A class of Gaussian processes with fractional spectral measures

arXiv:1009.0233v1 [math.PR] 1 Sep 2010 A CLASS OF GAUSSIAN PROCESSES WITH FRACTIONAL SPECTRAL MEASURES DANIEL ALPAY, PALLE JORGENSEN, AND DAVID LEVANONY Abstract. We study a family of stationary increment Gaussian processes, indexed by time. These processes are determined by certain measures σ (generalized spectral measures), and our focus here is on the case when the measure σ is a singular measure. We characterize the processes arising from when σ is in one of the classes of affine self-similar measures. Our analysis makes use of Kondratiev-white noise spaces. With the use of a priori estimates and the Wick calculus, we extend and sharpen (see Theorem 7.1) earlier computations of Ito stochastic integration developed for the special case of stationary increment processes having absolutely continuous measures. We further obtain an associated Ito formula (see Theorem 8.1). Contents 1. 2. 3. 4. 5. 6. 7. 8. 9. Introduction Measures with affine selfsimilarity The spaces L2 (dσ) A brief survey of white noise space analysis The operator Qσ The processes Xσ and Wσ The Wick-Ito integral An Ito formula Two examples 2 6 8 9 12 21 24 26 29 1991 Mathematics Subject Classification. Primary: 60G22, 60G15, 60H40. Secondary: 47B32. Key words and phrases. stationary increment processes, weighted symmetric Fock space, Kondratiev and white noise spaces, spectral pairs, singular measures. D. Alpay thanks the Earl Katz family for endowing the chair which supported his research. The research of the authors was supported in part by the Israel Science Foundation grant 1023/07. The work was done while the second named author visited Department of Mathematics, Ben Gurion University of the Negev, supported by a BGU distinguished visiting scientist program. Support and hospitality is much appreciated. We acknowledge discussions with colleagues there, and in the US, Dorin Dutkay, Myung-Sin Song, and Erin Pearse. 1 10. The operator Qσ for more general spectral measures dσ 11. Concluding Remarks References 31 36 37 1. Introduction There are two ways of looking at stochastic processes, i.e., random variables indexed by a continuous parameter (for example time): (i) One starts with a probability space, i.e., a sample space, a set Ω with a sigma algebra B of subsets, and a probability measure P on (Ω, B), and a system of random variables {X(t)} on the (Ω, B, P ). From this one may then compute quantities such as means, variances, co-variances, moments, etc, and then derive important spectral data . These in turn are used in various applications, such as in solving stochastic differential equations. Here we are concerned with the other direction: (ii) Given some a priori spectral data, how do we construct a suitable probability space (Ω, B, P ) and an associated process {X(t)} such that the prescribed spectral data is recovered from the constructed process? In other words, this is a version of an inverse spectral problem. For a number of reasons, it is useful in the study of the inverse problem to focus on the case of Gaussian processes. A zero mean Gaussian process {X(t)} on a probability space is said to be stationary increment if the mean-square expectation of the increment X(t) − X(s) is a function only of the time difference t − s. Then there is a measure σ such that the covariance function of such a process is of the form Z ∗ (1.1) E[X(t)X(s) ] = Kσ (t, s) = χt (u)χs (u)∗ dσ(u), t, s ∈ R, R where E is expectation, and we have set eitu − 1 . u The positive measure dσ is called the spectral measure, and is subject to the restriction Z dσ(u) (1.3) < ∞. 2 R u +1 (1.2) χt (u) = The covariance function Kσ (t, s) can be rewritten as Kσ (t, s) = r(t) + r(s)∗ − r(t − s), 2 where (1.4) r(t) = − Z n itu e R itu o dσ(u) −1− 2 , u +1 u2 When σ is even, r is real and takes the simpler form (1.5) r(t) = Z R 1 − cos(tu) dσ(u). u2 We note that some authors call spectral measure instead the measure u2 dσ(u) rather than the measure dσ(u). See [28, p. 25 (7)]. The literature contains a number of papers dealing with these processes, but our treatment here goes beyond this, offering two novelties: the inverse problem (see above), and an operator theory of singular measures. Both are motivated by the need to deal with families of singular measures σ (see (1.1) through (1.4)). Our focus is on families of purely singular measures σ with an intrinsic spatial selfsimilarity, typically with Cantor support, and with fractional scaling (and Hausdorff) dimension; see Section 2 below; these are measures with affine selfsimilarity. Note that this notion is different from self-similarity in the time-variable; the latter case includes fractional Brownian motion (fBm), studied in e.g., [1, 2, 3, 14, 26]. For the latter (fBm), it is known that the corresponding one-parameter family of measures σ consists of a scale of absolutely continuous measures. The derivative of a stationary increment process is a (possibly generalized) stationary process, with covariance function σ b(t − s), where σ b denotes the Fourier transform, possibly in the sense of distributions, of σ. For a function f , recall the Fourier transform Z b f (u) = eiux f (x)dx. R We note that second order stationary processes can be studied with the use of the theory of Hilbert spaces and of unitary one-parameter groups of operators in Hilbert space. One may then invoke the Stonevon Neumann spectral theorem, the spectral representation theorem, and a detailed multiplicity theory to study these processes. See for instance [27]. 