1Introduction DavidApplebaum JanvanNeerven SecondquantisationforskewconvolutionproductsofmeasuresinBanachspaces

El e c t ro nic J

o f

Pr

ob a bi l i t y

Electron. J. Probab.19(2014), no. 11, 1–17.

ISSN:1083-6489 DOI:10.1214/EJP.v19-3031

Second quantisation for skew convolution products of measures in Banach spaces

David Applebaum

Jan van Neerven

Abstract

We study measures in Banach space which arise as the skew convolution product of two other measures where the convolution is deformed by a skew map. This is the structure that underlies both the theory of Mehler semigroups and operator self- decomposable measures. We show how that given such a set-up the skew map can be lifted to an operator that acts at the level of function spaces and demonstrate that this is an example of the well known functorial procedure of second quantisation. We give particular emphasis to the case where the product measure is infinitely divisible and study the second quantisation process in some detail using chaos expansions when this is either Gaussian or is generated by a Poisson random measure.

Keywords: Skew convolution product, second quantisation, Ornstein-Uhlenbeck semigroup, Poisson random measure.

AMS MSC 2010:Primary 60B11, Secondary 47D03; 47D07; 60G51; 60H15; 60J35; 60G57.

Submitted to EJP on September 22, 2013, final version accepted on January 11, 2014.

1 Introduction

In recent years there has been considerable interest in skew-convolution semigroups of probability measures in Banach spaces and the so-called Mehler semigroups that they induce on function spaces. These objects arise naturally in the study of infinite dimensional Ornstein-Uhlenbeck processes driven by Banach-space valued Lévy pro- cesses. Such processes have attracted much attention as they are the solutions of the simplest non-trivial class of stochastic partial differential equations driven by additive Lévy noise (see [1, 7, 29]). The first systematic study of Mehler semigroups in their own right were [6] and [14] with the former concentrating on Gaussian noise while the latter generalised to the Lévy case. Harnack inequalities were obtained in [31] and the infinitesimal generators were found in [4]. From a different point of view, skew- convolution semigroups also appear naturally in the investigation of continuous state branching processes with immigration [11] and more general affine processes [10].

In this paper we focus on the representation of Mehler semigroups as second quan- tised operators. Such a result has been known for a long time in the Gaussian case.

∗Support: VICI subsidy 639.033.604 of the Netherlands Organisation for Scientific Research (NWO).

†Sheffield University, UK. E-mail:[email protected]

‡Delft University of Technology, The Netherlands. E-mail:[email protected]

It was first established for Hilbert space valued semigroups in [8] and then extended to Banach spaces in [23]. Once such a representation is known it can be put to good use in proving key properties of the semigroup such as compactness and smoothness [8], symmetry [9], analyticity [15, 22], and in the computation of theirL^p spectra [24].

When the semigroups act on Hilbert spaces, the desired second quantisation represen- tation was recently obtained in [28] in the pure jump case using chaotic decomposition techniques from [20], under the assumption that the Ornstein-Uhlenbeck process has an invariant measure. This paper extends that result to the Banach space case and ob- tains the second quantisation representation without needing to assume the existence of an invariant measure.

In fact, within the main part of our paper we dispense with Mehler semigroups altogether and work with a more general structure which we introduce herein. For this we require that there are measuresµ₁on a Banach spaceE₁andµ₂andρon a Banach spaceE₂which are related by the identity

µ2=T(µ1)∗ρ,

whereT :E1 →E2is a Borel mapping and∗is the usual convolution of measures. An operatorT that has such an induced action is precisely askew mapas featured in the abstract of this paper. Note that if E1 = E2 = E and µ1 = µ2 = µsay, then µis an operator self-decomposablemeasure and such objects have been intensely studied (see e.g. [18, 19, 34].) The invariant measures arising in [28] are precisely of this form. On the other hand a skew convolution semigroup of measures(µt, t≥0)with respect to a C0-semigroup(S(t), t≥0)is characterised by the relationsµs+t=S(t)µs∗µtand these are clearly also examples of our structure. At our more general level, the antecedent of a Mehler semigroup is a bounded linear operatorPT which acts from L²(E2, µ2)to L²(E₁, µ₁). Our main result is then to show that this operator can be seen as a second quantisation of the adjoint T^∗ : E₂ → E₁ in a natural way in the case whereµ₁ and µ2 are both infinitely divisible and either Gaussian or of pure jump type. With regard to the pure jump case, Surgailis [33] has found necessary and sufficient conditions for a contraction semigroup to second quantise to a Markov semigroup in Poisson Fock space. The emphasis in our paper is different, in that we work at a more general level, but it may be fruitful for future work to explore the confluence of these two streams of thought.

A key part of our approach is the use of a family of vectors that we call exponen- tial martingale vectors. We now explain how these arise and contrast them with the more familiarexponential vectors(see e.g. [3, 27]). Second quantisation is seen most naturally as a covariant functorΓwithin the category whose objects are Hilbert spaces and morphisms are contractions (see e.g. [27]). IfH is a Hilbert space andΓ(H)is the associated symmetric Fock space, the set of exponential vectors is linearly independent and total in Fock space. If we are given a Gaussian field overH then the exponential vectors correspond to the generating functions of the Hermite polynomials, and from the point of view of stochastic calculus they correspond both to the Doléans-Dade ex- ponentials and to the exponential martingales. When we consider Lévy processes, the latter symmetry is broken. Exponential vectors still correspond to Doléans-Dade expo- nentials (see [3]) but these are no longer exponential martingales. In this paper, we find that a natural context for defining second quantisation in a non-Gaussian context is to employ vectors that are natural generalisations of exponential martingales, rather than using exponential vectors themselves. Hence we call these exponential martingale vectors. In particular, as we show in Section 2 and the appendix, these are still both total and linearly independent.

