACE ANALYSIS AND MULTIPLICATIVE CHANGE OF MEASURES 5 action of ω on elements from S , denoted by ω ( s ) = h ω, s i for s ∈ S . We recall the Bochner–Minlos theorem (associated to Gelfand triples with S and S ′ ) as it appears in [33, AppendixA]; for a more general framework of the theorem associated to nuclear Gelfand triples, see[61]. Theorem 1.1 (Bochner–Minlos) . If g : S → C is continuous w.r.t the Fr´echet topology, g (0) = 1 and g is positive definite, in the sense of (1.1), then there exists a unique probabilitymeasure P on ( S ′ , B ( S ′ )) such that E P (cid:2) exp { i h· , s i} (cid:3) := Z S ′ exp { i h ω, s i} dP ( ω ) = g ( s ) , ∀ s ∈ S . We recall that L ( S ′ , P ) is the L − space for the white noise process. For every s ∈ S , we define the random variable X s on S ′ via the duality of S and S ′ , thatis by X s ( · ) = h· , s i , i.e., X s ( ω ) = h ω, s i = ω ( s )for every ω ∈ S ′ . One of the main studied objects in this paper is stochastic processes { X s } s ∈S of that form and in particular the cases where those processes are Gaussian, whilenoticing that using the Bochner–Minlos theorem, different positive definite functions pro-duce different probability measures on S ′ and hence different processes. Remark 1.2.
We use the framework of Gelfand triples which consist of an Hilbert spacetogether with S and S ′ , see [50] . There are other works in infinite dimensional analysis,which use different approaches; one of those is to use B and B ′ instead of S and S ′ , where B is a Banach space and B ′ is its dual space, see [26, 27] . Gaussian processes are one of the main objects in this paper, thereby we recall theexplicit definition of a stochastic process being Gaussian. A stochastic process { Y t } t ∈ I is called a Gaussian process (w.r.t a measure P ) , if for every t , . . . , t n ∈ I , the randomvector Y t ,...,t n = ( Y t , . . . , Y t n ) is a multivariate Gaussian random variable; sometimes it iscalled a jointly Gaussian process. This is equivalent to say that for every α , . . . , α n ∈ R and t , . . . , t n ∈ I , the random variable α Y t + . . . + α n Y t n has a univariate Gaussiandistribution. Another equivalent definition of { Y t } t ∈ I being a Gaussian process (w.r.t P ) —using the characteristic function ϑ X t ,...,X tn — is that for every t , . . . , t n ∈ I , there exist σ jℓ , µ ℓ ∈ R with σ jj >
0, such that ϑ X t ,...,X tn ( α , . . . , α n ) := E P h exp n i n X ℓ =1 α ℓ Y t ℓ oi = exp n i n X ℓ =1 µ ℓ α ℓ − − n X j,ℓ =1 σ jℓ α ℓ α j o , (1.2)here σ jℓ and µ ℓ can be shown to be the covariances and means of the variables in theprocess and the left hand side of (1.2) is the characteristic function of the random vector( Y t , . . . , Y t n ).Recall the definitions of the Hermite polynomials h n ( x ) = ( − n e x d n dx n (cid:0) e − x (cid:1) , n ≥ D. ALPAY, P. JORGENSEN, AND M. PORAT and the Hermite functions ξ n ( x ) = π − / (( n − − / e − x / h n − ( √ x ), for n ≥
1. The set { h n } n ≥ is an orthogonal basis for L ( R , µ ), where dµ ( x ) = (2 π ) − / e − x / dx , with Z R h n ( x ) h m ( x ) dµ ( x ) = (2 π ) − / Z R h n ( x ) h m ( x ) e − x / dx = n ! δ n,m henceforth the set (cid:8) ( n !) − / h n (cid:9) n ≥ is an orthonormal basis of L ( R , µ ). Finally, we adoptthe approach and notations as in [23], to recall the definition of the symmetric Fock spaceof an Hilbert space. For an Hibert (separable) space H with an orthonormal basis { e j } j ≥ ,let Γ( H ) be the full Fock space over H , given byΓ( H ) := ∞ M k =0 H ⊗ k = C ⊕ H ⊕ ( H ⊗ H ) ⊕ . . . , where H ⊗ k is the k ′ th tensor power of H for k ≥ H ⊗ = C ; the space H ⊗ k has theorthonormal basis { e j ⊗ . . . ⊗ e j k } j ,...,j k ≥ and the union of all such bases form a basis forΓ( H ). We are interested in the (boson) symmetric Fock space over H , that is the subspaceΓ sym ( H ) ⊆ Γ( H ) which consists of all symmetric tensors. More precisely, let S be thesymmetrizer which projects Γ( H ) onto Γ sym ( H ), that is given on H ⊗ k by S ( u ⊗ . . . ⊗ u k ) = ( k !) − X σ ∈ S k u σ (1) ⊗ . . . ⊗ u σ ( k ) , where S k is the permutation group of { , . . . , k } . Then, Γ sym ( H ) has the orthonormal basis { E α } α ∈ ℓ N , where E α := ( k !) / ( α !) − / S ( e α ⊗ e α ⊗ . . . ) , α = ( α , α , . . . ) ∈ ℓ N , (1.4)with | α | = P ∞ j =1 α j = k ; here ℓ N = { α = ( α , α , . . . ) : ∃ k ∈ N ∀ j > k, α j = 0 } and α ! = Q ∞ j =1 α j !. 2. Conditional Negative Definite Functions
Let N : S × S → R be a real CND kernel, continuous w.r.t the Fr´echet topology,such that N (0 ,
0) = 0. In this paper we focus on the case where N is stationary , in thesense that N ( s , s ) is a function of s − s , i.e., for every s , s and s in S , we have N ( s + s , s + s ) = N ( s , s ) = N ( s , s ). It is easily seen that all such CND kernels arecharacterized as N ( s , s ) = N ( s − s ) , ∀ s , s ∈ S where N : S → R is a CND function that is continuous w.r.t the Fr´echet topology, satisfying N (0) = 0. In the later discussions we begin from the function N rather than N .The theory of CND kernels goes back to Schoenberg and von Neumann, where theyanswered the question of when a metric space can be realized in a Hilbert space with anorm; see [46, 47, 48] and [49]. For a better understanding of what CND kernels are,we give a very wide family of conditional negative definite kernels, in the spirit of [20,Proposition 3.2], which corresponds nicely to the full answer given by Schoenberg and vonNeumann. HITE NOISE SPACE ANALYSIS AND MULTIPLICATIVE CHANGE OF MEASURES 7
Proposition 2.1.
Let X be a non-empty set. For any vector space V , a mapping Φ : X → V and a symmetric, bilinear positive semi-definite form h· , ·i V : V × V → C (not necessarilyinner product), the kernel N ( x, y ) := k Φ( x ) − Φ( y ) k V is a CND kernel on X × X , where k · k V is the semi-norm induced from h· , ·i V .Proof. Due to the properties of the product h· , ·i V , for every n ∈ N , x , . . . , x n ∈ X, and c , . . . , c n ∈ C with P nj =1 c j = 0, we have n X j,k =1 c j c k N ( x j , x k ) = n X j,k =1 c j c k k Φ( x j ) − Φ( x k ) k = n X j,k =1 c j c k k Φ( x j ) k V − n X j,k =1 c j c k h Φ( x j ) , Φ( x k ) i V + n X j,k =1 c j c k k Φ( x k ) k V = − D n X j =1 c j Φ( x j ) , n X k =1 c k Φ( x k ) E V ≤ . (cid:3) Following the theory by Schoenberg [46, 47, 48], while some of his results are well sum-marized in [20, Proposition 3.2], as N is a CND kernel with N ( s, s ) = 0 for all s ∈ S , thereis an embedding of S into an Hilbert space H (which can be thought as a linear subspaceof the space of all continuous mappings from S to R ), i.e., there exists a mapping s ϕ s from S to H such that N ( s − s ) = N ( s , s ) = k ϕ s − ϕ s k = k ϕ s − s k H (2.1)for any s , s ∈ S . We give some details on this building: it is an easy exercise to verifythat the kernel ϕ N ( s , s ) = 2 − [ N ( s ,
0) + N (0 , s ) − N ( s , s )] = 2 − (cid:2) N ( s ) + N ( s ) − N ( s − s ) (cid:3) is a positive definite kernel on S × S . Then, for every s ∈ S define the mapping ϕ s : S → R , by ϕ s ( e s ) = ϕ N ( s, e s ) , ∀ e s ∈ S and let H be the linear subspace of R S (the space of all functions from S to R ) that isgenerated by { ϕ s : s ∈ S} . On H we define an inner product by D n X j =1 α j ϕ s j , n X ℓ =1 β ℓ ϕ e s ℓ E := n X j =1 n X ℓ =1 α j β ℓ ϕ N ( s j , e s ℓ ) , (2.2)which makes H a pre-Hilbert space, hence there exists a (unique) continuation of H to anHilbert space H , in which H is dense and we have the equation N ( s , s ) = k ϕ s − ϕ s k H .Notice that the Hilbert space H is the reproducing kernel Hilbert space associated to thepositive definite kernel ϕ N ( · , · ) on S × S . The Hilbert space H plays an important role inour analysis which comes up later, especially when invoking its symmetric Fock space.If N is as above, it follows from [20, Theorem 2.2] that Q ( s , s ) = exp {− N ( s , s ) / } = exp {−N ( s − s ) / } is a PD kernel on S × S . D. ALPAY, P. JORGENSEN, AND M. PORAT
Remark 2.2.