3 An important role in the theory is played by the space M(σ) of functions in L2 (R, dx) such that Z |fb(u)|2 dσ(u) < ∞. R This space contains in particular the Schwartz space. In the paper [3], see also [2], the case where σ(u) is absolutely continuous with respect to Lebesgue measure, i.e., dσ(u) = m(u)du (where the Radon-Nikodym derivative m satisfies moreover some growth conditions) was considered. The study of [3] included in particular the case of the Brownian motion and of the fractional Brownian motion. A key role in that paper was played by the (in general unbounded) operator Tm on L2 (R, dx) defined by √ (1.6) Td mfb. mf = So Tm is a convolution operator in L2 (R, dx), i.e., √ Tm f = ( m)∨ ⋆ f, with ∨ denoting the inverse Fourier transform, in the sense of distributions. In this paper we focus on the case when the spectral measure is an affine iterated function-system measure (AIFSs). Among the AIFSmeasures there is a subfamily which admits an orthonormal family of Fourier frequencies. These are lacunary Fourier series studied first in a paper by one of the authors and Steen Pedersen in 1998, see [24]. A lacunary Fourier series is one in which there are large gaps between consecutive nonzero coefficients. AIFS-measures may be visualized as fractals in the small, while their Fourier expansions as dual fractals in the large. The spectral measure of such a process {X(t)} is important as it enters in a rigorous formulation of an associated Ito formula for functions f (X(t)) of the process. The main results of the paper may be summarized as follows: We construct a densely defined operator Qσ from L2 (dσ) into L2 (R, dx) such that (1.7) Z R χt (u)χs (u)∗ dσ(u) = hQσ (1[0,t] ), Qσ (1[0,s] )iL2 (R,dx) This operator Qσ is the counterpart of the operator Tm defined in (1.6) and introduced in [3]. We denote by f 7→ fe the natural isometric 4 imbedding of L2 (R, dx) into the white noise space; See Section 4 for details. The stochastic process {Xσ (t)} defined by Xσ (t) = Q^ σ (1[0,t] ), has covariance function ∗ E[Xσ (t)Xσ (s) ] = Z R t ∈ R, χt (u)χs (u)∗ dσ(u) Following [28], the measure σ in this expression will be called the spectral measure of the process. Its intuitive meaning is that of ”spectral densities”, not to be confused with ”power spectral measure” traditionally used for the much more restrictive family of stochastic process, the stationary processes. In the case of stationary processes, and when the power spectral measure is absolutely continuous with respect to Lebesgue measure, one speaks of power spectral density (psd). It is then the Fourier transform of the covariance function, a function of a single variable, namely, the time difference. When the process can be differentiated, its derivative is stationary and σ is absolutely continuous with respect to Lebesgue measure, its derivative is the psd of the derivative process. def. We show that Xσ (t) admits a derivative (Xσ (t)′ ) = Wσ (t) in the white noise space, which is moreover continuous in the white noise space norm. Furthermore E[Wσ (t)(Wσ (s))∗ ] = σ b(t − s). It is found that both the processes {Wσ (t)} and {Xσ (t)} are in the white noise space. We define a stochastic integral with respect to Xσ , and prove results similar to those of [1], but for a different class of processes studied here. The outline of the paper is as follows. The paper consists of nine sections besides the introduction. The first three small sections are of a review nature. In Section 2 we present some material on measures with affine selfsimilarity. In Section 3 we recall some properties of the associated L2 (dσ) spaces. Hida’s white noise space theory is based on a Hilbert space, and plays an important role in our work. Its main features are listed in Section 4. In Sections 4-9 we develop the new results of the paper. In Section 5 we construct an operator for which (1.7) holds. The corresponding process Xσ and its derivative are constructed in Section 6. The associated stochastic integral and a Ito formula are considered in Sections 7 and 8 respectively. To contrast 5 with the measures considered here, two examples of stationary increments Gaussian processes with measures with unbounded support are presented in Section 9. In Section 10 we briefly consider the case of a general measure σ. The last section is devoted to various concluding remarks. 2. Measures with affine selfsimilarity In understanding processes {X(t)} with stationary increments, one must look at the variety possibilities for measures σ, representing spectral measures, in the sense outlined above. Each measure-type for σ entails properties of the associated Ito formulas for {X(t)} as it enters into stochastic integration formulas. While earlier literature on stationary-increment processes has been focused on the case when σ was assumed to be absolutely continuous with respect to Lebesgue measure, or perhaps the case when it is singular but purely atomic; in this section we will focus instead on a quite different family of measures: purely singular and non-atomic. They share the following four features: (i) they are given by explicit recursive formulas; (ii) they posses an intrinsic affine selfsimilarity; see (2.1), (iii) they admit a harmonic analysis based on a lacunary Fourier expansion; see (2.4), and finally, (iv) the Fourier transform of σ admits an explicit infinite-product formula; see (2.3). Definition 2.1. A Borel probability measure σ on R is said to be an affine iterated function system measure (AIFS) if there is a finite family F of (usually contractive) affine transformations on R such that X 1 (2.1) σ= σ ◦ τ −1 card F τ ∈F holds i.e. Z XZ 1 f (x)dσ(x) = f (τ (x))dσ(x), card F τ ∈F for all bounded continuous functions on R. The simplest examples are Bernouilli convolutions. Then card F = 2 and there exists some fixed ρ > 0 such that (2.2) τ± (x) = ρ(x ± 1). 6 In that case, the measure dσρ satisfying (2.1), has Fourier transform of the form ∞ Y (2.3) σbρ (t) = cos(ρk t). k=1 Note that the function ∞ Y k=1 cos(ρk (t − s)) is positive definite on the real line, since each of the functions in that product cos(ρk (t − s)) = cos(ρk t) cos(ρk s) + sin(ρk t) sin(ρk s) is positive definite, and one can obtain σ from Bochner’s theorem. Cases with ρ of the form 1 , m = 2, 3, 4, . . . 2m will be of special interest here. Fix m and let σm (that is with ρ = be the corresponding Bernouilli measure. For t ∈ R, set ρ= 1 ) 2m et (u) = eitu . Let (2.4) Λm = 2π (N X 0 For instance, ) n mo . bj (2m) , where N ∈ N and bj ∈ 0, 2 j Λ2 = 2π {0, 1, 4, 5, 16, 17, 20, . . .}   3 21 Λ3 = 2π 0, , 9, , 18, . . . and 2 2 Λ4 = 2π {0, 2, 16, 18, 128, 130, . . .