Acta Applicandae Mathematicae 63: 333–347, 2000. 333 © 2000 Kluwer Academic Publishers. Printed in the Netherlands.

On Differential Operators in White Noise Analysis

Dedicated to Professor Takeyuki Hida on the occasion of his 70th birthday

JÜRGEN POTTHOFF Lehrstuhl für Mathematik V, Fakultät für Mathematik und Informatik, Universität Mannheim, D–68131 Mannheim, Germany

(Received: 5 February 1999) Abstract. White noise analysis is formulated on a general probability space which is such that (1) it admits a standard Brownian motion, and (2) its σ -algebra is generated by this Brownian motion (up to completion). As a special case, the white noise probability space with time parameter being the half-line is worked out in detail. It is shown that the usual differential operators can be defined on the smooth, finitely based functions of at most exponential growth via the chain rule, without supposing the existence of a linear structure (or translations) on the underlying probability space.

Mathematics Subject Classifications (2000): 60B05, 60B11, 60H07, 60H40.

Key words: white noise analysis, Malliavin calculus, differential operators.

1. Introduction In [DP97], T. Deck, G. Våge, and the author began the attempt to formulate T. Hida’s white noise analysis on a general probability space. One of the reasons that moti- vated the article [DP97] was that there are important applications – such as in nonlinear filtering – in which a Brownian motion and its noise are given,andit seems unnatural, if not wrong, to require that it is realized on a fixed probability space, such as the white noise space. Another reason was the wish to unify the bases of white noise analysis and the Malliavin calculus. It turned out that indeed at least two of the main features of white noise analysis can be formulated within a general framework without any loss: the theory of generalized random variables, and the calculus of differential operators. In the above-mentioned paper, however, the problem of showing that the differ- ential operators defined there are well-defined was left open, and the main purpose of the present paper is to give a proof of this fact. Our starting point here will be a general probability space which carries a Brownian motion whose time parameter domain is the half-line R+. The only addi- tional assumption is that the underlying σ -algebra is (up to completion) generated by the Brownian motion. This entails that the S-transform, which is one of the basic tools of white noise analysis, is injective. 334 JÜRGEN POTTHOFF

The framework and basic notions will be established in Section 2. In Section 3 one realization of the framework via the white noise space with time parameter domain R+ is provided in a rather detailed fashion, because this realization seems not to be so well known. Also, I reproduce there a result by S. Albeverio and M. Röckner [AR90] on quasi-invariant measures on Suslin spaces. This result im- plies that on a Suslin space with a (nontrivial) measure ν which is quasi-invariant with respect to translations of a dense linear subspace, a ν-class of random vari- ables can have at most one continuous representative. This fact will be essential for the proof of the well-definedness of the differential operators which is given in Section 4.

2. Framework Let (, B,P) a complete probability space which satisfies the following condi- tions:

(H.1) There exists a standard Brownian motion (Bt ,t∈ R+) on (, B,P); (H.2) The P -completion of σ(Bt ,t∈ R+) is equal to B. It is well known, e.g., [RY91], Lemma V.3.1, that condition (H.2) implies that the algebra generated by the (real, imaginary or complex) exponential functions in ∈ N ∈ R 2 the variables Bt1 ,...,Btn ,forn , t1,...,tn +, is dense in L (P ). In fact, both statements are equivalent, and it is obvious that the same holds just as well for the algebra of polynomials. Let us mention two realizations of these assumptions:

(1) Wiener space:  = C0(R+), P is Wiener measure, B is the completion of the σ -algebra generated by the cylinder sets, and Bt (ω) = ω(t) for ω ∈ . (2) White noise over the half line, which is discussed in detail in Section 3. 0 ∈ R ∈ R Let (Ft ,t +) be the filtration generated by (Bt ,t +). Denote by (Ft ,t ∈ R+) its standard µ-augmentation: if N is the ideal of the µ-null-sets := 04 ∈ R ∈ R in B, Ft Ft N , t +.Then(Ft ,t +) is right-continuous because (Bt ,t ∈ R+) is strongly Markov (cf. Proposition 2.7.7 in [KS88]). Actually, (Ft ,t ∈ R+) is continuous (e.g., [KS88], Corollary 2.7.8), and for all t, s ∈ R+ with t>s, Bt − Bs is independent of Fs (e.g., [KS88], Theorem 2.7.9), i.e., (Bt ,t∈ R+) is a Brownian motion relative to (Ft ,t∈ R+). Consider the linear mapping ∞ R −→ 2 X: Cc ( +) L (P ) f 7−→ Xf , R := where Xf R+ f(t)dBt . A trivial application of the Itô isometry shows that ∞ R |·| 2 R ≡ this mapping is continuous, if we give Cc ( +) the norm 2 of L ( +) 2 R R ∞ R L ( +, B( +), λ),whereλ is the Lebesgue measure. Since Cc ( +) is dense in 2 b 2 L (R+), this mapping extends to a continuous linear mapping X from L (R+) into ON DIFFERENTIAL OPERATORS IN WHITE NOISE ANALYSIS 335