A great contribution by Schoenberg, which is written and proved nicely in [20, Theorem 2.2] , actually tells us that N is a CND kernel if and only if Q t ( s , s ) :=exp {− tN ( s , s ) } is a PD kernel for every t > . Let us then define Q : S → R by Q ( s ) := Q ( s,
0) = exp {−N ( s ) / } , ∀ s ∈ S . (2.3)It is easy to see that Q ( · ) is continuous w.r.t the Fr´echet topology, with Q (0) = 1 and that Q is a PD function, that is n X j =1 n X k =1 c j c k Q ( s j − s k ) = n X j =1 n X k =1 c j c k exp {− N ( s j − s k , / } = n X j =1 n X k =1 c j c k exp {− N ( s j , s k ) / } = n X j =1 n X k =1 c j c k Q ( s j , s k ) ≥ n ∈ N , c , . . . , c n ∈ C , and s , . . . , s n ∈ S . The function Q given in (2.3) satisfiesthe conditions in Theorem 1.1, therefore there exists a unique probability measure P on S ′ such that E P (cid:2) exp { iX s } (cid:3) = Q ( s ) = exp {−N ( s ) / } , ∀ s ∈ S (2.4)where for every s ∈ S , the random variable X s on S ′ is defined via the duality of S and S ′ ,given by X s ( · ) = h· , s i , i.e., X s ( ω ) = h ω, s i = ω ( s )(2.5)for every ω ∈ S ′ and the notation E P stands for the expectation w.r.t P , i.e., E P (cid:2) f (cid:3) := R S ′ f ( ω ) d P ( ω ) for functions f on the space S ′ . From (2.1) it then follows that E P (cid:2) exp { iX s } (cid:3) = exp {−k ϕ s k H / } , ∀ s ∈ S . (2.6)A main issue of this paper (cf. Subsection 3.1) lies on the fact that { X s } s ∈S is a Gaussianprocess , hence determined by its mean value and covariance functions, heavily depended onthe measure that we assign to S ′ . However the stochastic process { X s } s ∈S is not alwaysGaussian. In the following proposition we show that the Gaussian property of this processis equivalent to some scaling property of the CND function N : Proposition 2.3.
Let N : S → R be a CND function, that is continuous w.r.t the Fr´echettopology, with N (0) = 0 and let P be the corresponding measure on S ′ that is defined in (2 . . Then the stochastic process { X s } s ∈S is a (centered) Gaussian process (w.r.t P ) if andonly if N satisfies N ( αs ) = α N ( s ) , ∀ α ∈ R , s ∈ S . (2.7) In that case, N ( s ) is the variance of X s (w.r.t P ).Proof. First, observe that for every α , . . . , α n ∈ R , s , . . . , s n ∈ S , and ω ∈ S ′ , we have (cid:16) n X ℓ =1 α ℓ X s ℓ (cid:17) ( ω ) = n X ℓ =1 α ℓ ω ( s ℓ ) = ω (cid:16) n X ℓ =1 α ℓ s ℓ (cid:17) = X P nℓ =1 α ℓ s ℓ ( ω ) , HITE NOISE SPACE ANALYSIS AND MULTIPLICATIVE CHANGE OF MEASURES 9 i.e., α X s + . . . + α n X s n = X α s + ... + α n s n . Thus { X s } s ∈S is (jointly) Gaussian if and onlyif X s is a Gaussian random variable for every s ∈ S . The characteristic function of X s isgiven by ϑ X s ( α ) := E P (cid:2) exp { iαX s } (cid:3) , hence — as X αs = αX s — ϑ X s ( α ) = E P (cid:2) exp { iX αs } (cid:3) = exp {−N ( αs ) / } . Therefore, X s is a Gaussian random variable with covariance N ( s ) if and only if ϑ X s ( α ) =exp {− α N ( s ) / } , i.e., if and only if (2.7) holds.As for the Gaussian process being centered, we need to show that E P (cid:2) X s (cid:3) = 0 for every s ∈ S . For every α ∈ R and s ∈ S , we have αX s = X αs , therefore by using (2.7) we obtainthat Z S ′ exp { iαX s ( ω ) } d P ( ω ) = exp {− α N ( s ) / } , ∀ α ∈ R . (2.8)By invoking the Taylor expansions (as functions of α ∈ R ) of both sides of (2.8), togetherwith the fact that on the right hand side the function is odd, as N ( − s ) = ( − N ( s ) = N ( s ), we get that all the even coefficients must vanish. However, the coefficient of α inthe expression on the left hand side of (2.8) is equal to i E P (cid:2) X s (cid:3) , hence E P (cid:2) X s (cid:3) = 0. (cid:3) We finish this section by showing two well-studied examples, which arise naturally fromparticular choices of N . In all of the cases mentioned below, we deal with a real CNDfunction, that is continuous w.r.t the Fr´echet topology and satisfies condition (2.7).To present those examples, one must recall an important family of (tempered) measureson R , which includes the Lebesgue measure, that is defined by M := n µ : µ is a positive measures on R , such that Z R dµ ( u ) u + 1 < ∞ o . (2.9)For every µ ∈ M we get a dual pair of path space measures P µ and P b µ on S ′ , presented belowin Examples 2.4 and 2.5; one we may think of them as an infinite dimensional Fourier duality(for path-space measures). In many examples, e.g. in financial math, for the correspondingGaussian process { X s } s ∈S as part of a given dual pair, we will find that { X s } s ∈S will havefat tail w.r.t one of the path-space measures ( P b µ in our case) in a dual pair; as compared tothe other. In financial math application, realization of fat tails is important; see e.g. [38]. Example 2.4.
For any measure µ ∈ M , let N µ ( s ) := Z R | s ( u ) | dµ ( u ) = k s k µ . (2.10) This corresponds to Proposition 2.1 by choosing X = S , V = S , Φ :
S → S to be theidentity mapping and the product h s , s i µ := R R s ( u ) s ( u ) dµ ( u ) . In that case the measure P , obtained from (2.6), is denoted by P µ and the covariance function of the Gaussian process { X s } s ∈S (w.r.t P µ ) is given by E P µ (cid:2) X s X s (cid:3) = Z S ′ X s ( ω ) X s ( ω ) d P µ ( w ) = Z R s ( u ) s ( u ) dµ ( u ) , ∀ s , s ∈ S . Then the Gaussian process can be re-indexed in R + instead of S , by the following rule: forany t , t > , let s ( u ) = 1 [0 ,t ] ( u ) and s ( u ) = 1 [0 ,t ] ( u ) , thus the covariance function isgiven by Z S ′ h w, [0 ,t ] ih w, [0 ,t ] i d P µ ( w ) = µ (cid:0) [0 , min { t , t } ] (cid:1) . (2.11) See [7, 8, 12] for more details.
To present the next example, we recall the definition of the Fourier transform b s ( t ) = Z ∞−∞ e − itu s ( u ) du, ∀ s ∈ L ( R ) . (2.12) Example 2.5.
For any measure µ ∈ M , let N b µ ( s ) := Z R | b s ( u ) | dµ ( u ) = k b s k µ , (2.13) where b s is the Fourier transform of s (cf. (2.12)). This corresponds to Proposition 2.1 bychoosing X = S , V = S , Φ :
S → S to be the identity mapping and the product h s , s i b µ := Z R b s ( u ) b s ( u ) dµ ( u ) . In that case the measure P is denoted by P b µ and the covariance function of the Gaussianprocess { X s } s ∈S (w.r.t P b µ ) is given by E P b µ (cid:2) X s X s (cid:3) = Z S ′ X s ( ω ) X s ( ω ) d P b µ ( w ) = Z R b s ( u ) b s ( u ) dµ ( u ) , ∀ s , s ∈ S . Then the Gaussian process can be re-indexed in R + instead of S , by the following rule: forany t , t > , let s ( u ) = 1 [0 ,t ] ( u ) and s ( u ) = 1 [0 ,t ] ( u ) , thus the covariance function isgiven by Z S ′ h w, [0 ,t ] ih w, [0 ,t ] i d P b µ ( w ) = Z R e it u − u e − it u − u dµ ( u );(2.14) see [9, 15] for more details. We remind the reader that — in view of the two examples above — for any µ ∈ M , weobtain two stochastic processes which are both given by the functions { X s } s ∈S , howeverthey are taken to be w.r.t the different path-space measures P µ and P b µ . Notice that thelatter process is the generalized Fourier transform of the first process. Example 2.6.