} . It is known that the set Λm makes { eλ | λ ∈ Λm } into an orthonormal basis (ONB) in L2 (dσm ); we say that (σm , Λm ) is a spectral pair. Let us formalize this notion: Definition 2.2. A Borel finite measure σ on R is said to have a spectrum Λ ⊂ R if Λ is a discrete set and the set {eλ , λ ∈ Λ} is an orthonormal basis in L2 (dσ). Then (σ, Λ) is called a spectral pair. 7 By the Fourier basis property mentioned above, we refer to the presence of a Fourier orthonormal basis (ONB) in the Hilbert space L2 (dσ); and our discussion below is restricted to the case when σ is assumed to be a finite measure. The study of these singular measures was initiated by one of the authors in collaboration with co-authors, see [24, 36, 23, 22, 20, 21, 19, 5, 18, 6, 8, 7]. The Fourier expansion in L2 (dσ) for fractal measures σ differs from standard Fourier series (for periodic functions) in that the fractal Fourier expansion is local, much like wavelet expansions; see [37] for details. While the family of these singular measures is extensive, we found it helpful to focus our discussion below on one of the simplest cases, the first occurring in the literature, see [24]. It has Hausdorff dimension= scaling dimension= 1/2, and its support is a Cantor-subset of the real axis. It is proved in [24] that the support of dσm is inside the closed interval [−1/2, 1/2], and has Lebesgue measure 0. We note however, there are also spectral pairs (σm , ∆m ) where the measure dσm is not compactly supported. 3. The spaces L2 (dσ) For later use, we review two results from [5, 6, 7, 8] and [24]. Recall that, for t ∈ R, et (u) = eitu . In general one does not assume that σ has compact support. When the support of σ is compact, it has a well defined Fourier transform, which is an entire function and not merely a distribution. We have the following result, proved in [22, 23, 8]. Theorem 3.1. Let σ be a finite positive Borel measure and let Λ ⊂ R be a discrete set. Then, (σ, Λ) is a spectral pair if and only if (3.1) X λ∈Λ |b σ (t − λ)|2 = 1, ∀t ∈ R. Lemma 3.2. Let t, s ∈ R. It holds that (3.2) ket − es kL2 (dσ) ≤ K|t − s| where K= Z u2 dσ(u). [− 12 , 12 ] 8 Proof: We have ket − es k2L2 (dσ) = = Z Z [−1/2,1/2] |eitu − eisu |2 dσ(u) [−1/2,1/2] |1 − eiu(t−s) |2 dσ(u)   = ≤ Z Z 2 [−1/2,1/2] [−1/2,1/2] sin 4u2   u(t−s) 2    (t − s)2 2 dσ(u) · 4 u(t−s) 2  u2 dσ(u) · (t − s)2 .  As already mentioned, the measures σ we consider are such that an orthonormal basis of L2 (dσ) is of the form eiλn u , n = 0, 1, . . . , where λn ∈ πN0 for all n ∈ N0 . 4. A brief survey of white noise space analysis In this section we present some technical details required in the subsequent sections, taken from Hida’s white noise space theory. We refer the reader to [12], [13], [14] for more information. The facts reviewed here are essential for our analysis of certain stochastic integrals (Section 7), and our Ito formula (Section 8). While convergence questions for stochastic integrals traditionally involve integration in probability spaces of paths, in our approach, the sample space will instead be a space of tempered distributions S ′ derived from a Gelfand triple construction; but there is a second powerful tool involved, a completion called the Kondratiev-Wick algebra. We briefly explain the justification for this approach below. The second system of duality spaces are called Kondratiev spaces, see Section 4 for details. Further, there is a particular Kondratiev space, endowed with a product, the Wick product and an algebra under this product. It serves as a powerful tool in building stochastic integrals because, as we show, the stochastic integral takes place in the KondratievWick algebra; and we can establish convergence there; see Theorem 7.1. Moreover (see Theorem 8.1), the stochastic integration making up our 9 Ito formula lives again in the Kondratiev-Wick algebra. Let S denote the Schwartz space of real-valued C ∞ (R) functions such that ∀p, q ∈ N0 , lim xp f (q) (x) = 0 x→±∞ For s ∈ S, let ksk denote its L2 (R) norm. The function K(s1 − s2 ) = e− ks1 −s2 k2 2 is positive definite for s1 , s2 running in S. By the Bochner-Minlos theorem (see [32], [11, Théorème 3, p. 311]), there exists a probability measure P on S ′ such that Z ′ (4.1) K(s) = e−ihs ,si dP (s′), S′ where hs′ , si denotes the duality between S and S ′ . Henceforth, we set Ω = S ′ . The real Hilbert space W = L2 (Ω, F , dP ), where F is the Borelian σ-algebra, is called the white noise space. For s ∈ S and ω ∈ Ω we set (4.2) s(ω) = hω, si e From (4.1) follows that the map s 7→ e s is an isometry from S endowed with the L2 (R, dx) norm into W. This isometry extends to all of L2 (R, dx), and we will denote the extension by the same symbol. We now present an orthogonal basis of W. We set ℓ to be the space of sequences (α1 , α2 , . . .), whose entries are in N0 = {0, 1, 2, 3, . . .} , where αk 6= 0 for only a finite number of indices k. Furthermore, we denote by h0 , h1 , . . . the Hermite polynomials. The functions ∞ Y hαk (hek (ω)), α ∈ ℓ, Hα = Hα (ω) = k=1 form an orthogonal base of the white noise space (the ω-dependence will be omitted throughout, unless specifically required). Furthermore, one has (4.3) kHα k2W = α!, where we have used the multi-index notation α! = α1 !α2 ! · · · , 10 The Wick product ♦ in W is defined by the formula Hα ♦Hβ = Hα+β , α, β ∈ ℓ, on the basis (Hα )α∈ℓ , and is extended by linearity to W as X X (4.4) F ♦G = ( fα gβ )Hγ , γ∈ℓ α+β=γ P P where F = α∈ℓ fα Hα and G = α∈ℓ gα Hα . See [14, Definition 2.4.1, p. 39]. The Wick product F ♦G reduces to multiplication by a constant when one of the elements F or G is non random. The Wick product is not everywhere defined in W, and one may remedy this by viewing W as the middle part of a Gelfand triple. The first element in the triple is the Kondratiev space S1 of stochastic test functions, defined as the P intersection of the Hilbert spaces Hk , k = 1, 2, . . ., of series f = α∈ℓ fα Hα such that X def. (α!)2 |fα |2 (2N)kα < ∞. (4.5) |kf k|2k = α∈ℓ The third element in the Gelfand triple is the Kondratiev space S−1 of stochastic distributions. It is a nuclear space, and is defined as ′ the inductive limit of the P increasing family of Hilbert spaces Hk , k = 1, 2, . . . of formal series α∈ℓ fα Hα such that X def. (4.6) kf k2k = |fα |2 (2N)−kα < ∞, α∈ℓ where, for β ∈ ℓ, (2N)±β = 2±β1 (2 × 2)±β2 (2 × 3)±β3 · · · . See [14, §2.3, p. 28]. The Wick product is stable both in S1 and in S−1 . Moreover, Våge’s inequality (see [14, Proposition 3.3.2, p. 118]) makes precise the fact that F ♦G ∈ S−1 for every choice of F and G in S−1 : Let l and k be natural numbers such that k > l + 1. Let h ∈ Hl′ and u ∈ Hk′ . Then, (4.7) kh♦ukk ≤ A(k − l)khkl kukk , where (4.8) A(k − l) = X (2N)(l−k)α α∈ℓ 11 !1/2 . 