2 ∈ 2 R ∞ R L (P ).Iff L ( +) is not in Cc ( +), we mean by Xf any representative of b 2 the P -class Xf . It is clear, that the family (Xf ,f ∈ L (R+)) forms a centered 2 Gaussian family with covariance given by the inner product of L (R+). I think it is appropriate to call the mapping Xb (Gaussian) white noise,andan essential part of white noise analysis is really the analysis of this mapping and its extensions. Consider the Schwartz space S(R+) over the half-line as defined in 2 [DP97], cf. also Section 3, and the dense continuous embedding S(R+) ⊂ L (R+). S(R+) is by construction nuclear, and therefore we have the canonically associated Gel’fand–Hida triple (e.g., [KL96])

2 ∗ (S+) ⊂ L (P ) ⊂ (S+) , ∗ where (S+) is a space of generalized random variables. Therefore, we may con- b ∞ R ∗ sider now X as a mapping from Cc ( +) into (S+) , and it has a continuous linear 0 extension to the Schwartz space of tempered distributions S (R+) over the half– ∈ R b ∈ ∗ line (cf. Section 3). In particular, we have for t +, Xδt (S+) (δt is the Dirac distribution at t), and this is white noise at time t. It is customary to denote ˙ this generalized random variable by Bt , and it is not hard to see that it is indeed the time derivative of Brownian motion at time t, the derivative being taken in the ∗ strong topology of (S+) . ∈ 2 ∞ R Let Z L (P ). Define its S-transform SZ as a function on Cc ( +) by
− | |2 ∞ := 1/2 f 2 E Xf ∈ R SZ(f) e Z e ,f Cc ( +). Among other purposes, the factor in front of the expectation serves to normalize the S-transform so that S1 = 1. By what has been said above, it is clear that for Z ∈ L2(P ), SZ has a continuous 2 extension to L (R+), and we shall not distinguish the two mappings in the sequel. The S-transform was first introduced in white noise analysis by I. Kubo and S. Takenaka [KT80], and it is closely related to the Segal–Bargman transform on Fock space, cf., e.g., [KL96]. The S-transform is very useful for a number of reasons. Two are: (i) It ‘diagonalizes’ the Itô integral [DP97], and thereby allows to handle and extend ‘multiplication by Gaussian (white) noise’ very efficiently. ∗ (ii) Spaces of generalized random variables, like (S+) , can be characterized via the S-transform in a way (e.g., [KL96] and literature quoted there) which is quite convenient for theoretical work as well as applications.

3. Schwartz Spaces and White Noise on the Half-Line

In this section we construct a white noise probability space over the half-line R+, which – according to Section 2 – is to be interpreted as the domain of the time parameter. The construction and properties are very similar to those of the usual white noise probability space (e.g., [Hi80, HK93]). 336 JÜRGEN POTTHOFF

2 Let C∞(R+) denote the set of twice continuously differentiable functions f on (0, +∞), which are such that (i) f and its first two derivatives vanish rapidly (i.e., faster than any power) at infinity, and (ii) the first two right derivatives of f exist at 0, and they coincide with f 0(0+) and f 00(0+), respectively. Consider the following differential operator 1 d d 1 Af (t ) := − t f(t)+ t f (t), t ∈ R+, 2 dt dt 4 2 2 acting on C∞(R+). It is obvious that A is symmetric on the dense subspace C∞(R+) 2 of L (R+). (As usual, we talk about the elements of this Hilbert space as if they were functions. There will be no danger of confusion.) Introduce the set of La- guerre functions

−t/2 lk(t) := e Lk(t), t ∈ R+,k∈ N0, where Lk is the Laguerre polynomial of order k ∈ N0 (e.g., [Sa77]). It is well- 2 known that (lk,k ∈ N0) is a complete orthonormal system in L (R+).TheLa- 2 guerre functions lk obviously belong to C∞(R+), and an elementary calculation shows that they are the eigenfunctions of A:   1 Al = k + l ,k∈ N . k 2 k 0 Therefore it is plain (for example by an application of Nelson’s analytic vector 2 theorem [RS75]), that A is essentially self-adjoint on C∞(R+), and its unique self- adjoint extension has domain  X X  := ∈ 2 R ; = 2 2 +∞ D(A) f L ( +) f fk lk with fk k < . k∈N0 k∈N0 We see that on the half-line A plays a role which is very similar to the one of the Hamiltonian of the harmonic oscillator on the full line, and the Laguerre functions play the role of the Hermite functions. Now we can proceed as in the case of the full line (e.g., [Hi80, RS72, Si71]). Define \ n S(R+) := D(A ),