The special case of the fractional Brownian motion — which has stationaryindependent increments but not independent increments — is considered, if one takes µ ( u ) = | u | − H du for some H ∈ (0 , and then (after re-indexing from S to R as in Example 2.5)the covariance function of the Gaussian process { X s } s ∈S (w.r.t P b µ ) is given by | t | H + | t | H − | t − t | H . For more details, see [7, 16] . Example 2.7.
The two functions N µ and N b µ coincide when µ is taken to be the Lebesguemeasure (which is clearly in M ). This will correspond to the classical Brownian motion.For more details, see [30] . A One-Parameter Family of Gaussian Measures
In this section we study some behaviours of what happen when we do a simple scaling by λ ∈ R + . Let N : S → R be a real CND function that is continuous w.r.t the Fr´echet topol-ogy, which satisfies condition (2.7) and recall that we obtained the existence of an Hilbertspace H (cf. equation (2.1)), with the special property that is N ( s ) = k ϕ s k H for every s ∈ S . HITE NOISE SPACE ANALYSIS AND MULTIPLICATIVE CHANGE OF MEASURES 11
A one-parameter family of Hilbert spaces.
For every λ ∈ R + , we define the (scaled) Hilbertspace H λ = ( H , h· , ·i H λ ), that is the space H equipped with the (scaled) inner product h h , h i H λ := λ h h , h i H , ∀ h , h ∈ H . (3.1) A one-parameter family of measures on S ′ . For every λ ∈ R + , define Q λ : S → R by Q λ ( s ) := Q ( λs ) = exp {−N ( λs ) / } , which — due to (2.7) — can be written as Q λ ( s ) = exp {− λ N ( s ) / } . (3.2)Due to [20, Theorem 2.2] and as it was explained in Remark 2.2, Q λ is PD thus we can applythe Bochner–Minlos theorem for Q λ . By doing so we get the existence (and uniqueness) ofa probability measure P λ on S ′ , such that E P λ (cid:2) exp { iX s } (cid:3) = Q λ ( s ) = exp (cid:8) − λ k ϕ s k H / (cid:9) = exp (cid:8) − k ϕ s k H λ / (cid:9) . (3.3)Notice that when λ = 1, we have that P = P is the measure obtained in (2.4). Remark 3.1.
Since λs ∈ S for every s ∈ S and λ ∈ R + , we have the following obviousrelation between any two measures from { P λ } λ ∈ R + , that is E P λ (cid:2) exp { iλ X s } (cid:3) = Q λ ( λ s ) = exp (cid:8) − λ λ N ( s ) / (cid:9) = Q λ ( λ s ) = E P λ (cid:2) exp { iλ X s } (cid:3) , for every λ , λ ∈ R + and s ∈ S . Remark 3.2.
It is possible to have corresponding definitions for λ < , however the answersfor all the questions we study in this paper depend only on the value of λ . For every λ ∈ R + , we define the λ − white noise space L ( S ′ , P λ ) := n f : S ′ → C | Z S ′ | f ( ω ) | d P λ ( ω ) < ∞ o , (3.4)which is an inner product space w.r.t the inner product h f, g i L ( S ′ , P λ ) := Z S ′ f ( ω ) g ( ω ) d P λ ( ω ) , ∀ f, g ∈ L ( S ′ , P λ ) . (3.5) Lemma 3.3.
The space L ( S ′ , P λ ) contains all the functions { X s } s ∈S , while their momentsare given in formulas (3.6) and (3.7). As { X s } s ∈S is a centered Gaussian process, its odd moments vanish, while its even2 n − moments are equal to (2 n − σ ns , where σ s = N ( λs ) is the variance of X s ; for aproof, see e.g. [13, Proposition 6.2]. Nevertheless, we provide the readers a short proof. Proof. As X αs = αX s and k ϕ αs k H λ = α k ϕ s k H λ for every α ∈ R and s ∈ S , we applyequation (3.3) to obtainexp (cid:8) − α k ϕ s k H λ / (cid:9) = Q λ ( αs ) = E P λ (cid:2) exp { iX αs } (cid:3) = Z S ′ exp { iαX s ( ω ) } d P λ ( ω ) . Thus, by invoking the Taylor expansions which correspond to the exponents in both sides,we have ∞ X n =0 ( − n k ϕ s k n H λ n n ! α n = ∞ X n =0 i n n ! (cid:16) Z S ′ X s ( ω ) n d P λ ( ω ) (cid:17) α n , so for every n ∈ N and s ∈ S , the (2 n + 1) − moment of X s w.r.t P λ is given by Z S ′ X s ( ω ) n +1 d P λ ( ω ) = 0(3.6)and the 2 n − moment of X s w.r.t P λ is given by(3.7) Z S ′ X s ( ω ) n d P λ ( ω ) = (2 n − k ϕ s k n H λ , where (2 n − · · · · (2 n −
1) = (2 n )!2 n n ! . The last equation can be rewritten as (cid:13)(cid:13) X ns (cid:13)(cid:13) L ( S ′ , P λ ) = (2 n − k ϕ s k n H λ , ∀ n ∈ N , s ∈ S . In particular, for any s ∈ S , we obtained that X s ∈ L ( S ′ , P λ ) with(3.8) k X s k L ( S ′ , P λ ) = k ϕ s k H λ , while E P λ (cid:2) X s (cid:3) = 0 and the variance of X s (w.r.t the measure P λ ) is equal to E P λ (cid:2) X s (cid:3) = k X s k L ( S ′ , P λ ) = k ϕ s k H λ = λ N ( s ) . (cid:3) Remark 3.4.
As we complete S to the Hilbert space H λ , we use the Itˆo isometry to deter-mine that once we index the Gaussian process by elements h ∈ H λ , the mapping h X h isan isometry from the Hilbert space H λ to L ( S ′ , P λ ) . In other words, the Gaussian process { X s } s ∈S is now extended to the Gaussian process { X h } h ∈H λ , with E P λ (cid:2) | X h | (cid:3) = k h k H λ which is the extension of (3.8). Next, we introduce the function Γ λ : S × S → R that is defined byΓ λ ( s , s ) := E P λ [ X s X s (cid:3) = h X s , X s i L ( S ′ , P λ ) (3.9)for every s , s ∈ S ; that is the covariance function of the Gaussian process { X s } s ∈S w.r.tthe measure P λ , so it is obviously PD. Notice that equation (3.8) connects the two Hilbertspaces L ( S ′ , P λ ) and H λ , via the norm preserving mapping X s ↔ ϕ s . Proposition 3.5.
The covariance function of the Gaussian process { X s } s ∈S w.r.t the mea-sure P λ admits the formula Γ λ ( s , s ) = 4 − λ (cid:0) N ( s + s ) − N ( s − s ) (cid:1) , ∀ s , s ∈ S . (3.10) The mapping X s → ϕ s (from a subspace of L ( S ′ , P λ ) into H λ ) is an isometry if and onlyif N ( s ) + N ( s ) = 2 − (cid:0) N ( s + s ) + N ( s − s ) (cid:1) . (3.11)This is an important fact for us, which will come into play in Subsection 3.1. Proof.
By using the parallelogram law in L ( S ′ , P λ ) and the fact that X s + s = X s + X s for every s , s ∈ S , we know that h X s , X s i L ( S ′ , P λ ) = 4 − (cid:0) k X s + s k L ( S ′ , P λ ) − k X s − s k L ( S ′ , P λ ) (cid:1) = 4 − (cid:0) k ϕ s + s k H λ − k ϕ s − s k H λ (cid:1) = 4 − λ ( N ( s + s ) − N ( s − s )) . HITE NOISE SPACE ANALYSIS AND MULTIPLICATIVE CHANGE OF MEASURES 13
Therefore, the covariance function of the Gaussian process { X s } s ∈S w.r.t the measure P λ admits the formula in 3.10. Moreover, as the inner product in H λ is defined as follows (cf.equation (2.2)) h ϕ s , ϕ s i H λ = λ ϕ ( s , s ) = 2 − λ (cid:0) N ( s ) + N ( s ) − N ( s − s ) (cid:1) , we obtain that h X s , X s i L ( S ′ , P λ ) = h ϕ s , ϕ s i H λ if and only if the function N meets condition(3.11). So the mapping X s → ϕ s is always norm preserving (as k X s k L ( S ′ , P λ ) = k ϕ s k H λ = λ N ( s )), however it is isometry if and only if N satisfies (3.11). (cid:3) Finally, we recall a general formula for the joint distribution of X ξ , . . . , X ξ n , where ξ , . . . , ξ n ∈ S are linearly independent, which holds as the process { X s } s ∈S is Gaussian. Lemma 3.6.