5. The operator Qσ As we saw, the construction of a process {X(t)} from a fixed spectral measure σ depends on properties of a certain operator in L2 (R, dx). If σ is assumed absolutely continuous, with Radon-Nikodym derivative m, then this operator Tm was studied earlier and it is a convolution operator with the square root of m. See (1.6) and [3]. In this section we introduce the counterpart of the operator Tm in the present setting. Recall that h0 , h1 , . . . denote the Hermite functions. We define a unitary map W from L2 (dσ) onto L2 (R, dx) via the formula W (eλn ) = hn , n = 0, 1, . . . Let Mu denote the operator of multiplication by the variable u in L2 (dσ). The formula T = W Mu W ∗ (5.1) defines a bounded self-adjoint operator from L2 (R, dx) into itself. Furthermore, for ψ ∈ S we set b )h0 . Qσ (ψ) = ψ(T Lemma 5.1. Let ψ ∈ S. Then, it holds that:  ∞ Z X (5.2) (Qσ ψ)(x) = σ b(y − λn )ψ(y)dy hn (x) R n=0 Proof: Using the functional calculus we have from (5.1) that, b ) = W ψ(M b u )W ∗ ψ(T and hence b )h0 = W ψ(M b u )W ∗ h0 ψ(T b u )1 = W ψ(M b = W ψ(u). Moreover, since ψb ∈ L2 (dσ) we have b ψ(u) = = ∞ X n=0 b eλ iL (dσ) eλ (u) hψ, n n 2 ∞ Z X n=0 R so that W ∗ ψb is given by (5.2).  −iuλn b ψ(u)e dσ(u) eλn (u), 12  The operator Qσ is typically an unbounded operator in the Hilbert space L2 (R, dx), but it is well defined on a dense domain which consists of the Schwartz space S ⊂ L2 (R, dx). These facts are elaborated upon in the next theorem. In the statement, note that the Fréchet topology of S is stronger than that of the L2 (R, dx)-norm. Theorem 5.2. Let ψ ∈ S. Then, it holds that, (5.3) kQσ (ψ)k2L2 (R,dx) = Z R 2 b |ψ(u)| dσ(u). In particular, Qσ is a continuous operator from S into L2 (R, dx). More precisely, (5.4) Z 2 Z 2 !1/2 √ kQσ ψkL2 (R,dx) ≤ K |ψ(x)|dx + |ψ ′ (x)|dx , R R where (5.5) K= Z R dσ(u) < ∞. 1 + u2 Remark 5.3. For some stochastic processes it is important to relax condition (5.5) to (5.6) Kp = Z R dσ(u) <∞ 1 + |u|2p for some p ∈ N. In this case the estimate in (5.4) becomes (5.7) 2 Z 2 !1/2 Z p kQσ ψkL2 (R,dx) ≤ Kp |ψ(x)|dx + |ψ (p) (x)|dx . R R 13 Proof of Theorem 5.2: Let ψ ∈ S. We have: Z ∞ X 2 σ b(y − λn )ψ(y)dy kQσ ψkL2 (R,dx) = = 2 R n=0 ∞ X ZZ n=0 eiu(y−λn ) ψ(y)dσ(u)dy 2 Z Z ∞ X = ( ψ(y)eiuy dy)e−iuλn dσ(u) 2 n=0 = Z ∞ X −iuλn b ψ(u)e dσ(u) 2 n=0 Z 2 b |ψ(u)| dσ(u), = R where we have used Fubini’s theorem for the third equality, and Parseval’s equality for the last equality. We now prove (5.4). This will prove the continuity of Qσ from S endowed with its Fréchet topology into L2 (R, dx). For ψ ∈ S we have Z 2 2 b kQσ ψkL2 (R,dx) = |ψ(u)| dσ(u) R Z 2 dσ(u) b = (1 + u2 )|ψ(u)| 1 + u2 R 2 b ≤ K max(1 + u2 )|ψ(u)| u∈R  Z Z ♯ ′ ♯ ′ |ψ ⋆ ψ |(x)dx + |ψ ⋆ (ψ ) |(x)dx ≤K R R Z  Z ! 2 ≤K R |ψ(x)|dx where ψ ♯ (x) = ψ(−x) and ψ ′ = and Z dψ . dx 2 + R Furthermore, we have used 2 \ b ψ ⋆ ψ ♯ (u) = |ψ(u)| . ♯ R |ψ ⋆ ψ |(x)dx ≤ = Z Z R R |ψ ′ (x)|dx  Z  ♯ |ψ|(x)dx |ψ |(x)dx 2 |ψ|(x)dx . 14 R , Since the expression on the right hand side in this estimate is one of the Fréchet semi-norms of S, the continuity assertion in Theorem 5.2 follows. The estimate (5.4) further gives an exact rate of continuity.  From equation (5.3) we can extend the domain of definition of Qσ to a wider set, which in particular include the functions 1[0,t] . This is explicited in the following proposition. We remark that such a result may be extended to more general measures σ’s, see Section 10. Proposition 5.4. Let f ∈ L2 (dσ) be such that, for some sequence (sn )n∈N of Schwartz functions, lim |f − sbn |∞ = 0. (5.8) n→∞ Then the sequence (Qσ sn )n∈N is a Cauchy sequence in L2 (R, dx). Its limit is the same for all sequences which satisfy (5.8), and will be denoted by def. Qσ f = lim Qσ sn . n→∞ Proof: From (5.3) follows that for every ǫ > 0 there is an N ∈ N such that n, m ≥ N =⇒ |sbn − sc m |∞ ≤ ǫ. Thus for such n and m Z 2 2 |sbn (u) − sc kQσ sn − Qσ sm kL2 (R,dx) = m (u)| dσ(u) R (5.9) 2 ≤ σ(R) · |sbn − sc m |∞ ≤ σ(R) · ǫ2 . Therefore, limn→∞ Qσ sn in the norm of L2 (R, dx). Call this limit q1 , and assume that, for another sequence (tn )n∈N satisfying (5.8), we obtain another limit, say q2 . Note that (5.10) lim |sbn − tbn |∞ ≤ lim |sbn − f |∞ + lim |f − tbn |∞ = 0. n→∞ n→∞ n→∞ Then, kq1 − q2 kL2 (R,dx) ≤ kq1 − Qσ sn kL2 (R,dx) + + kQσ sn − Qσ tn kL2 (R,dx) + kQσ tn − q2 kL2 (R,dx) ≤ kq1 − Qσ sn kL2 (R,dx) + p + σ(R) · |sbn − tbn |∞ + kQσ tn − q2 kL2 (R,dx) , which goes to 0 as n → ∞ by definition of q1 and q2 and due to (5.10). 15 We now verify that the function eitu − 1 u can be approximated in the supremum norm by Schwartz functions. The function χt vanishes at infinity, and hence can be approximated in the supremum norm by continuous functions with compact support; see [35, Theorem 3.17, p. 70]. These in turn can be approximated by functions in S, using approximate identities, as for instance Step 6 in the proof of Theorem 6.1 in [1]. We sketch the argument for completeness. Let   1 x2 (5.11) kǫ (x) = √ exp − 2 . 2ǫ 2πǫ χt (u) = kǫ is an N (0, ǫ2 ) density, and therefore Z (5.12) kǫ (x)dx = 1, R and, for every r > 0 (5.13) lim ǫ→0 Z kǫ (x)dx = 0. |x|>r Indeed, for |x| > r > 0, Z ∞ Z ∞ 2 1 1 x − x22 ǫ2 − x2 2ǫ √ e 2ǫ dx e dx = √ x ǫ 2π r ǫ 2π r ǫ2 Z ∞ 2 x ǫ x − 2 ≤ √ e 2ǫ dx r 2π r ǫ2 r2 ǫ = √ e− 2ǫ2 r 2π −→ 0 as ǫ → 0. Theses properties express the fact that kǫ is an approximate identity. Applying [10, Theorem 1.2.19, p. 25] we see that, for every continuous function with compact support, lim kkǫ ∗ f − f k∞ = 0. ǫ→0 To conclude, one proves by induction on n that the n-th derivative (kǫ ∗ f )(n) (x) is a finite sum of terms of the form Z 1 (u − x)2