n∈N0 and equip this space as usual with the projective limit topology defined by the n Hilbert spaces D(A ). Due to the spectral properties of A, it is obvious that S(R+) is a nuclear countable Hilbert space, and as such a Fréchet and a Montel space (e.g., [Tr67]). In [DP97] an elementary argument was given which shows that the functions f in S(R+) are infinitely differentiable on (0, +∞),andthatf together ON DIFFERENTIAL OPERATORS IN WHITE NOISE ANALYSIS 337 with all its derivatives vanishes rapidly at infinity. It is also straightforward to check that all right derivatives of f ∈ S(R+) exist at zero, and that they coincide with the right limits of the derivatives of f at 0. That is, f ∈ S(R+) can be considered as the restriction of a function in S(R) to R+. 0 We denote the dual of S(R+) by S (R+), and call it Schwartz space of tempered distributions over the half-line. 0 Let us equip S (R+) with some of the various common topologies, and discuss its topological properties. The arguments below are all completely standard, and given here for the convenience of the interested reader. 0 If we give S (R+) the strong topology (i.e., the topology of uniform conver- 0 gence on bounded subsets of S(R+)), then S (R+) is reflexive since S(R+) is Montel (e.g., [Tr67], p. 376, Corollary). For example, from the Lemma, p. 166, in [RS72] we can conclude that the strong and the Mackey topology coincide. (The Mackey topology is the topology of uniform convergence on the weakly compact, 0 convex subsets of S(R+), and it is the finest dual topology on S (R+), i.e., the 0 finest locally convex Hausdorff topology on S (R+) so that S(R+) is its dual in this 0 topology. The weak topology on S (R+) is the topology of pointwise convergence, 0 and it is the coarsest dual topology on S (R+). The Mackey–Arens theorem (e.g., [RS72]) states that all locally convex dual topologies are between the weak and the 0 Mackey topology. Furthermore, the strong and the Mackey topology on S (R+) also coincide with the topology of uniform convergence on compact subsets of S(R+) (e.g., [Tr67], p. 357, Proposition 34.5, but it also follows directly from the definition of a Montel space). 0 Now we can use Theorem 7 in [S73], p. 112, to conclude that S (R+) equipped with the strong topology is a Lusin space. (One only has to make the trivial remark 0 that the compact-open topology used for S (R+) = L(E, F ) there, with E = S(R+) and F = R, is just the topology of uniform convergence on compact subsets 0 of S(R+), which had already been identified with the strong topology on S (R+) above.) It is evident from the definition of a Lusin space (e.g., [S73], p. 94), that if 0 we equip S (R+) with any topology weaker than the strong topology, it remains a 0 Lusin space. In particular, S (R+) equipped with any locally convex dual topology 0 is a Lusin space, and we consider this case from now on. As a Lusin space, S (R+) is also a Suslin space ([S73], p. 96), and it is strongly Lindelöf ([S73], p. 104, 0 Proposition 3), i.e., every open cover of any open subset of S (R+) has a countable subcover. 0 Moreover, since S (R+) is Suslin for any dual topology, Corollary 2, p. 101, in [S73] entails that they all generate the same Borel σ -algebra, which is the σ -algebra generated by the cylinder sets, because the weak topology is a dual topology on 0 S (R+). Let us denote this σ -algebra by B0. Consider a total subset T in S(R+). As a Fréchet space S(R+) is barreled. 0 Therefore we have the Banach–Steinhaus theorem which implies that on S (R+) the topology of pointwise convergence on T coincides with the weak topology. 338 JÜRGEN POTTHOFF