Let n ∈ N and ξ , . . . , ξ n ∈ S be linearly independent. Define the (truncatedcovariance) matrix C ( λ ) n ∈ R n × n by ( C ( λ ) n ) ij = Γ λ ( ξ i , ξ j ) = E P λ (cid:2) X ξ i X ξ j (cid:3) . Then the jointdistribution of X ξ , . . . , X ξ n w.r.t P λ is given by the function g ( λ ) n ( x ) := (det C ( λ ) n ) − / exp n x T (cid:0) C ( λ ) n (cid:1) − x o (3.12) where x = ( x , . . . , x n ) , i.e., for every f ∈ L ( R n , g ( λ ) n ( x ) dx ) we have E P λ (cid:2) f ( X ξ , . . . , X ξ n ) (cid:3) = Z R n f ( x ) g ( λ ) n ( x ) dx. It is easily seen from (3.10) that for any λ , λ ∈ R + , we have the following relationbetween the covariance functions, that is λ − Γ λ ( s , s ) = λ − Γ λ ( s , s ) , which implies the following relation between the covariance matrices λ − C ( λ ) n = λ − C ( λ ) n . The last equality implies that the way that the covariance matrix C ( λ ) n depends on λ is justby a diagonal matrix, and also that the joint distributions of the Gaussian processes (w.r.t P λ and P λ ) admit the following relation λ n g ( λ ) n ( λ x , . . . , λ x n ) = λ n g ( λ ) n ( λ x , . . . , λ x n ) . Remark 3.7.
It is obvious from the formula (3.12), that if C ( λ ) n is a diagonal matrix, thenthe function g ( λ ) n is of the special form g ( λ ) n ( x ) = α ,λ exp { α ,λ x } · · · exp { α n,λ x n } , where α ,λ , α ,λ , . . . , α n,λ ∈ R . This will allow us to provide an easy way to build an orthonormalbasis for the space L ( S ′ , P λ ) . However, in order for C ( λ ) n to be diagonal, we need to choose ξ , . . . , ξ n ∈ S such that Γ λ ( ξ i , ξ j ) = 0 , i.e., N ( ξ i + ξ j ) = N ( ξ i − ξ j ) , ∀ i = j. (3.13) In general, we know that Γ λ ( s , s ) is a symmetric, positive semi-definite bilinear mappingon S × S , therefore we might use the Gram–Schmidt algorithm to get a set { ξ i } ni =1 in S suchthat Γ λ ( ξ i , ξ j ) = δ ij . Mutual singularity of the measures.
In [30, Chapter 3] the author studies thebehaviour of the space L ( S ′ , µ λ ), where the measure µ λ is the one obtained from theBochner–Minlos theorem applied to the PD functionexp (cid:8) − λ k s k / (cid:9) = E µ λ (cid:2) exp { iX s } (cid:3) ;here k s k stands for the L ( R ) norm, that is k s k = R R s ( u ) du . In particular, it is shown(see [30, Proposition 3.1]) that whenever λ , λ ∈ R + and λ = λ , the measures µ λ and µ λ are mutually singular. This fits exactly into our settings, simply by choosing the CNDfunction N ( s ) = k s k L ( R ) , thus the measure µ λ which appears in [30, Chapter 3] coincideswith the measure P λ introduced earlier (in (3.3)).The purpose of this subsection is to generalize [30, Proposition 3.1] to the case wherethe measures { P λ } λ ∈ R + correspond to a function N : S → R which is a CND function,continuous w.r.t the Fr´echet topology, and satisfies condition (2.7). To do so, we will usethe results from [39], which highly depend on the theory of RKHSs, whereas the leading ideais to compare between the covariance functions of a Gaussian process w.r.t two differentmeasures and establish a condition which determines whether the measures are equivalentor singular.For the convenience of the reader, we first recall the settings from [39]. Let (Ω , F ) bea measurable space, where F is the σ − algebra generated by a class of random variables { X ( t ) : t ∈ T } and T is assumed to be an interval, or more generally a separable metricspace. Assume e P and P are probability measures on (Ω , F ), such that { X ( t ) : t ∈ T } areGaussian processes with mean value functions e m ( t ) and 0, and covariance functions e Γ( s, t )and Γ( s, t ), respectively. Then, as they are PD kernels, the covariance functions e Γ( s, t ) andΓ( s, t ) generate RKHSs H ( e Γ) and H (Γ), with the RK e Γ( s, t ) and Γ( s, t ), respectively. If { g k } ∞ k =1 is a complete orthonormal system in H (Γ), then the RK has the following formΓ( s, t ) = ∞ X k =1 g k ( s ) g k ( t );for a proof see e.g. [39, Proposition 3.8]. For a more elegant and general presentation ofthe RKHSs H (Γ) see [50], where Hilbert spaces of tempered distributions are discussed aswell. We will use the following result: Theorem 3.8 (Theorem 4.4 in [39]) . The measures e P and P are mutually equivalent if andonly if the following hold: e m ( · ) ∈ H (Γ) , e Γ has a representation e Γ( s, t ) = ∞ X k =1 β k g k ( s ) g k ( t ) , (3.14) where { g k } is a complete orthonormal system in H (Γ) , with ∞ X k =1 (1 − β k ) < ∞ and β k > for all k. (3.15)Our interpretation of the results from [39] is well summarized in the proof of the followingtheorem, where we consider the Gaussian process { X s } s ∈S and the distinct path-spacemeasures are any pair of measures from the family { P λ } λ ∈ R + , introduced in (3.3). HITE NOISE SPACE ANALYSIS AND MULTIPLICATIVE CHANGE OF MEASURES 15
Theorem 3.9.
Let N : S → R be a CND function, that is continuous w.r.t the Fr´echettopology and satisfies condition (2.7). Then the measures P λ and P λ are mutually singular,for every λ , λ ∈ R + with λ = λ .Proof. Fix λ , λ ∈ R + . Let our measurable space (Ω , F ) be given by Ω = S ′ and F = B ( S ′ ),while the probability measures be e P = P λ and P = P λ , as obtained in (3.3), hence thedependence in N . Let T = S which is a separable metric space and consider the randomvariables X ( s ) = X s for every s ∈ S . The stochastic process { X s } s ∈S is Gaussian w.r.t both P λ and P λ , due to Proposition 2.3. Then, using (3.6) and (3.7), the mean value functionsare given by e m ( s ) = E P λ (cid:2) X s (cid:3) = 0 = E P λ (cid:2) X s (cid:3) = m ( s ) , ∀ s ∈ S while from (3.10) the covariance functions are given by e Γ( s , s ) = Γ λ ( s , s ) = 4 − λ (cid:0) N ( s + s ) − N ( s − s ) (cid:1) , Γ( s , s ) = Γ λ ( s , s ) = 4 − λ (cid:0) N ( s + s ) − N ( s − s ) (cid:1) , therefore e Γ( s , s ) = λ λ − Γ( s , s ) . Notice that the space H (Γ) consists of real functions on S , which are continuous w.r.t theFr´echet topology and S is a separable metric space, thus we can use [32, Lemma 4.10] tojustify the fact that H (Γ) is separable as well. Then, by letting { g k } ∞ k =1 be a completeorthonormal system in H (Γ), one can writeΓ( s , s ) = ∞ X k =1 g k ( s ) g k ( s ) = ⇒ e Γ( s , s ) = ∞ X k =1 β k g k ( s ) g k ( s )where β k = λ λ − > k ≥
1. Finally, we apply Theorem 3.8 to conclude that themeasures P λ and P λ are equivalent if and only if ∞ X k =1 (1 − β k ) = ∞ X k =1 (cid:0) − λ λ − (cid:1) < ∞ , i.e., if and only if λ = λ . Thus, whenever λ = λ , the measures P λ and P λ are notequivalent, however due to [39, Theorem 4.3], it means they are mutually singular. (cid:3) Remark 3.10.
In the proof we use the same ideas as the first and second authors used inthe proof of [10, Corollary 7.5] . Remark 3.11. In [41] the authors study a question that is quite different than our question,yet their condition for having a noise signal being detected or not is very similar to thecondition we have; they use classical ideas on convergence of measures and some analysisinvolved with the Radon–Nikodym derivative, see for example [41, Lemma 3.3] . For relatedresults on Radon–Nikodym–Girsanov for Gaussian Hilbert spaces, see e.g., [21, 45] . Our present framework is motivated by applications. Indeed, a variety of families ofmutually singular systems of path-space measures, and associated RKHSs (see Theorems3.9 and 4.8) arise in diverse applications. The following includes a number of such distinctcontexts: stochastic analysis, stochastic differential equations, analysis on fractals withscaling symmetry, and leaning theory models; see e.g., [9, 11], [35], and [17, 18].
To finish this section we present another interesting one-parameter family of CND func-tions on S which reproduces a one-parameter family of mutually singular measures on S ′ ,using the same machinery as in Theorem 3.9. Example 3.12.