√ exp − p(x − u)f (u)du, 2ǫ2 2πǫ R 16 where p is a polynomial. All limits, lim xm (kǫ ∗ f )(n) (x) = 0 |x|→∞ are then shown, using the dominated convergence theorem, and all the functions Z 1 (u − x)2 (kǫ ∗ f )(x) = √ exp(− )f (u)du ǫ2 2πǫ R are in the Schwartz space.  We now compute the adjoint operator Q∗σ . Note that it is an operator from L2 (R, dx) into S ′ , and therefore lies outside L2 (R, dx). We begin with a notation and a preliminary computation. For φ ∈ L2 (R, dx) set Z cn (φ) = φ(y)hn (y)dy. R Then (cn (φ))n∈N is in ℓ2 . Indeed k(cn (φ))n∈N k2ℓ2 = kψk2L2 (R,dx) . We introduce the operator from L2 (R, dx) into L2 (dσ): (Tσ φ)(u) = ∞ X cn (φ)eiλn u . n=0 Clearly kφkL2(R,dx) = kTσ φkL2 (dσ) . Theorem 5.5. Let ψ ∈ S and φ ∈ L2 (R, dx). Then, Z b (5.14) hQσ ψ, φiL2 (R,dx) = ψ(u)(T σ (φ))dσ(u) sup σ Z (5.15) = ψ(y)X(φ)(y))dy, R where (5.16) (X(φ))(y) = ∞ X n=0 hhn , φiL2(R,dx) σ b(y − λn ). 17 Proof: We first prove (5.14). In view of the formula (5.2) for Qσ , we have  Z  ∞ Z X hQσ ψ, φiL2 (R,dx) = σ b(y − λn )ψ(y)dy φ(x)hn (x)dx n=0 = ∞ Z Z X R = n=0 ∞ Z X n=0 = Z R = Z R R b ψ(u) Z R R iu(y−λn ) e  dσ(u) ψ(y)dy R iuy e −iuλn ψ(y)dy e R ∞ X  e−iλn u Z R n=0  Z dσ(u) R  Z  φ(x)hn (x)dx R  φ(x)hn (x)dx ! φ(x)hn (x)dx dσ(u) b · Tσ (φ)dσ(u), ψ(u) where we have used Fubini’s theorem for the third equality, and the continuity of the inner product for the fourth equality. We now turn to the second formula. The sequence (hhn , φiL2(R,dx) )n∈N0 is in ℓ2 . In view of (3.1), the Cauchy-Schwarz inequality implies that X(φ) in (5.16) converges pointwise for every real y. We note that, in general, X(φ) 6∈ L2 (R, dx). For φ and ψ as in (5.14) we have: !  Z X ∞ Z σ b(y − λn )ψ(y)dy hn (x) φ(x)dx hQσ ψ, φiL2(R,dx) = = R n=0 ∞ X Z n=0 = Z R R ψ(y) R σ b(y − λn )ψ(y)dy ∞ X n=0 σ b(y − λn ) Z  Z R R  hn (x)φ(x)dx ! φ(x)hn (x)dx dy. To obtain the second equality, we note the following: Write Z Z σ b(y − λn ) 2 σ b(y − λn )ψ(y)dy = ((y + 1)ψ(y))dy. y2 + 1 R R Using the Cauchy-Schwarz inequality, we see that  Z  Z Z |b σ (y − λn )|2 2 2 2 2 (y + 1) )|ψ(y)| dy . σ b(y − λn )ψ(y)dy ≤ 2 2 R R R (y + 1) 18 In view of (3.1), ∞ Z X n=0 R  σ b(y − λn )ψ(y)dy hn (x) belongs to L2 (R, dx), and we use the continuity of the scalar product. Furthermore we have used the dominated convergence theorem to obtain the third equality.  Equations (5.14) and (5.15) allow to compute Q∗σ : Theorem 5.6. The domain of Q∗σ is the Lebesgue space L2 (R, dx). For φ ∈ L2 (R, dx), Q∗σ φ is the tempered distribution defined through Z ∗ b hQσ (φ), ψiS ′,S = ψ(u)(T σ (φ))dσ(u). sup σ Equivalently, Q∗σ (φ) is the tempered distribution defined by the function X(φ), that is, with some abuse of notation Q∗σ (ψ)(y) = ∞ X n=0 hhn , φiL2 (R,dx) σ b(y − λn ). The second representation for Q∗σ has an important consequence: Theorem 5.7. It holds that ker Q∗σ = {0} . It follows that the operator Q∗σ Qσ is a continuous and bounded operator from S into S ′ . We now provide two formulas for this operator. Theorem 5.8. Let ψ and φ be in S. Then, h(Q∗σ Qσ )φ, ψiS,S ′ =  Z  ∞ Z X −iλ u n −iλ u b b n dσ(u) = φ(u)e dσ(u) ψ(u)e R n=0 Z b b ψ(u)dσ(u). = φ(u) R R 19 Proof: By definitions of Qσ φ and of Q∗σ we have Z ∗ b (Tσ (Qσ φ)) h(Qσ Qσ )(φ), ψiS,S ′ = ψ(u) = = ∞ Z X Z R b ψ(u) n=0 R ∞ Z X n=0 But R R Z R σ b(y − λn )φ(y)dy eiλn u dσ(u) −iλn u dσ(u) b ψ(u)e σ b(y − λn )φ(y)dy = =  Z Z Z R i(y−λn )v e  σ b(y − λn )φ(y)dy .  dσ(v) φ(y)dy R R R  Z −iλn v b φ(y)e dσ(v) where we have used Fubini’s theorem. This concludes the proof.  Remark 5.9. While the operator Qσ (see Theorems 5.6 through 5.8) is well defined as an unbounded linear operator in the Hilbert space L2 (R, dx), the other two operators Q∗σ and Q∗σ Qσ are not. The reason is that the range of Q∗σ is not contained in L2 (R, dx). In fact, (Q∗σ hn )(x) = σ b(x − λn ), where σ b is the infinite product expression (2.3). It can be shown that x 7→ σ b(x) is not in L2 (R, dx); so as an L2 (R, dx)-operator, Qσ is not closable (its adjoint, computed in L2 (R, dx), does not have dense domain! Nonetheless Q∗σ in the extended sense maps L2 (R, dx) into S ′ ). The use of the ambient space S ′ of tempered distributions is essential. We illustrate the above discussion with the following diagrams: i i∗ Qσ Q∗σ S ֒−−→ L2 (R, dx) ֒−−→ S ′ ց . ր i∗ i S ֒−−→ L2 (R, dx) ֒−−→ S ′ In the following diagram, dom Q∗σ is only a small subspace of L2 (R, dx): Q∗σ Dom restricted ր i Q∗σ ֒−−→ L2 (R, dx) L2 (R, dx) 20 i∗ ֒−−→ Q∗σ ր unbounded S′ . 6. The processes Xσ and Wσ In this section we build the Gaussian process with covariance function Kσ . First recall that, thanks to Proposition 5.4, the domain of the operator Qσ has been extended to include the functions 1[0,t] . We begin with: Theorem 6.1. Let Qσ be as (5). Then, for every t, s ∈ R, Z iut e − 1 e−ius − 1 hQσ 1[0,t] , Qσ 1[0,s] iL2 (R,dx) = dσ(u). u u R Proof: We take (sn )n∈N and (tn )n∈N two sequences of elements of S with Fourier transforms (sbn )n∈N and (tbn )n∈N converging in the supremum norm respectively to χt and χ∗s . Then Q1[0,t] and Q1[0,s] are the limit in L2 (R, dx) of the sequences (Qsn )n∈N and (Qtn )n∈N respectively. Hence, hQσ 1[0,t] , Qσ 1[0,s]iL2 (R,dx) = lim hQσ sn , Qσ tm iL2 (R,dx) n,m→∞ Z = lim sbn (u)tbn dσ(u) n,m→∞ R Z iut e − 1 e−ius − 1 dσ(u). = u u R  Recall that we have denoted by fe the natural isometric imbedding (4.2) of L2 (R, dx) into the white noise space. We set (6.1) fn , Zn = h n = 0, 1, . . . The Zn are independent, identically distributed N (0, 1) random variables, and represented in white noise space. We arrive at the following decomposition: (6.2) Xσ (t) = Q^ σ (1[0,t] )  ∞ Z t X = σ b(y − λn )dy Zn . n=0 0 Theorem 6.2. It holds that (6.3) kXσ (t) − Xσ (s)kW ≤ |t − s|, 21 t, s ∈ R. The function t 7→ Xσ (t) is differentiable in W (white noise space), and its derivative is given by (6.4) Wσ (t) = ∞ X n=0 σ b(t − λn )Zn Remark 6.3. Formulas (6.2) and (6.4) are analogous to, but different from Karhunen-Loève expansions; see e.g., [25]. Proof of Theorem 6.2: We first prove (6.3). We have Eσ [|Xσ (t) − Xσ (s)|2 ] = kXσ (t) − Xσ (s)k2W Z t ∞ X σ b(y − λn )dy = ≤ n=0 ∞ X n=0 2 s ( Z t 2 1 dy)( s Z t s = (t − s) Z s 2 = (t − s) , ∞ t X ( n=0 |b σ (y − λn )|2 dy) |b σ (y − λn )|2 )dy where we have used the Cauchy-Schwarz inequality and (3.1) for the second and fourth equalities, respectively. We remark that, in view of (3.1), Wσ (t) ∈ W for every real t. We have Xσ (t) − Xσ (s) − Wσ (t) = t−s P∞ R t n=0 − = = s ∞ X n=0 σ b(y − λn )dyZn − t−s σ b(t − λn )dyZn P∞ nR t n=0 s P∞ nR t n=0 s 22 o (b σ (y − λn ) − σ b(t − λn )) dy Zn t−s o hey − et , eλn iL2 (dσ) dy Zn t−s . Hence, using Parseval’s equality and (3.2), we obtain Xσ (t) − Xσ (s) − Wσ (t) t−s 2 = W ≤ P∞ n=0 ∞ X n=0 Rt s 1 t−s Z t 2 hey − et , eλn iL2 (dσ) dy (t − s)2 Z t |hey − et , eλn iL2 (dσ) |2 dy s 1 key − et k2L2 (dσ) dy t−s s Z t 1 (y − t)2 dy ≤ t−s s (t − s)2 ≤ . 