Consequently, the σ -algebra σ(T ) := σ(h·,fi; f ∈ T ) is equal to B0.For example, we may choose for T the set of Laguerre functions. As usual, we may use Minlos’ theorem (e.g., [Hi80]) to introduce the centered 0 Gaussian measure µ on (S (R+), B0) via the relation Z ihω,f i − 1 |f |2 e µ(dω) = e 2 2 ,f∈ S(R+), 0 S (R+) 0 where h·, ·i stands for the dual pairing of S (R+) and S(R+),and|·|2 for the 2 norm of L (R+). We shall call µ the white noise measure on the half line. It shares almost all properties with the usual white noise measure, and the analysis of 2 0 L (S (R+), B0,µ)can be done just as in, e.g., [Hi80, HK93]. One aspect, however, should be recorded here for later use, namely that µ is quasi-invariant under the 2 translations by elements in L (R+) (which we embed in the canonical way into 0 S (R+) so that it becomes a dense linear subspace). Here is a quick way to see m m this: give the spaces D(A ), m ∈ N, the norms |A ·|2 so that they and their duals D(Am)∗ become Hilbert spaces. Choose m ∈ N so that the injection of m 2 D(A ) into L (R+) is Hilbert–Schmidt. Notice that Minlos’ theorem implies that ∗ m 0 2 ∗ m µ is carried by D (A ) ⊂ S (R+). Thus (L (R+), D (A ), µ) forms an abstract Wiener space, and we can use the well-known Cameron–Martin formula derived, e.g., in [Ku75]. Our next step is to quote a very useful result by S. Albeverio and M. Röckner [AR90]. Consider a Suslin topological vector space E over the reals, equipped with its Borel σ -algebra, denoted again by B0. Assume that ν is a measure on (E, B0) which is not identically zero. Define an open subset V of E as the union of all open subsets which have ν-measure zero. Since by construction V is covered by open subsets of zero measure, the fact that E is strongly Lindelöf (s.a.) tells us, that V has a countable subcover by ν-null sets, and consequently has itself ν-measure zero. Hence V is the largest open subset of E which has zero ν-measure. Albeverio and Röckner define supp(ν) := E \ V . Let k ∈ E. ν is called k-quasi-invariant if ν is quasi-invariant w.r.t. translations by the elements of the one-dimensional subspace generated by k, i.e., for every A ∈ B0,andallλ ∈ R, ν(A) = 0 implies ν(A + λk) = 0. Albeverio and Röckner prove in [AR90] the following

PROPOSITION. Let K := {k ∈ E; ν is k-quasi-invariant}.ThenK is a linear space. If K is dense in E then supp(ν) = E.

For the convenience of the reader – and with the permission of the authors – I repeat the proof given in [AR90] here: The fact that K is linear follows directly from the definition of k-quasi-invariance. Assume that K is dense in E,andletz ∈ supp(ν).(Suchz exists, otherwise ν would be the zero measure.) We want to show then that every element z00 ∈ E of ON DIFFERENTIAL OPERATORS IN WHITE NOISE ANALYSIS 339 the form z00 = z + αz0, z0 ∈ E, belongs to supp(ν), and consequently also every 00 00 z ∈ E. First we show this for z = z + k with k ∈ K:LetVz+k be an open neighborhood of z + k.ThenVz := Vz+k − k is an open neighborhood of z. Thus we must have ν(Vz)>0, and k-quasi-invariance of ν implies that ν(Vz+k)>0. 0 Hence z + k ∈ supp(ν).Nowletz ∈ E, and choose a net (ki ,i∈ I) in K which 0 0 converges to z .Thenz + αz = limi (z + αki ).Wehavez + αki ∈ supp(ν) for all i ∈ I. supp(ν) is closed, and thus we get z + αz0 ∈ supp(ν). 2 In other words: if ν is quasi-invariant with respect to the translations of a dense linear subset, then every nonempty open set has nonvanishing ν-measure. Assume now that f and g are continuous mappings from E into a Hausdorff topological space F , which are ν-a.e. equal. If ν is k-quasi-invariant for all k in a dense set K, then they must be equal: The set {ω; f(ω) 6= g(ω)} is open, and the assumption that it is nonempty leads to a contradiction. For the white noise measure µ on 0 (S (R+), B0) we get therefore the following conclusion.

0 COROLLARY. Let f and g be two mappings from S (R+) into a Hausdorff topo- 0 logical space which are continuous w.r.t. any dual topology on S (R+), and which are such that f = gµ-a.s. Then f = g. In particular, a µ-class of real or com- 0 plex valued random variables on (S (R+), B0,µ) can only have one continuous representative.