Fix a tempered measure µ ∈ M such that S ⊂ L ( µ ) ⊂ S ′ . For every u ∈ [0 , , we adapt the notations from Examples 2.4 and 2.5, to define N µ,u : S → R by N µ,u ( s ) = u N µ ( s ) + (1 − u ) N b µ ( s ) = u k s k µ + (1 − u ) k b s k µ , (3.16) that is a convex combination of N µ, = N µ and N µ, = N b µ . It is easily seen that N µ,u is aCND function which is continuous w.r.t the Fr´echet topology and satisfies condition (2.7), sothere exists a probability measures P ( u ) µ on S ′ such that E P ( u ) µ (cid:2) exp { iX s } (cid:3) = exp {−N µ,u ( s ) / } for every s ∈ S . With respect to the measure P ( u ) µ , the stochastic process { X s } s ∈S is Gaussian(due to Proposition 2.3) with the covariance function being equal to Γ µ,u ( s , s ) = E P ( u ) µ [ X s X s ] = 4 − ( N µ,u ( s + s ) − N µ,u ( s − s ))= u Z R s ( t ) s ( t ) dµ ( t ) + (1 − u ) Z R b s ( t ) b s ( t ) dµ ( t ) , as it follows from formula (3.10) with λ = 1 . Let { h n } n ≥ be an orthonormal basis of L ( µ ) ,therefore it is readily checked that Γ µ,u ( s , s ) = u ∞ X n =1 g n ( s ) g n ( s ) + (1 − u ) ∞ X n =1 b g n ( s ) b g n ( s ) , (3.17) where g n ( s ) = R R s ( t ) h n ( t ) dµ ( t ) and b g n ( s ) = R R b s ( t ) h n ( t ) dµ ( t ) for every s ∈ S and n ≥ .Under the assumption (which fails for µ being the Lebesgue measure on R ) that the system { u / g n } n ≥ ∪ { (1 − u ) / b g n } n ≥ is a Parseval frame (not necessarily orthogonal) in H (Γ µ,u ) ,we get that Γ µ,v ( s , s ) = vu − ∞ X n =1 u / g n ( s ) u / g n ( s )+(1 − v )(1 − u ) − ∞ X n =1 (1 − u ) / b g n ( s )(1 − u ) / b g n ( s ) for every v ∈ [0 , and u ∈ (0 , such that v = u , hence Theorem 3.8 yields that themeasures P ( u ) µ and P ( v ) µ are mutually singular, as ∞ X n =1 (cid:0) − u v − (cid:1) + ∞ X n =1 (cid:0) − (1 − v ) (1 − u ) − (cid:1) = ∞ . In conclusion, we obtain the one-parameter family { P ( u ) µ } ≤ u ≤ of mutually singular measureson S ′ . Remark 3.13.
We used Theorem 3.8 (that is Theorem 4.4 in [39] ) under the assumptionthat our system is only a Parseval frame system and not necessarily orthogonal completesystem, as stated in the theorem itself. Note that the time of Jorsboe’s paper predates muchlater systematic studies of frame systems, i.e., the study of varieties of non-orthogonalexpansions, which play an important rule in signal processing, spectral theory and waveletstheory; see [36, 28, 29] . HITE NOISE SPACE ANALYSIS AND MULTIPLICATIVE CHANGE OF MEASURES 17
Remark 3.14. As exp {−N µ,u ( s ) / } = exp {− u N µ ( s ) / } exp {− (1 − u ) b N µ ( s ) / } and inview of (3.3), one can think of the space L ( S ′ , P ( u ) µ ) as the tensor product L s ( S ′ , P ( u ) µ ) ∼ = L ( S ′ , P µ, √ u ) ⊗ L ( S ′ , P b µ, √ − u ) , where P µ, √ u and P b µ, √ − u are the probability measures on S ′ obtained from (3.3) by consider-ing the initial measures P µ and P b µ (instead of P ), with the scalars λ = √ u and λ = √ − u ,respectively. Isometric Isomorphisms and Intertwining Operators
The symmetric Fock space.
In view of (3.1) and (3.8), the mapping ψ on H thatis defined by ψ ( ϕ s ) = X s , satisfies k ψ ( ϕ s ) k L ( S ′ , P λ ) = λ N ( s ) = λ k ϕ s k H = k ϕ s k H λ for every λ ∈ R + and hence — as H is a pre Hilbert space w.r.t the inner product givenin (2.2), which is completed to the Hilbert space H — can be (uniquely) extended to anisometry between the Hilbert spaces ψ λ : H λ → L ( S ′ , P λ )such that k ψ λ ( h ) k L ( S ′ , P λ ) = k h k H λ for every h ∈ H λ . We proceed by showing that theHilbert space H λ has an important role in our analysis of the λ − white noise space, that isthat Γ sym ( H λ ) is isometrically isomorphic to L ( S ′ , P λ ) by an explicit transformation givenbelow. To do that we have to use the following lemma, which generalizes a result fromHida’s book [30] and will be used again later in the proof of Theorem 4.8. Lemma 4.1.
Let N : S → R be a CND function, that is continuous w.r.t the Fr´echettopology and satisfies condition (2.7). Then the set span (cid:0) exp { X s } : s ∈ S (cid:1) is dense in L ( S ′ , P λ ) , for every λ ∈ R + . A closer analysis on L ( S ′ , P λ ), which includes the proof of the lemma, requires theHermite polynomials; see (1.3) for their precise definition and [12, 9, 11, 13] for more details. Theorem 4.2.
For every λ ∈ R + , the spaces L ( S ′ , P λ ) and Γ sym ( H λ ) are isometricallyisomorphic. Moreover, an explicit isomorphism W λ : Γ sym ( H λ ) → L ( S ′ , P λ )(4.1) is presented in the proof in (4.3). Before the proof, we recall some facts on the symmetric Fock space (cf. Section 1); formore supplementary facts see [34, Chapter 3-4]. It is well known that the symmetric Fockspace Γ sym ( H λ ) is generated by the set span (cid:0) ε ( h ) : h ∈ H (cid:1) , where ε ( h ) := ∞ X n =0 h ⊗ n √ n ! . Moreover, exp {h h , h i H λ } = h ε ( h ) , ε ( h ) i Γ sym ( H λ ) for every h , h ∈ H and in particular k ε ( h ) k sym ( H λ ) = exp {k h k H λ } , ∀ h ∈ H λ . (4.2) Proof. As H = span (cid:0) ϕ s : s ∈ S (cid:1) is dense in ( H and hence in) H λ , we know that span (cid:0) ε ( ϕ s ) : s ∈ S (cid:1) is dense in span (cid:0) ε ( h ) : h ∈ H λ (cid:1) and hence the symmetric Fock spaceΓ sym ( H λ ) is generated by span (cid:0) ε ( ϕ s ) : s ∈ S (cid:1) . Therefore in order to define a mapping onΓ sym ( H λ ), it is enough to define the mapping on these generators from { ε ( ϕ s ) : s ∈ S} . Forany s ∈ S , define W λ ( ε ( ϕ s )) = ∞ X n =0 ( n !) − − n/ X ns = exp (cid:8) X s / √ (cid:9) . (4.3)Thus, we have (cid:13)(cid:13) W λ ( ε ( ϕ s )) (cid:13)(cid:13) L ( S ′ , P λ ) = (cid:10) exp { X s / √ } , exp { X s / √ } (cid:11) L ( S ′ , P λ ) = Z S ′ exp {√ X s ( ω ) } d P λ ( ω )= ∞ X n =0 ( n !) − n/ Z S ′ X s ( ω ) n d P λ ( ω ) = ∞ X m =0 ((2 m )!) − m Z S ′ X s ( ω ) m d P λ ( ω )+ ∞ X m =0 ((2 m + 1)!) − m +1 / Z S ′ X s ( ω ) m +1 d P λ ( ω ) , and by applying equations (3.6) and (3.7), we get (cid:13)(cid:13) W λ ( ε ( ϕ s )) (cid:13)(cid:13) L ( S ′ , P λ ) = ∞ X m =0 ((2 m )!) − m (2 m − k ϕ s k m H λ = ∞ X m =0 ( m !) − k ϕ s k m H λ = exp {k ϕ s k H λ } = k ε ( ϕ s ) k sym ( H λ ) , when the last equality is due to (4.2). Therefore for every s ∈ S , we have W λ ( ε ( ϕ s )) ∈ L ( S ′ , P λ ) and k W λ ( ε ( ϕ s )) k L ( S ′ , P λ ) = k ε ( ϕ s ) k Γ sym ( H λ ) . Next, as the set span (cid:0) ε ( ϕ s ) : s ∈ S (cid:1) generates Γ sym ( H λ ), the mapping W λ can be extended(uniquely) to a mapping W λ : Γ sym ( H λ ) → L ( S ′ , P λ ). Finally, the extended mapping isonto L ( S ′ , P λ ), as we know from Lemma 4.1 that the set span (cid:0) exp { X s/ √ } : s ∈ S (cid:1) = span (cid:0) exp { X s } : s ∈ S (cid:1) generates the space L ( S ′ , P λ ). (cid:3) Remark 4.3.