3 =  Theorem 6.4. The derivative process Wσ is continuous in the k · kW norm. It is furthermore stationary and of constant variance, E[|Wσ (t)|2 ] ≡ 1. (6.5) Proof: kWσ (t) − Wσ (s)k2W = ∞ X n=0 = ∞ X n=0 |b σ (t − λn ) − σ b(s − λn )|2 |het − es , eλn iL2 (dσ) |2 = ket − es k2L2 (dσ) = 2(1 − het , es iL2 (dσ) ) = 2(1 − σ b(t − s))  ! ∞ Y t−s cos =2 1− . 4n n=1 Continuity and stationarity follow from the above chain of inequalities, together with the fact that {Wσ (t)} is a Gaussian process. Equation (6.5) follows from (3.1).  We now turn to another type of representation for Xσ : 23 Theorem 6.5. Let H(ω, u) = ∞ X n=0 Then (Xσ (t))(ω) = fn (ω)eiλn u ∈ W ⊗ L2 (dσ). h Z H(ω, u) R e−iut − 1 dσ(u) u Proof: Using Fubini’s theorem we have Z Z Z t −iut −1 iλn u e iλn u e dσ(u) = e ( e−iuv dv)dσ(u) u R R 0 Z tZ ei(λn −v)u dσ(u) = 0 R Z t = σ b(λn − v). 0 This together with (6.2) leads to the required conclusion.  7. The Wick-Ito integral In this section we establish a stochastic integration formula for the general class of stationary increment processes considered here. An extension of Ito’s formula is addressed in section 8. More precisely, see Theorem 7.1, with the use of a priori estimates and of a Wick calculus (from weighted symmetric Fock spaces), we extend and sharpen earlier computations of Ito-stochastic integration developed originally only for the special case of stationary increment processes having absolutely continuous spectral measures. We further obtain in the subsequent section an associated Ito formula (Theorem 8.2). The main result of this section is the following theorem, which is the counterpart of [1, Theorem 5.1]. The fact that the derivative process Wσ (t) is W-valued (rather than lying in the larger Kondratiev space) allows to get a sharper statement. The proof follows the strategy of [1] and hence is only outlined. Theorem 7.1. Let Y (t), t ∈ [a, b] be an S−1 -valued function, continuous in the strong topology of S−1 . Then, there exists a p ∈ N such that the function t 7→ Y (t)♦Wσ (t) is Hp′ -valued, and Z b n−1 X Y (tk , ω)♦ (Xσ (tk+1 ) − Xσ (tk )) , Y (t, ω)♦Wσ (t)dt = lim a |∆|→0 k=0 24 where the limit is in the Hp′ norm, with ∆ : a = t0 < t1 < · · · < tn = b a partition of the interval [a, b] and |∆| = max0≤k≤n−1 (tk+1 − tk ). Proof: We proceed in a number of steps. STEP 1: There exists a p ∈ N such that Y (t) ∈ Hp′ for all t ∈ [a, b], being uniformly continuous from [a, b] into Hp′ . This is proved in [1, STEP 2 of the Proof of Theorem 5.1]. STEP 2: The function t 7→ Y (t)♦Wσ (t) is continuous over [a, b]. def. Here and in the sequel, we set k · kp = k · kH′p to simplify notation. Using Våge’s inequality (4.7), it follows that, for p > 1, kY (t)♦Wσ (t) − Y (s)♦Wσ (s)kp ≤ ≤ k(Y (t) − Y (s))♦Wσ (t)kp + kY (s)♦(Wσ (t) − Wσ (s))kp ≤ A(p)kY (t) − Y (s)kp kWσ (s)k0 + + A(p)kY (s)kp kWσ (t) − Wσ (s)k0 where A(p) is defined by (4.8). We conclude the proof of STEP 2 by observing that k · kH0 ≤ k · kW , which implies the continuity of the function Y (t)♦Wσ (t) in the norm k · kW . Rb In view of Step 2, the integral a Y (t)♦Wσ (t)dt makes sense as a Riemann integral of a continuous Hilbert space valued function. STEP 3: Let ∆ be a partition of the interval [a, b]. We now compute an estimate for Z b n−1 X Y (t)♦Wσ (t)dt − Y (tk )♦ (Xσ (tk+1 ) − Xσ (tk )) = a k=0 = n−1 Z X k=0 tk+1 tk  (Y (t) − Y (tk ))♦Wσ (t)dt . As for steps 1 and 2, we closely follow [1]. Let p be as in Step 2, and set ǫ > 0. Since Y is uniformly continuous on [a, b], there exists an η > 0 such that |t − s| < η =⇒ kY (t) − Y (s)kp < ǫ. 25 Set C̃ = max kWσ (s)k0 s∈[a,b] and A = A(p − N − 3). Let ∆ be a partition of [a, b] with |∆| = max {|tk+1 − tk |} < η. We then have: n−1 Z X tk+1 tk k=0 ≤  (Y (t) − Y (tk ))♦Wσ (t)dt n−1 Z tk+1 X tk k=0 ≤A n−1 Z X k=0 ≤ C̃A k=0  k(Y (t) − Y (tk ))♦Wσ (t)kp dt tk+1 tk n−1 Z X p ≤  k(Y (t) − Y (tk ))kp kWσ (t)k0 dt tk+1 tk k(Y (t) − Y (tk ))kp dt ≤ ǫC̃A(b − a), which completes the proof of Step 3 and the proof of the Theorem.  8. An Ito formula We extend the classical Ito’s formula to the present setting. Our present wider context for these stochastic processes entails important analytical points: Singular measures of fractal dimension, and singular operators, extending beyond the Hilbert space L2 (R, dx). This in turn brings to light new aspects of Ito calculus which we detail below. In addition to the examples in Section 2 (affine IFS measures), we further offer two examples in sect 9 below: the periodic Brownian bridge, and the Orenstein-Uhlenbeck processes. Lemma 8.1. The function r(t) = kQσ 1[0,t] k2L2 (R) is absolutely continuous with respect to the Lebesgue measure. 26 Proof: By (5.3) we have kQσ 1[0,t] k2L2 (R) = Z =2 R Z |χt (u)|2dσ(u) R 1 − cos(tu) dσ(u). u2 Since the support of dσ is bounded, the dominated convergence theorem allows to show that r(t) is differentiable and that its derivative is given by Z sin(tu) ′ dσ(u). r (t) = 2 u R  Theorem 8.2. Let f : R → R be a C 2 (R) function. Then Z t f (Xσ (t)) = f (Xσ (t0 )) + f ′ (Xσ (s))♦Wσ (s)ds+ t0 (8.1) Z t 1 + f ′′ (Xσ (s))r ′ (s)ds, t0 < t ∈ R, 2 t0 where the equality is in the P -almost sure sense. Proof: We prove for t > t0 = 0. The proof for any other interval in R is essentially the same. We divide the proof into a number of steps. Step 1-Step 5 are constructed so as to show that (8.1) holds, ∀t > 0, for Schwartz functions. This enables the extension to C 2 functions with compact support, with the equality holding in the Hp′ sense. This implies its validity in the P -a.s. sense (actually, holding ∀ω ∈ Ω), hence, setting the ground for the concluding step, in which the result is extended to hold for all C 2 functions f . STEP 1: For every (u, t) ∈ R2 , it holds that eiuXσ (t) ∈ W, and (8.2) eiuXσ (t) ♦Wσ (t) ∈ H2′ . Indeed, since Xσ is real, we have |eiuXσ (t) | ≤ 1, ∀u, t ∈ R, and hence eiuXσ (t) ∈ W. Since W ⊂ H2′ , we have in particular that Wσ (t) ∈ H2′ for all t ∈ R, it follows from Våge’s inequality (4.7) that 27 (8.2) holds. The aim of the following two steps is formula (8.1) for exponential functions. For α ∈ R we set: g(x) = exp(iαx). The proofs are as in [1], taking into account that r(t) is absolutely continuous with respect to Lebesgue measure and are omitted. STEP 2: It holds that (8.3) 1 g ′ (Xσ (t)) = iαg(Xσ (t))♦Wσ (t) + (iα)2 g(Xσ (t))r ′ (t). 