0 Our next task is to introduce a Brownian motion on (S (R+), B0,µ),andwe shall follow the usual constructions. One way to do this is as follows: Consider b 2 b the linear mapping X·: S(R+) → L (µ) where Xf is the µ-class of the random 0 variable given by Xf (ω) := hω,fi for f ∈ S(R+), ω ∈ S (R+). It follows directly from the definition of µ that this mapping is continuous, if S(R+) is equipped 2 with the norm of L (R+). Therefore, it has a unique linear continuous extension to 2 b L (R+), which we continue to denote by X·, and by Xf we mean any representative b 2 2 of Xf if f ∈ L (R+) \ S(R+). It is clear that (Xf ; f ∈ L (R+)) is a centered 2 Gaussian family whose covariance is given by the inner product in L (R+). Choose ∈ R = e := for t +, f 1[0,t], and set Bt X1[ ] . Then it is plain to check, that the e 0,t process (Bt ; t ∈ R+) has the same finite-dimensional distributions as a Brownian motion. Therefore, we can apply the Kolmogorov–Chentsov lemma to find a mod- e ification of (Bt ; t ∈ R+) with a.s. continuous sample paths. This modification is a standard Brownian motion, and it will be denoted by (Bt ; t ∈ R+). Alternatively, we can mimic the Lévy–Ciesielski construction, e.g., [MK69]. To this end, we consider L2([0, 1]) (with Lebesgue measure) as embedded into 2 n−1 L (R+).Let(sn,k; n ∈ N0,k = 1, 3,...,2 −R1) be the system of Schauder [ ] ∈ R := t functions on 0, 1 .Thatis,fort +, sn,k(t) 0 fn,k(s) ds,wherefn,k is the Haar function of index (n, k),whichis2(n−1)/2 on the interval [(k − 1)/2n,k/2n), −2(n−1)/2 on [k/2n,(k+ 1)/2n) and zero otherwise. Recall that the system of Haar functions forms a complete orthonormal system on L2([0, 1]) (e.g., [MK69]). It is clear that sn,k is continuous on [0, 1]. 340 JÜRGEN POTTHOFF

0 ; ∈ N = n−1 − 2 [ ] Now choose any CONS (en,k n 0,k 1, 3,...,2 1) of L ( 0, 1 ) ∞ [ ] := h 0 i ; ∈ N = in Cc ( 0, 1 ).Let
Nn,k(ω) ω,en,k . Then the family Nn,k n 0,k 1, 3,...,2n−1 − 1 is an i.i.d. system of everywhere defined standard normal ran- dom variables. Define X Bt := tN0,1 + sn,k(t) Nn,k,t∈[0, 1]. n,k Then we are in the same situation as in [MK69], p. 7, and can use the Borel– Cantelli lemma to prove that this series converges a.s. absolutely and uniformly in t ∈[0, 1]. Therefore, the process Bt , t ∈[0, 1], has a.s. continuous sample paths. It is obvious that Bt is centered Gaussian, and to check that the covariance of Bs, Bt is s ∧ t is an easy exercise. Thus (Bt ,t∈[0, 1]) is a standard Brownian motion over [0, 1]. For any N ∈ N we can now repeat the same construction on the interval [N,N+ ] N ∈ N = n−1 − 2 [ + ] 1 with a CONS (en,k,n 0,k 1, 3,...,2 1) of L ( N,N 1 ) in ∞ [ + ] Cc ( N,N 1 ), and obtain an independent standard Brownian motion indexed by t ∈[N,N + 1]. Gluing these Brownian motions continuously together, we obtain a standard Brownian motion indexed by the half-line. Let us denote the µ-completion of B0 by B. (The extension of µ to B will be denoted by the same symbol.) Consider our first construction of Brownian motion (Bt ,t∈ R+), i.e., by modi- e fication of the random variables Bt , t ∈ R+. We want to show that the µ-completion µ σ (Bt ,t∈ R+) of σ(Bt ,t∈ R+) generated by it is equal to B. First we prove the following result which is almost obvious.

LEMMA. Let T be any subset of S(R+) which is total in S(R+) with respect 2 to the norm of L (R+) restricted to S(R+). Then the µ-completion of σ(T ) := σ(Xf ,f∈ T ) coincides with B.

Proof. Since all linear combinations of random variables of the form Xf with f ∈ T are σ(T )-measurable, we may assume without loss of generality that T is a dense linear subspace of S(R+) with respect to |·|2.Letf ∈ S(R+), and choose ∈ N | − | → a sequence (fn,n 0) in T so that f fn 2 0. Then Xfn converges in mean square to Xf . By choosing a subsequence if necessary, we may assume that Xfn converges to Xf µ-a.s. This implies that Xf has a modification, say e := Xf lim sup Xfn , n∈N which is measurable with respect to the σ -algebra generated by the random vari- ∈ N e ables Xfn , n . In particular, Xf is σ(T )-measurable, and therefore Xf is measurable with respect to the µ-completion σ µ(T ) of the σ -algebra σ(T ).On the other hand, the random variables Xf , f ∈ S(R+), generate B0,sothatwehave the following inclusions