Another way to obtain an isometric isomorphism between Γ sym ( H λ ) and L ( S ′ , P λ ) , as in [37] , is by considering the mapping f W λ ( ε ( ϕ s )) := exp (cid:8) X s − k ϕ s k H λ / (cid:9) , while it is easily seen that (cid:13)(cid:13)f W λ ( ε ( ϕ s )) (cid:13)(cid:13) L ( S ′ , P λ ) = k ε ( ϕ s ) k Γ sym ( H λ ) for every s ∈ S . Not only that we have an isometric isomorphism W λ between Γ sym ( H λ ) and L ( S ′ , P λ ),for each λ ∈ R + , but we can also learn that some operators on one side become interestingoperators on the other side. The annihilation and creation operators on the symmetric Fockspace (of H λ ) are well studied and their relations and connections to physics, as well as theirgeometric description, are well known; see for example [25, Chapter 6.3] and [23]. On theother hand we can understand what they become after we use the intertwining mapping W λ , and then get a corresponding system of operators on L ( S ′ , P λ ).The following proposition admits a nice relation between the stochastic properties in L ( S ′ , P λ ) and the operator theoretic properties in Γ sym ( H λ ). HITE NOISE SPACE ANALYSIS AND MULTIPLICATIVE CHANGE OF MEASURES 19
Proposition 4.4.
The operator W λ intertwines the pair of operators ( a λ , a ∗ λ ) on Γ sym ( H λ ) and the pair of operators ( d λ , m λ ) on L ( S ′ , P λ ) , i.e., d λ ( h ) = W λ a λ ( h ) W ∗ λ and m λ ( h ) = W λ a λ ( h ) ∗ W ∗ λ , ∀ h ∈ H λ . (4.4)Here { a λ ( h ) , a ∗ λ ( h ) } h ∈H λ is the Fock representation of annihilation and creations operatorson Γ sym ( H λ ), while m λ ( h ) stands for the multiplication operator by X h and d λ ( h ) = m ∗ λ ( h )stands for the abstract infinite dimensional Malliavin derivative operator, both on the space L ( S ′ , P λ ). For a proof see [37, Theorem 3.30 & Section 4]. Remark 4.5.
The intertwining operator W λ is important, because it makes the connectionbetween two problems we study in this paper. The first problem is the mutual singularity ofthe Gaussian measures { P λ } λ ∈ R + (cf. Subsection 3.1), while the second is on representationsof the CCR algebra being disjoint (cf. Subsection 4.3). The link between these two problemsis the intertwining operator W λ (cf. the proof of Corollary 4.19). Remark 4.6.
There is a functor for symmetric Fock spaces, in the sense that every operatoron a one-particle Hilbert space, can be lifted to the symmetric Fock space of that Hilbert space.Moreover, if the operator (on the one-particle) is contractive, then the corresponding liftedoperator, called the second quantization is a bounded operator on the symmetric Fock space.This is applicable for us only when λ ≤ , when we let S : H λ → H λ be the operatordefined as multiplication by λ — which is now a contraction — thus the second quantizationoperator Γ( S ) : Γ sym ( H sym ) → Γ sym ( H sym ) , given by Γ( S )( ε ( h )) := ε ( S ( h )) , is a boundedoperator. Generalized infinite Fourier transform.
In [30, Chapter 4.3] the author presentsan infinite dimensional generalized Fourier transform on the space L ( S ′ , e P ), where e P is themeasure obtained from the Bochner–Minlos theorem which corresponds in our settings tothe case we choose the CND function to be N ( s ) = k s k L . In this subsection we will followthe ideas from [30] and generalize some of the results to our case, in which N : S → R isa real CND function that is continuous w.r.t the Fr´echet topology and satisfies condition(2.7); by doing so, we cover a much bigger family of Gaussian processes.Another difference between our approach and the one in [30] is that we define our gen-eralized Fourier transform in a more direct way (cf. equation (4.5)), that is connected tothe way we achieved the Gaussian process from an application of a Gelfand triple and theBochner–Minlos theorem. A reproducing kernel Hilbert space.
For every λ ∈ R + , recall that Q λ ( s ) = exp (cid:8) − λ N ( s ) / (cid:9) is a positive definite function on S , which corresponds to the positive definite kernel Q λ ( s , s ) = Q λ ( s − s ) on S × S . There exists a (unique) RKHS, denoted H ( Q λ ) , whichconsists of real functions on S , where Q λ ( · , · ) is its RK. For every s ∈ S , define the function Q λ,s ∈ H ( Q λ ) by Q λ,s ( e s ) := Q λ ( s, e s ) = Q λ ( s − e s ) = exp {− λ N ( s − e s ) / } , ∀ e s ∈ S and notice that H ( Q λ ) is generated by the subspace n n X j =1 c j Q λ,s j : n ∈ N , c , . . . , c n ∈ R , s , . . . , s n ∈ S o . Then for every f ∈ H ( Q λ ), we know that h f, Q λ,s i H ( Q λ ) = f ( s ); in particular h Q λ,s , Q λ, e s i H ( Q λ ) = Q λ,s ( e s ) = Q λ, e s ( s ) = Q λ ( s − e s ) . A generalized Fourier transform.
For every F ∈ L ( S ′ , P λ ), we define a real function T λ ( F )on S , by the following rule( T λ F )( s ) := E P λ (cid:2) F exp { iX s } (cid:3) = Z S ′ F ( ω ) exp { iX s ( ω ) } d P λ ( ω ) = h F, exp { iX s }i L ( S ′ , P λ ) (4.5)The transform T λ is our analog of the classical Fourier transform, in infinite dimensions. Lemma 4.7.
For every F ∈ L ( S ′ , P λ ) , we have T λ ( F ) ∈ H ( Q λ ) . To prove the lemma we use the following general fact (see [19]): If H is a RKHS offunctions on a set X , with the RK K ( · , · ), then f : X → R belongs to H if and only if thereexists C f > n ∈ N , c , . . . , c n ∈ R , and x , . . . , x n ∈ X we have (cid:12)(cid:12)(cid:12) n X j =1 c j f ( x j ) (cid:12)(cid:12)(cid:12) ≤ C f (cid:13)(cid:13)(cid:13) n X j =1 c j K x j (cid:13)(cid:13)(cid:13) H where K x ∈ H is defined by K x ( · ) = K ( x, · ). Proof.
Let F ∈ L ( S ′ , P λ ), then for every n ∈ N , c , . . . , c n ∈ R , and s , . . . , s n ∈ S , we have (cid:12)(cid:12)(cid:12) n X j =1 c j (cid:0) T λ ( F ) (cid:1) ( s j ) (cid:12)(cid:12)(cid:12) = (cid:12)(cid:12)(cid:12) n X j =1 c j E P λ (cid:2) F exp { iX s j } (cid:3)(cid:12)(cid:12)(cid:12) = (cid:12)(cid:12)(cid:12) E P λ (cid:2) F n X j =1 c j exp { iX s j } (cid:3)(cid:12)(cid:12)(cid:12) = (cid:12)(cid:12)(cid:12)D F, n X j =1 c j exp { iX s j } E L ( S ′ , P λ ) (cid:12)(cid:12)(cid:12) ≤ C F (cid:13)(cid:13)(cid:13) n X j =1 c j exp { iX s j } (cid:13)(cid:13)(cid:13) L ( S ′ , P λ ) by the Cauchy-Schwarz inequality, where C F = k F k L ( S ′ , P λ ) < ∞ . So, (cid:12)(cid:12)(cid:12) n X j =1 c j (cid:0) T λ ( F ) (cid:1) ( s j ) (cid:12)(cid:12)(cid:12) ≤ C F n X j,k =1 c j c k E P λ (cid:2) exp { iX s j − s k } (cid:3) = C F n X j,k =1 c j c k Q λ ( s j − s k )= C F n X j,k =1 c j c k h Q λ,s j , Q λ,s k i H ( Q λ ) = C F (cid:13)(cid:13)(cid:13) n X j =1 c j Q λ,s j (cid:13)(cid:13)(cid:13) H ( Q λ ) , as needed, concluding T λ ( F ) ∈ H ( Q λ ). (cid:3) In view of Lemma 4.7, the image of T λ is contained in H ( Q λ ). We next establish severalfurther properties of the operator T λ : L ( S ′ , P λ ) → H ( Q λ ) and its adjoint T ∗ λ : H ( Q λ ) → L ( S ′ , P λ ), which lead to the conclusion that T λ is an isometric isomorphism. Theorem 4.8.