2 STEP 3: Equation (8.1) holds for exponentials. In the following two steps, we prove (8.1) to hold for Schwartz functions. STEP 4: The function (u, t) 7→ eiuXσ (t) ♦Wσ (t) is continuous from R2 into H2′ . We first recall that the norm in Hp′ is denoted by k · kp . See (4.6). The particular case p = 0 in (4.6) gives in particular X kf k20 = |fα |2 . α∈ℓ The structure of H0′ has been studied in [4, Section 7]. Recall now that the function t 7→ Xσ (t) is continuous, and even uniformly continuous, from R into W, and hence from R into Hp′ for any p ∈ N0 since kukp ≤ kukW , for u ∈ W. The function (u, t) 7→ eiuXσ (t) is in particular continuous from R2 into H2′ . Furthermore, using Våge’s inequality (4.7) we have: keiu1 Xσ (t1 ) ♦Wσ (t1 ) − eiu2 Xσ (t2 ) ♦Wσ (t2 )kH′2 ≤ ≤ k(eiu1 Xσ (t1 ) − eiu2 Xσ (t2 ) )♦Wσ (t1 )kH′2 + + keiu1 Xσ (t1 ) ♦(Wσ (t2 ) − Wσ (t1 ))kH′2 ≤ A(2)k(eiu1 Xσ (t1 ) − eiu2 Xσ (t2 ) )kH′2 · kWσ (t1 )kH′0 + + A(2)keiu1 Xσ (t1 ) kH′2 · k(Wσ (t2 ) − Wσ (t1 ))kH′0 , 28 where A(2) is defined by (4.8). This completes the proof of STEP 4 since k · k2 ≤ k · k0 ≤ k · kW and in particular t 7→ Wσ (t) is continuous in the norm of H0′ and (u, t) 7→ eiuXσ (t) is continuous in the norm of H2′ . STEP 5: (8.1) holds for f in the Schwartz space. The reminder of the proof is exactly as Steps 6-9 in the corresponding proof of [1, Theorem 6.1], and hence omitted.  9. Two examples In this section, to contrast the measures considered above, we now consider two example where the measure σ has an unbounded support. Example (The periodic Brownian bridge). Take (9.1) σ(u) = ∞ X n=0 δ(u − 2n), that is the measure with support on the even positive integers with mass equal to 1 at each of these points. Then, it follows from the formula (1.5) on r(t) that r(t) = π ∞ X 1 − cos(2nt) (2n)2 n=1 . Note that, t(π − t) = π ∞ X 1 − cos(2nt) n=1 (2n)2 , t ∈ [0, 2π]. In view of the preceding equality, we call the associated process {Xσ (t)} the periodic Brownian bridge over [0, π]. We have r ∞ π X sin(nt) Zn , Xσ (t) = 2 n=1 n fn . where, as in (6.1), Zn = h 29 We note that, by construction, for every t, Xσ (t) belongs to the white noise space. On the other hand, the power series r ∞ πX Wσ (t) = cos(nt)Zn 2 n=1 converges only in the Kondratiev space. More precisely, Theorem 9.1. Let σ be given by (9.1). For every t, we have that Wσ (t) ∈ H2′ and in the topology of H4′ (9.2) Xσ (t) − Xσ (s) = Wσ (t). s→t t−s lim Proof: The fact that Wσ (t) belongs to H2′ follows from definition (4.6) since kZn k22 = (2n)−2 , and since the Zn are mutually orthogonal in H2′ . We now turn to (9.2). We have ! Rt ∞ X (cos(nu) − cos(nt))du Xσ (t) − Xσ (s) s Zn . − Wσ (t) = t−s t−s n=0 But Rt Rt (cos(nu) − cos(nt))du (nu − ns)n sin(nv)du s ≤ s , t−s t−s Rt (u − s)du ≤ n2 s t−s 2 n |t − s| = 2 and hence the limit goes to 0 in the H4′ norm since for some v ∈ (s, t) kZn k24 = (2n)−4 , and the Zn are mutually orthogonal in H4′ .  Example (The Ornstein-Uhlenbeck process). This is the solution of ths stochastic differential equation dX(t) = θ(µ − X(t))dt + αdB(t), t ≥ 0, where B is a Brownian motion and θ 6= 0, µ and σ are parameters. We have α X(t) = µ + √ W (e2θt )e−θt , 2θ 30 and E[X(t)] = e−θt + µ(1 − e−µt ), E[(X(t) − E(X(t))(X(s) − E(X(s))] = α2 −θ(t+s) 2θs∧t e (e − 1). 2θ Theorem 9.2. The centered Ornstein-Uhlenbeck process {X(t) − E[X(t)]} is a stationary increment Gaussian process. Furthermore α2 (1 − e−2θt ) 2θ Z 1 − cos(tu) dσ(u), = u2 R E[|X(t) − E[X(t)]|2 ] = with α2 θu2 du . dσ(u) = 2πθ θ2 + u2 10. The operator Qσ for more general spectral measures dσ The purpose of this section is to the contrast the difference between a singular measure or not. This is a crucial distinction in passing from the given measure σ to the associated process Xσ (t) ; i.e., in solving the inverse problem. The two cases are: (i) σ is assumed absolutely continuous with respect to Lebesgue measure; versus: (ii) σ is singular. This distinction results in a dichotomy for the induced operators in L2 (R, dx), so for the two operator questions. In case (i) we have a Radon-Nikodym derivative m , and the induced operator is Tm , a selfadjoint convolution operator in the Hilbert space L2 (R, dx) with the Schwartz space S as dense domain; see (1.6). In the second case (ii) it is not, even existence is subtle. Now rather the induced operator is our operator Qσ from Section 5. But it is much more subtle, and below we address some of the technical points omitted in Section 5: (a) There is now more than one choice for Qσ ; and (b) none of the choices will be closable operators, referring just to the Hilbert space L2 (R, dx). (c) Nonetheless, by working within the environment of the Gelfand triples (of Gaussian random fields), we are still able to make precise the two operators Qσ and the corresponding adjoint Q∗σ . For the singular case, i.e., case (ii), the Gelfand triple is thus essential in justifying our construction of the process {Xσ (t)}, existence and related properties. One common point between the previous work [3] and the present work is the construction of an operator Qσ from a subspace of L2 (R, dx) into 31 itself (denoted by Tm in [3]; see (1.6)) such that (10.1) Kσ (t, s) = hQσ (1[0,t] ), Qσ (1[0,s])iL2 (R,dx) = E[Xσ (t)Xσ (s)∗ ]. The natural isometry from L2 (R, dx) into the white noise space allows then to proceed by doing analysis in the Gelfand triple associated with the Kondratiev spaces, see Section 7. Although there are numerous possible other Gelfand triples, Våge’s inequality, see (4.7), is a feature