µ B0 ⊂ σ (T ) ⊂ B. ON DIFFERENTIAL OPERATORS IN WHITE NOISE ANALYSIS 341

Consequently σ µ(T ) = B. 2

∈ ∞ R Let f Cc ( +), and consider the random variable Xf . Define the following sequence X∞
n n X := f(s ) B n − B n ,n∈ N, f k sk+1 sk k=0 n = n ∈ N where sk k/2 . Observe that for each n , the series above has only a finite n number of terms. It is straightforward to check that Xf converges to Xf in mean square sense. By an argument similar to the one in the proof of the lemma, we find µ that Xf is measurable with respect to σ (Bt ,t∈ R+). This holds for every f in ∞ R R |·| the subspace Cc ( +) of S( +), which is dense w.r.t. the norm 2.Usingthe = ∞ R lemma with T Cc ( +) we therefore get the following inclusions:

∞ R ⊂ µ ∈ R ⊂ = µ ∞ R σ Cc ( +) σ (Bt ,t +) B σ Cc ( +) , ∞ R where the last σ -algebra is the µ-completion of σ(Cc ( +)). Hence we can con- clude that

µ σ (Bt ,t∈ R+) = B.

For the Brownian motion defined by the Lévy–Ciesielski construction we obtain the same result – more easily – in a similar way. Thus we have completed our discussion of a realization of the hypotheses (H.1), (H.2) in the setting of the white noise space over the half-line. We conclude this section with the following remark. Instead of S(R+) we could as well have chosen the smaller space S0(R+) which is defined as the closure of ∞ R R Cc ( +) in S( ) (or in S(R+), which leads to the same space). It is clear that S(R+) is the subspace of those functions of S(R) which are supported in R+ (or those functions in S(R+) which together with all their derivatives vanish at the origin). As a linear subspace of S(R) or S(R+), S0(R+) is again nuclear. There- fore, we can repeat the preceding discussion almost word by word, with obvious modifications here and there.

4. Differential Operators Consider a random variable Y on (, B,P).If is a topological vector space (or admits some suitable notion of translations), like in the case of the two realizations mentioned in Section 2, it is obvious how to define directional derivatives. Since in the general case such a structure is not available, we introduce directional deriv- atives by imitating the chain rule. This has been done in [DP97], but the question whether the derivatives are well-defined had been left open. We start with this question here. 342 JÜRGEN POTTHOFF

∈ N ∞ Rn For n , denote by Ce ( ) the space of infinitely often continuously dif- ferentiable functions on Rn which – together with all their derivatives – have at most exponential growth. It will be convenient to denote the ith partial derivative ∈ ∞ Rn ∈ Rn of f Ce ( ) at x by f,i(x). ∞ By Cfi,e() we mean the space of all random variables Y for which there exist ∈ N ∈ ∞ R ∈ ∞ Rn n , h1,...,hn Cc ( +),andf Ce ( ) so that = Y f(Xh1 ,...,Xhn ). ∞ ⊂ p > Note that Cfi,e() L (P ) for all p 1, and these inclusions are dense. ∈ ∞ LEMMA 1. Assume that Y Cfi,e() has two representations of the form = Y f(Xh1 ,...,Xhn ) = g(Xk1 ,...,Xkm ) ∈ N ∈ ∞ R ∈ ∞ Rn ∈ with n, m , h1,...,hn, k1,...,km Cc ( +), and f Ce ( ), g ∞ Rm ∈ 2 R Ce ( ). Then for all u L ( +), Xn 2 f,i(Xh1 ,...,Xhn )(u,hi)L (R+) i=1 Xm = 2 g,j (Xk1 ,...,Xkm )(u,kj )L (R+),P-a.s. j=1 ∈ ∞ R Proof. By assumption, we have for all l Cc ( +),
− Xl = f(Xh1 ,...,Xhn ) g(Xk1 ,...,Xkm ) e 0, and in particular the law of this random variable is the Dirac measure ε0 at 0.

Now consider the Gaussian random variables Zh1 ,...,Zhn , Zk1 ,...,Zkm , Zl in 0 the white noise realization on (S (R+), B,µ)of Section 3. That is, Zη(ω) =hω,ηi ∈ R ∈ 0 R for η S( +), ω S ( +). The random variables Zh1 ,...,Zl have the same joint distribution as Xh ,...,Xl. Therefore also the random variable 1
− Zl f(Zh1 ,...,Zhn ) g(Zk1 ,...,Zkm ) e ∈ ∞ R has ε0 as its law. Since this is true for every l Cc ( +), this implies that the S-transform of − f(Zh1 ,...,Zhn ) g(Zk1 ,...,Zkm ) is zero. Because of the injectivity of the S-transform we conclude that = f(Zh1 ,...,Zhn ) g(Zk1 ,...,Zkm ), µ-a.s. 0 By construction, both sides of the last equality are weakly continuous on S (R+). The corollary of Section 3 entails that these two random variables are equal every- 0 where on S (R+). Hence we may compute the directional derivative of both sides ON DIFFERENTIAL OPERATORS IN WHITE NOISE ANALYSIS 343