Fix λ ∈ R + . (1) For every s ∈ S , we have T ∗ λ ( Q λ,s ) = exp {− iX s } and T λ (exp {− iX s } ) = Q λ,s . (2) T ∗ λ is an isometry, i.e., kT ∗ λ ( ψ ) k L ( S ′ , P λ ) = k ψ k H ( Q λ ) for every ψ ∈ H ( Q λ ) . (3) T λ is onto H ( Q λ ) , with ker( T λ ) = { } . As both T and T ∗ are isomorphic, we say that T defines an isometric isomorphism from L ( S ′ , P λ ) onto H ( Q λ ) . HITE NOISE SPACE ANALYSIS AND MULTIPLICATIVE CHANGE OF MEASURES 21
Proof.
1. Fix s ∈ S . For every F ∈ L ( S ′ , P λ ), we have hT λ ( F ) , Q λ,s i H ( Q λ ) = (cid:0) T λ ( F ) (cid:1) ( s ) = E P λ (cid:2) F exp { iX s } (cid:3) = h F, exp {− iX s }i L ( S ′ , P λ ) which means that T ∗ λ ( Q λ,s ) = exp {− iX s } . Also, by the definition of T λ , for every e s ∈ S wehave (cid:0) T λ (exp {− iX s } ) (cid:1) ( e s ) = E P λ (cid:2) exp { iX e s − s } (cid:3) = Q λ ( e s − s ) = Q λ,s ( e s ) , i.e., T λ (exp {− iX s } ) = Q λ,s .2. For every n ∈ N , c , . . . , c n ∈ R and s , . . . , s n ∈ S , we have (cid:13)(cid:13)(cid:13) T ∗ λ (cid:16) n X j =1 c j Q λ,s j (cid:17)(cid:13)(cid:13)(cid:13) L ( S ′ , P λ ) = (cid:13)(cid:13)(cid:13) n X j =1 c j exp { iX s j } (cid:13)(cid:13)(cid:13) L ( S ′ , P λ ) = n X j,k =1 c j c k Q λ ( s j − s k ) = (cid:13)(cid:13)(cid:13) n X j =1 c j Q λ,s j (cid:13)(cid:13)(cid:13) H ( Q λ ) , however the subspace n P nj =1 c j Q λ,s j : n ∈ N , c , . . . , c n ∈ R , s , . . . , s n ∈ S o is densein the Hilbert space H ( Q λ ) and hence we get that kT ∗ λ ( ψ ) k L ( S ′ , P λ ) = k ψ k H ( Q λ ) for every ψ ∈ H ( Q λ ). That is of course equivalent to say that T λ T ∗ λ = I H ( Q λ ) .3. Suppose that ψ ∈ H ( Q λ ) is orthogonal to the image of T λ on L ( S ′ , P λ ). Then0 = (cid:10) ψ, T λ (exp {− iX s } ) (cid:11) H ( Q λ ) = h ψ, Q λ,s i H ( Q λ ) = ψ ( s ) for every s ∈ S , i.e., ψ = 0, whichmeans that the image of T λ on L ( S ′ , P λ ) is equal to H ( Q λ ).Finally, let F ∈ ker( T λ ), then h F, exp { iX s }i L ( S ′ , P λ ) = ( T λ ( F ))( s ) = 0 for any s ∈ S , i.e., F is orthogonal to the subspace { exp { iX s } : s ∈ S} of L ( S ′ , P λ ). However, it follows fromLemma 4.1, that this subspace is dense in L ( S ′ , P λ ), thus F = 0. (cid:3) Remark 4.9.
Some of the results in this subsection, such as the formula for the Fouriertransform and its being isometrically isomorphism, can be obtained from [13, Section 3.4] as well.
Remark 4.10.
This kind of infinite dimensional Fourier transform is used in applicationsto statistics as well, see e.g. [22, 57] where the studied transform is the Esscher transform.
We built two isometric isomorphisms W λ : Γ sym ( H λ ) → L ( S ′ , P λ ) and T λ : L ( S ′ , P λ ) → H ( Q λ ) , both involve with the Gaussian measure P λ . However, it is only natural to take the com-position of these two isometric isomorphisms, that is R λ : Γ sym ( H λ ) → H ( Q λ ) , R λ := T λ ◦ W λ (4.6)which is an isometric isomorphism by itself. Remark 4.11.
Both the symmetric Fock space Γ sym ( H λ ) and the RKHS H ( Q λ ) are obtaineddirectly from the CND function N : S → R that we start from, independently of the measure P λ . Hence this mapping R λ described below appears more naturally in the context of [39] . Corollary 4.12.
The spaces Γ sym ( H λ ) and H ( Q λ ) are isometrically isomorphic, via theexplicit isometric isomorphism R λ between the two, that is given in (4.7). In the proof we get an explicit formula for the mapping, while using the complexificationof the Schwartz class S — which follows the same ideas as in [30, Chapter 6.2] and is possibledue to (2.5) — and in particular the relation X s + is ′ := X s + iX s ′ for every s, s ′ ∈ S . Proof.
Clearly, R λ := T λ ◦ W λ is an isometric isomorphism, as both W λ and T λ are. Moreover,we have the following explicit way to write its formula. For every s ∈ S , the function R λ ( ε ( ϕ s )) = T λ (cid:0) W λ ( ε ( ϕ s ) (cid:1) ∈ H ( Q λ )satisfies (cid:0) R λ ( ε ( ϕ s )) (cid:1) ( s ′ ) = (cid:16) T λ (cid:0) exp (cid:8) X s / √ (cid:9)(cid:1)(cid:17) ( s ′ ) = Z S ′ exp { X s / √ } exp { iX s ′ } d P λ = Z S ′ exp { iX s ′ − is/ √ } d P λ = Q λ ( s ′ − is/ √
2) = Q λ,is/ √ ( s ′ ) , for every s ′ ∈ S . Thus we have the explicit formula (on basis elements), which is R λ ( ε ( ϕ s )) = Q λ,is/ √ . (4.7) (cid:3) The canonical commutation relations.
Let L be a Hilbert space. The algebra CCR ( L ) is generated axiomatically by a system { a ( ℓ ) , a ∗ ( ℓ ) } ℓ ∈L , whereas a ( ℓ ) , a ∗ ( k ) areoperators (on an Hilbert space), subject to[ a ( ℓ ) , a ( k )] = 0 , ∀ ℓ, k ∈ L (4.8)and [ a ( ℓ ) , a ∗ ( k )] = h ℓ, k i L . (4.9)A system { a ( ℓ ) , a ∗ ( ℓ ) } ℓ ∈L which satisfies (4.8) and (4.9) is called a representations of theCCR algebra CCR ( L ). Some representations, such as the Fock representations, might berealized in the symmetric Fock space Γ sym ( L ); e.g., see Example 4.18. Definition 4.13.
Let { a ( ℓ ) , a ∗ ( ℓ ) } ℓ ∈L and { b ( ℓ ) , b ∗ ( ℓ ) } ℓ ∈L be two representations of CCR ( L ) ,w.r.t the Hilbert spaces L , L (i.e., a ( ℓ ) : L → L and b ( ℓ ) : L → L for every ℓ ∈ L ).We say that the representations are unitarily equivalent if there exists a unitary operator U : L → L such that U a ( ℓ ) U − = b ( ℓ ) for every ℓ ∈ L . Segal’s work ([51, 52, 53]) was motivated by quantum mechanics, while Bargmann motiva-tion for the case of finite number of degrees of freedom was in complex analysis (of multivari-able complex functions). The result by Stone and von Neumann (see [58, 59, 62, 63])treatsthis case of finite number of degrees of freedom, that is the uniqueness — up to unitarilyequivalence — of families of unitary operators which are irreducible and satisfy the Weylcommutation relations.
Theorem 4.14 (Stone non-Neumann) . If L is a finite dimensional Hilbert space, then anytwo representations of the CCR algebra CCR ( L ) are unitarily equivalent. In his book [30], Hida gives a nice survey proof for this fact and an explantation to whathappens when going to infinite number of degrees of freedom. The question of how bad canit be when one goes to an infinite dimensional Hilbert space arises. It turns out that inaddition to the case of equivalent representations, there is one (and only one) more optionand that is that the representations are disjoint.
HITE NOISE SPACE ANALYSIS AND MULTIPLICATIVE CHANGE OF MEASURES 23
Definition 4.15.
Two representations of the CCR algebra are said to be disjoint , if theonly operator that intertwines one with the other is zero. In other words, two representa-tions of
CCR ( L ) , say { a ( ℓ ) , a ∗ ( ℓ ) } ℓ ∈L and { b ( ℓ ) , b ∗ ( ℓ ) } ℓ ∈L are disjoint if for every unitaryoperator U : L → L such that U a ( ℓ ) = b ( ℓ ) U for every ℓ ∈ L , we must have U = 0 . Theorem 4.16. If L is infinite dimensional, then any two irreducible representations ofthe CCR algebra CCR ( L ) are either equivalent or disjoint. Remark 4.17.