which seems characteristic of this triple and is very useful in the computations. In the present section we present some general results on the existence and properties of the operator Qσ . To develop the associated stochastic integral and Ito’s formula, one has to relate the properties of σ and of Q. Not all the arguments go through for general σ’s. Details will be presented in forthcoming publications. Theorem 10.1. Let σ be a positive measure subject to Z dσ(u) (10.2) < ∞, 2 R 1+u and assume that dim L2 (dσ) = ∞. There exists a possibly unbounded operator Q from L2 (dσ) into L2 (R, dx), with domain containing the Schwartz space, such that (10.1) holds: and Kσ (t, s) = hQ(1[0,t] ), Q(1[0,s])iL2 (R,dx) , ^ Xσ (t) = Q1 [0,t] . The operator Q in the preceding theorem is not unique. In the previous cases, a specific choice of Q was made by recipe, which depended on the special structure of L2 (dσ). In general there seems no natural way to chose a specific Q. We have dropped therefore the index σ in the notation. Proof of Theorem 10.1: We proceed in a number of step. STEP 1: Let W be a unitary map from L2 (dσ) onto L2 (R, dx), and let f1 = W b1 , where b1 denotes the function 1 ∈ L2 (dσ). 1 + u2 Z dσ(u) 2 < ∞. = kb1 kL2 (dσ) = 2 R u +1 b1 (u) = √ Then: kf1 k2L2 (R,dx) 32 This is clear from the unitarity of W . STEP 2: The operator Mu of multiplication by the variable u is a priori an unbounded operator in L2 (dσ). It is densely defined and self-adjoint in L2 (dσ). This follows from (5.4), which is true for any measure σ satisfying (10.2). It follows from the preceding step that the operator T = W Mu W ∗ . is a self-adjoint operator in L2 (R, dx). The operator Q √ b )f1 (10.3) Qψ = 1 + T 2 ψ(T can therefore be computed using the spectral theorem. We claim Q satisfies (10.1) and (5.3). This is done in the next two steps. STEP 3: (5.3) holds. Indeed, using the spectral theorem we have √ b )f1 k2 kQψk2L(R,dx) = k( I + T 2 ψ(T L(R,dx) √ ∗ b )W b1 k2 = kW I + T 2 ψ(T L2 (dσ) Z √ b √ 1 |2 dσ(u) = | 1 + u2 ψ(u) 2 1+u ZR 2 b = |ψ(u)| dσ(u). R STEP 4: Set ^ Xσ (t) = Q1 [0,t] . Then E[Xσ (t)Xσ (s)∗ ] = hQ1[0,t] , Q1[0,s] iL(R,dx) . Indeed, the function 1[0,t] belongs to the domain of Q and the claim is a direct consequence of the isometric imbedding of L(R, dx) inside W.  33 Corollary 10.2. Let σ be a positive measure subject to Z dσ(u) (10.4) < ∞, R 1 + |u| and let {Xσ (t)} be the associated process as in Theorem 10.1. Then, {Xσ (t)} has a continuous version. (By this we mean [34] that {Xσ (t)} agrees a.e. with some time-continuous process.) Proof: We will prove this as an application of Kolmogorov’s test for the existence of a continuous version, [34, p. 14]. Because {Xσ (t)} is Gaussian, there exists K independent of t, s such that
2 E[|Xσ (t) − Xσ (s)|4 ] = K E[|Xσ (t) − Xσ (s)|2 ] . On the other hand, 2 E[|Xσ (t) − Xσ (s)| ] = 2Re r(t − s) = 2 Using (10.4) we now show that Re r(t) ≤ C|t|, (10.5) Z R 1 − cos((t − s)u) dσ(u). u2 |t| ∈ [0, 1]. The result will then follow from Kolmogorov’s continuity criterion (see for instance [34, Theorem 2.2.3, p. 14] for the latter). To prove (10.5) we proceed in a way similar as in [1] as follows. We compute first
Z 1 Z 1 tu 2 1 − cos(tu) 2 sin 2 dσ(u) = 2 dσ(u) t u2 t2 u2 0 0 ≤ K1 t2 (K1 > 0 independent of t) ≤ K1 |t| for |t| ∈ [0, 1]. Furthermore, using the mean-value theorem for the function u 7→ cos(tu) we have Thus 1 − cos(tu) = t2 u sin(tξt ), Z 1 ∞ 1 − cos(tu) dσ(u) = t2 u2 Z ξt ∈ [0, u]. ∞ Z1 ∞ sin(tξt ) dσ(u) u dσ(u) u 1 2 ≤ K2 t , where we use (10.4), ≤ t2 ≤ K2 |t| for |t| ∈ [0, 1], 34 where K2 > 0 is independent of t. Inequality (10.5) follows and hence the result.  In Section 5 we defined a specific operator Qσ on the Schwartz functions, and then defined Q1[0,t] by approximation. Here we have used the spectral theorem. Still it is possible to compute Q1[0,t] via approximating sequences. We note that, in view of (10.2), the measure dµ(u) = dσ(u) 1 + u2 satisfies the following property: ∀ǫ > 0, ∃K compact and such that µ(R \ K) ≤ ǫ. (10.6) The arguments in Proposition 5.4 can be adapted as follow. We take as a special sequence sn = 1[0,t] ⋆ k1/n where k1/n is defined via (5.11). Then u2 sbn (u) = χt (u) · e− n2 . Instead of (5.9) we write (for the special sequence (sn )n∈N at hand) kQσ sn − Qσ sm k2L2 (R,dx) = = = Z ZR ZR 2 |sbn (u) − sc m (u)| dσ(u) u2 u2 u2 u2 |χt (u)|2 (e− n2 − e− m2 )2 dσ(u) |χt (u)|2 (e− n2 − e− m2 )2 dσ(u)+ K Z u2 u2 + |χt (u)|2 (e− n2 − e− m2 )2 dσ(u). R\K where K is a compact to be determined. We first focus on the second integral in the last equality above. Set u2 u2 sup |χt (u)|2 (1 + u2 )(e− n2 − e− m2 )2 = M < ∞, m,n 35 . Then and recall that dµ(u) = dσ(u) 1+u2 Z u2 u2 |χt (u)|2 (e− n2 − e− m2 )2 dσ(u) = R\K Z u2 u2 = |χt (u)|2(1 + u2 )(e− n2 − e− m2 )2 dµ(u) R\K ≤ Mµ(R \ K). For a preassigned ǫ > 0, chose now the compact K such that µ(R \ K) ≤ Then (10.7) ǫ2 . 2M Z u2 u2 ǫ |χt (u)|2 (e− n2 − e− m2 )2 dσ(u) ≤ . 2 R\K Since K is compact there exists N such that, for n, m larger than N, the integral Z u2 u2 ǫ (10.8) |χt (u)|2(e− n2 − e− m2 )2 dσ(u) ≤ . 2 K Therefore, for n, m ≥ N, kQσ sn − Qσ sm kL2 (R,dx) ≤ ǫ, and the sequence (Qσ sn )n∈N is a Cauchy sequence in the norm of L2 (R, dx). This provides a constructive way to compute Q1[0,t] . To see that the obtained limit gives the same value as the one obtained from the spectral theorem, it suffices to let n go to infinity in (10.7) and (10.8). 11. Concluding Remarks We conclude with comments comparing our approach with the literature. 1. As in [1], no adaptability of the integrand with respect to an underlying filtration has been made. In this sense, one may regard the integral defined here in fact as a Wick-Skorohod integral. 2. 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(DA) Department of Mathematics Ben Gurion University of the Negev P.O.B. 653, Be’er Sheva 84105, ISRAEL E-mail address: [email protected] (PJ) Department of Mathematics 14 MLH The University of Iowa Iowa City, IA 52242-1419 USA E-mail address: [email protected] (DL) Department of Electrical Engineering Ben Gurion University of the Negev P.O.B. 653, Be’er Sheva 84105, ISRAEL E-mail address: [email protected] 39