2 0 in direction u ∈ L (R+) ⊂ S (R+) as follows: Xn = 2 Du f(Zh1 ,...,Zhn ) f,i(Zh1 ,...,Zhn )(u,hi )L (R+) i=1 = Du g(Zk1 ,...,Zkm ) Xm = g,j (Zk1 ,...,Zkm )(u,kj )L2(R+), j=1

0 because for η ∈ S(R+), ω, u ∈ S (R+),

d Du Zη(ω) = hω + λu, ηi dλ λ=0 = (u, η)L2(R+). ∈ ∞ R Consequently, we have for all l Cc ( +) that

Xn 2 − f,i(Zh1 ,...,Zhn )(u,hi )L (R+) i=1 ! Xm − Zl g,j (Zk1 ,...,Zkm )(u,kj )L2(R+) e j=1 has ε0 as its law. Now we go back to the original probability space, i.e., replace the Gaussian random variables Zh1 ,etc.,byXh1 , etc. Then we obtain that the S- transform of Xn Xm 2 − 2 f,i(Xh1 ,...,Xhn )(u,hi)L (R+) g,j (Xk1 ,...,Xkm )(u,kj )L (R+) i=1 j=1 is zero, and hence this random variable is P -a.s. zero. 2

The preceding lemma justifies the following

∈ ∞ DEFINITION. Let Y Cfi,e() be given by = Y f(Xh1 ,...,Xhn ), ∈ N ∈ ∞ R ∈ ∞ Rn ∈ 2 R with n , h1,...,hn Cc ( +),andf Ce ( ).Forallu L ( +),the ∞ random variable in Cfi,e() defined by Xn := 2 DuY f,i(Xh1 ,...,Xhn )(u,hi )L (R+) i=1 is called the derivative of Y in direction u. 344 JÜRGEN POTTHOFF

2 2 For every u ∈ L (R+), Du is a densely defined operator on L (P ). Thus it has ∗ ∗ ∞ an adjoint Du with domain D(Du). It is not very difficult to check that Cfi,e() is ∗ ∈ ∞ a subset of D(Du), and that on Y Cfi,e() it acts as ∗ = − DuY Xu Y DuY. 2 In particular, Du is closable on L (P ), and we denote its closure by (Du, D(Du)). At this point we can repeat all arguments, as, e.g., given in [HK93], to show that actually = ∗ = · = 1/2 D(Du) D(Du) D(Xu ) D(N ), where N is the number operator, which can – for example – be defined on the core ∞ Cfi,e() by the formula X N = D∗ D , ek ek k∈N

2 with a CONS (ek,k∈ N) of L (R+). The following result will be useful.

2 LEMMA 2. For every u ∈ L (R+), Du commutes with the S-transform in the 1/2 2 sense that for all Y ∈ D(N ), f ∈ L (R+),

S(DuY )(f ) = Du(SY )(f ),

2 where the right-hand side is the usual Gâteaux derivative of a function on L (R+). Proof. Compute
S(D Y )(f ) = E D Y : eXf : u u
∗ = E YD : eXf : u

Xf = E Y Xu − (u, f ) : e : , where we have set

X X −1/2 |f |2 : e f :=e f 2 . On the other hand, we have

∂ Du(SY )(f ) = (SY )(f + λu) ∂λ λ =0
∂ X + = E Y : e f λu : ∂λ λ=0
∂ X +λX −1/2 |f +λu|2 = E Y e f u 2 ∂λ

λ=0 Xf = E Y Xu − (f, u) : e : , ON DIFFERENTIAL OPERATORS IN WHITE NOISE ANALYSIS 345 where the interchange of the derivative w.r.t. λ and the expectation is readily justi- fied, e.g., by an application of the dominated convergence theorem. 2

∈ ∞ Consider again the formula for Du Y with Y Cfi,e(). We may write it as follows Z

DuY = (∂t Y)u(t)dt, R+ where we have put Xn := ∂t Y f,i(Xh1 ,...,Xhn )hi(t). i=1 The integral above could of course be interpreted pointwise on , but we shall take in the sense of an L2(P )-valued Pettis integral (e.g., [HP57]). Therefore we have for all Z ∈ L2(P ) the relation  Z  Z
E Z (∂t Y)u(t)dt = E Z(∂t Y) u(t) dt. R+ R+

1 2 ∞ = − | | ∈ R+ Choosing in particular Z exp(Xf 2 f 2) with f Cc ( ),wefindthat Z