The Stone von-Neumann theorem (cf. Theorem 4.14) and Theorem 4.16illustrate one difference between the finite and infinite dimensional cases. So is the resultin Theorem 3.9, regarding the mutual singularity of the measures, which is true only inthe infinite dimensional case. Another difference is that the group of unitary operators onthe Hilbert space L induces a group of measure preserving transformations on L space of aGaussian, and that action is ergodic if and only if L is of infinite dimension. This explainednicely in [55, 56] and goes back to ideas of Segal in the papers [51, 52] . There are many ways to construct unitarily inequivalent representations of the CCR (w.r.tan infinite dimensional Hilbert space). One example of an explicit way of finding infinitelymany unitarily inequivalent representations is given in [51, 52] and presented (shortly) here:
Example 4.18.
On the symmetric Fock space Γ sym ( H ) we define the operators A j : Γ sym ( H ) → Γ sym ( H ) , A j ( E α ) = √ α j E α − j , ∀ α ∈ ℓ N and A ∗ j : Γ sym ( H ) → Γ sym ( H ) , A ∗ j ( E α ) = p α j + 1 E α +1 j , ∀ α ∈ ℓ N where the basis elements { E α } α ∈ ℓ N are defined in (1.4). The family { A j , A ∗ j } ∞ j =1 is a repre-sentation of the CCR algebra and for any j ≥ the operators A j and A ∗ j are adjoints in Γ sym ( H ) . Using the family { A j , A ∗ j } ∞ j =1 , we can build another family of operators dependingon a scalar λ ∈ R + , in the following way- let A j,λ : Γ sym ( H ) → Γ sym ( H ) , A j,λ ( E α ) = λ √ α j E α − j , ∀ α ∈ ℓ N and A ∗ j,λ : Γ sym ( H ) → Γ sym ( H ) , A ∗ j,λ ( E α ) = p α j + 1 λ E α +1 j , ∀ α ∈ ℓ N , which are actually given by A j,λ = λA j and A ∗ j,λ = λ − A ∗ j . The family { A j,λ , A ∗ j,λ } ∞ j =1 is alsoa representation of the CCR algebra and in Segal’s papers it is shown that this representationis unitarily inequivalent to the representation { A j , A ∗ j } ∞ j =1 , as long as λ = 1 . In many works the connections between the equivalence of the symmetric Fock spacesand the equivalence of the prospective measures are stressed out. We finish this paperby presenting a family of unitarily inequivalent representations of the CCR, which followsfrom the influence of the simple operation of multiplication by λ , on our white noise space,symmetric Fock space and the corresponding RKHS, which appear in earlier stages of thepaper. Corollary 4.19.
Let π = { a ( h ) , a ∗ ( h ) } h ∈H be a Fock representation of the CCR over H .For every λ ∈ R + define the λ − Fock representation π λ := { a λ ( h ) , a λ ( h ) ∗ } h ∈H λ , with a λ ( h ) = λ a ( h ) and a ∗ λ ( h ) = λ − a ∗ ( h ) . (4.10) Then any two representations from the family (cid:8) π λ : λ ∈ R + (cid:9) are disjoint, i.e., π λ and π λ ′ are disjoint (irreducible) representations for every λ, λ ′ ∈ R + with λ = λ ′ . The idea of the proof is that there are two notions of equivalence; one is equivalence ofGaussian measures and another is equivalence of irreducible representations. There is a nontrivial theorem saying that the two notions of equivalence are equivalent. It is hinted (notexplicitly) in several places in [25, 44, 42, 30, 37, 34], so we only sketch some of the ideas ofthe proof of Corollary 4.19.
Proof.
Let λ, λ ′ ∈ R + be such that λ = λ ′ . Suppose that the representations π λ and π λ ′ areunitarily equivalent, i.e., suppose there exists a unitary mapping U : Γ sym ( H λ ) → Γ sym ( H λ ′ )(4.11)such that U a λ ( h ) = a λ ′ ( h ) U and U a ∗ λ ( h ) = a ∗ λ ′ ( h ) U, ∀ h ∈ H . (4.12) Step 1: Obtain a family of representations (of operators on L ( S ′ ) ): Recall the map-ping W λ : Γ sym ( H λ ) → L ( S ′ , P λ )that is an isometric isomorphism (cf. Theorem 4.2), with W λ a λ ( h ) = d λ ( h ) W λ and W λ a ∗ λ ( h ) = m λ ( h ) W λ , where d λ and m λ are given in (4.4). Define the mapping V = W λ ′ U W ∗ λ : L s ( S ′ , P λ ) → L ( S ′ , P λ ′ ) , (4.13)then V is unitary and satisfies the intertwining relations V m λ ( h ) = m λ ′ ( h ) V and V d λ ( h ) = d λ ′ ( h ) V, ∀ h ∈ h ∈ H . (4.14)The justification for (4.14) is because V m λ ( h ) = (cid:0) W λ ′ U W ∗ λ (cid:1) m λ ( h ) W λ W ∗ λ = W λ ′ U (cid:0) W ∗ λ m λ ( h ) W λ (cid:1) W ∗ λ = W λ ′ (cid:0) U a ∗ λ ( h ) (cid:1) W ∗ λ = W λ ′ (cid:0) a ∗ λ ′ ( h ) U (cid:1) W ∗ λ = W λ ′ (cid:0) W ∗ λ ′ m λ ′ ( h ) W λ ′ (cid:1) U W ∗ λ = m λ ′ ( h ) V, and thus by taking the adjoints and then multiplying by V on both sides we get m λ ( h ) ∗ V ∗ = V ∗ m ∗ λ ′ ( h ) = ⇒ V m ∗ λ ( h ) = m ∗ λ ′ ( h ) V = ⇒ V d λ ( h ) = d λ ′ ( h ) V. Thus instead of looking at the intertwining operator U and the representations π λ and π λ ′ ,we can consider the intertwining operator V and the representations Π λ and Π λ ′ , which aregiven byΠ λ = (cid:8) d λ ( h ) , m λ ( h ) (cid:9) h ∈H λ , where d λ ( h ) = W λ a λ ( h ) W ∗ λ and m λ ( h ) = W λ a ∗ λ ( h ) W ∗ λ (4.15)andΠ λ ′ = (cid:8) d λ ′ ( h ) , m λ ′ ( h ) (cid:9) h ∈H λ ′ , where d λ ′ ( h ) = W λ ′ a λ ′ ( h ) W ∗ λ ′ and m λ ′ ( h ) = W λ ′ a ∗ λ ′ ( h ) W ∗ λ ′ (4.16)notice that d λ ( h ) and m λ ( h ) are operators on the space L ( S ′ , P λ ). Step 2:
We obtained that the multiplication operators { m λ ( h ) } h ∈H λ on the space L ( S ′ , P λ )are equivalent to the multiplication operators { m λ ′ ( h ) } h ∈H λ ′ on the space L ( S ′ , P λ ′ ), in thesense of the existence of an intertwining operator between these families. This implies that HITE NOISE SPACE ANALYSIS AND MULTIPLICATIVE CHANGE OF MEASURES 25 the measures P λ and P λ ′ must have a Radon–Nikodym derivative, see for example [42, Chap-ter 6]. However the two measures P λ and P λ ′ on S ′ are singular, as λ = λ ′ (cf. Theorem3.9), therefore we obtained a contradiction, as needed. Step 3:
From steps 1 and 2 it follows that the representations π λ and π λ ′ are unitarilyinequivalent whenever λ = λ ′ . To conclude that the two representations are disjoint, wewill show they are both irreducible and then use Theorem 4.16. The proof of the irreducibil-ity of the representation π λ can be obtained by repeating steps 1 and 2, for the case where λ = λ ′ and eventually use the fact that the dual pair combined system { m λ ( h ) , d λ ( h ) } h ∈H λ isirreducible and so any operator which commutes with the combined system of multiplicationoperators and their duals must be a scalar operator (the system { m λ ( h ) } h ∈H λ is a maximalabelian algebra of multiplication operators, hence the only operator which commutes withthose multiplication operators must be a multiplication operator by itself, but then as it alsocommutes with the system of derivatives { d λ ( h ) } h ∈H λ , it must be a scalar operator). (cid:3) Acknowledgment.
Daniel Alpay thanks the Foster G. and Mary McGaw Professorshipin Mathematical Sciences, which supported this research. Palle Jorgensen and Motke Poratthank Chapman University for their hospitality for many fruitful visits over the last 3years. Motke Porat thanks the University of Iowa for a one week visit, which was veryhelpful for the creation of this paper. Lastly, the research of the third named authorwas partially supported by the US-Israel Binational Science Foundation (BSF) Grant No.2010432, Deutsche Forschungsgemeinschaft (DFG) Grant No. SCHW 1723/1-1, and IsraelScience Foundation (ISF) Grant No. 2123/17.
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