S(DuY )(f ) = S(∂t Y )(f ) u(t) dt. R+

We are interested to compute the S-transform of ∂t Y . To this end, let us prove first the following result

∈ ∞ R ∈ ∞ LEMMA 3. For all f Cc ( +), Y Cfi,e(), the mapping

u 7→ Du(SY )(f )

2 from L (R+) into R is linear and continuous. ∈ ∞ 7→ Proof. That for all Y Cfi,e(), the mapping u DuY is linear and continuous 2 2 from L (R+) into L (P ), is obvious from the definition. On the other hand, the S- 2 − 1 | |2 transform of a random variable is its L (P )-inner product with exp(Xf 2 f 2), and hence it is clear that this is a continuous linear map from L2(P ) into R. Thus 2 u 7→ S(Du Y )(f ) is linear and continuous from L (R+) into R. The proof is finished by an application of Lemma 2. 2

∈ ∞ ∈ ∞ R From Lemma 3 we conclude that for given Y Cfi,e(), f Cc ( +),there 2 exists an element in L (R+) which we denote by δ t 7→ SY(f) δf (t ) 346 JÜRGEN POTTHOFF

2 so that for all u ∈ L (R+), Z δ Du(SY )(f ) = SY (f ) u(t) dt. R+ δf (t ) δSY(f)/δf(t)is also called the Fréchet functional derivative of SY at f .Ifweuse again the fact that Du and S commute, we arrive at the following (slightly informal) intertwining relation for ∂t and S δ ∂ = S−1 S, t δf (t ) which is essentially T. Hida’s original definition of ∂t in the ground-breaking paper [Hi75].

Acknowledgement It is a pleasure to thank Thomas Deck and Gjermund Våge for many stimulating discussions. The author is very grateful to S. Albeverio and M. Röckner for their advice and their permission to reproduce here one of their results.

References [AR90] Albeverio, S. and Röckner, M.: New developments in the theory and applications of Dirichlet forms, In: S. Albeverio et al. (eds), Stochastic Processes, Physics and Geometry, World Scientific, Singapore, 1990. [DP97] Deck, T., Potthoff, J. and Våge, G.: A review of white noise analysis from a probabilistic standpoint, Acta Appl. Math. 48 (1997), 91–112. [Hi75] Hida, T.: Analysis of Brownian Functionals, Carleton Math. Lecture Notes, No. 13, 1975. [Hi80] Hida, T.: Brownian Motion, Springer, New York, 1980. [HK93] Hida, T., Kuo, H.-H., Potthoff, J. and Streit, L.: White Noise – An Infinite Dimensional Calculus, Kluwer Acad. Publ., Dordrecht, 1993. [HP57] Hille, E. and Phillips, R. S.: Functional Analysis and Semigroups,Amer.Math.Soc. Colloq. Publ., Amer. Math. Soc., Providence, 1957. [KS88] Karatzas,I.andShreve,S.E.:Brownian Motion and Stochastic Calculus, Springer, New York, 1988. [KL96] Kondratiev, Yu. G., Leukert, P., Potthoff, J., Streit, L. and Westerkamp, W.: Generalized functionals on Gaussian spaces – the characterization theorem revisited, J. Funct. Anal. 141 (1996), 301–318. [KT80] Kubo, I. and Takenaka, S.: Calculus on Gaussian white noise I, Proc. Japan Acad. 56 (1980), 376–380. [Ku75] Kuo, H.-H.: Gaussian Measures in Banach Spaces, Springer, New York, 1975. [MK69] McKean,H.P.,Jr.:Stochastic Integrals, Academic Press, New York, 1969. [RS72] Reed, M. and Simon, B.: Methods of Modern Mathematical Physics I: Functional Analysis, Academic Press, New York, 1972. [RS75] Reed, M. and Simon, B.: Methods of Modern Mathematical Physics II: Fourier Analysis, Self-Adjointness, Academic Press, New York, 1975. [RY91] Revuz, D. and Yor, M.: Continuous Martingales and Brownian Motion, Springer, New York, 1991. ON DIFFERENTIAL OPERATORS IN WHITE NOISE ANALYSIS 347

[Sa77] Sansone, G.: Orthogonal Functions, Robert Krieger Publ. Co., 1977. [Si71] Simon, B.: Distributions and their Hermite expansions, J. Math. Phys. 12 (1971), 140–148. [S73] Schwartz, L.: Radon Measures on Arbitrary Topological Vector Spaces and Cylindrical Measures, Oxford Univ. Press, London, 1973. [Tr67] Trèves, F.: Topological Vector Spaces, Distributions and Kernels, Academic Press, New York, 1967.