Notation. Throughout this article,Eis a real Banach space. The space of all bounded

linear operators on E is denoted byL(E) and the dual of E is denoted byE^∗. The action ofE^∗ onE is represented byx^∗(x) =hx, x^∗i. Whenever we consider measures on a Banach space E, they are defined on the Borelσ-algebra B(E). If µ is a Borel measure onEandT :E →F is a Borel mapping fromEinto another Banach spaceF we frequently writeT(µ)to denote the Borel measureµ◦T⁻¹. The Dirac measure based atx∈Eis denoted byδx. The Banach space (with respect to the supremum norm) of all bounded Borel measurable functions onEwill be denotedB_b(E;K), whereK^{is either} R^orC. If both choices are permitted we simply writeB_b(E).

2 Skew convolution of measures and associated skew maps

Let ν be a finite Radon measureon a Banach space E, that is, ν is a finite Borel measure onE with the property that for all ε > 0 there exists a compact setK inE such thatν(E\K)< ε.Recall that ifEis separable, then every finite Borel measure is Radon.

Thecharacteristic functionofνis the mappingbν:E^∗→C^{defined by}

bν(x^∗) = Z

E

exp(ihx, x^∗i)ν(dx),

for all x^∗ ∈ E^∗. The mapping bν is continuous with respect to the topology of uniform convergence on compact subsets ofE. More generally, for a measurable functionφ : E→Rwe may define

bν(φ) = Z

E

exp(iφ(x))ν(dx).

Definition 2.1. Let µ₁ and µ₂ be Radon probability measures on the Banach spaces E₁and E₂, respectively, withµc₂(x^∗)6= 0for allx^∗ ∈E₂^∗ (e.g. this condition is fulfilled, whenµ2is infinitely divisible). A Borel mappingT :E1→E2is called askew mapwith respect to the pair(µ1, µ2)if there exists a Radon probability measureρonE2such that

T(µ1)∗ρ=µ2,

and we say thatµ₂is the skew-convolution product(with respect toT) ofµ₁ andρ. If T is also a bounded linear operator betweenE1andE2we call it askew operatorwith respect to(µ1, µ2).

Given the pair(µ₁, µ₂), the measureρis easily seen to be unique. Indeed, the iden- tity T\(µ₁)(x^∗)ρ(xb ^∗) = cµ₂(x^∗) 6= 0 forces T\(µ₁)(x^∗) 6= 0, and therefore ρbis uniquely determined byT(µ1)and µ2. We callρtheskew convolution factorassociated withT and the pair(µ1, µ2).

Proposition 2.2. Suppose that T : E₁ → E₂ is a skew map with respect to the pair (µ1, µ2), wherecµ2(x^∗) 6= 0for all x^∗ ∈ E₂^∗. Let ρbe the associated skew convolution factor. For all1≤p <∞the linear mappingPT :Bb(E2)→Bb(E1)defined by

PTf(x) :=

Z

E₂

f(T(x) +y)dρ(y), x∈E1, extends uniquely to a linear contractionPT :L^p(E2, µ2)→L^p(E1, µ1).

Proof. Fix1≤p <∞. By the Hölder inequality, for allf ∈B_b(E₂)we have kPTfk^p_Lp(E1,µ1)=

Z

E₁

Z

E₂

f(T(x) +y)dρ(y)

p

dµ1(x)

≤ Z

E1

Z

E2

|f(T(x) +y)|^pdρ(y)dµ1(x)

= Z

E₂

Z

E₂

|f(y⁰+y)|^pdρ(y)dT(µ₁)(y⁰)

= Z

E2

|f(z)|^pd(T(µ1)∗ρ)(z)

= Z

E₂

|f(z)|^pdµ₂(z) =kfk^p_Lp(E

2,µ2), and the required result follows.

Example 2.3(Skew Convolution Semigroups). Let(S(t), t≥0) be aC0-semigroup on a Banach space E. A skew convolution semigroup is a family (µt, t ≥ 0) of Radon probability measures on E for which µ_s+t = S(t)µ_s∗µ_t for all s, t ≥ 0. Then S(t) is a skew operator with respect to the pair (µ_s, µ_s+t), In this case we write P_t for the linear operatorP_S(t). Then (Pt, t ≥0) is a semigroup in that P0 =I and Ps+t =PsPt

for all s, t ≥ 0, and is called a Mehler semigroup (see e.g. [4, 6, 10, 11, 14]). Such objects arise naturally in the study of linear stochastic partial differential equations with additive noise of the form:

dY(t) =AY(t) +dL(t), (2.1)

whereAis the infinitesimal generator of (S(t), t ≥0) and (L(t), t ≥0) is an E-valued Lévy process. IfE is a real Hilbert space then it is well-known (see e.g. [1, 7] and the recent book [29]) that this equation has a unique mild (equivalently weak) solution (Y(t), t ≥0) which is a Markov process given by the generalised Ornstein-Uhlenbeck process:

Y(t) =S(t)Y(0) + Z t

0

S(t−u)dL(u), (2.2)

(where the initial conditionY(0)is assumed to be independent of(L(t), t ≥ 0).) Then µt is the law of the E-valued random variable Rt

0S(t−u)dL(u)and (Pt, t ≥ 0) is the transition semigroup of(Y(t), t≥0). On a Banach space we may define the stochastic convolution in (2.2) by using integration by parts as in [19]. Quite general necessary and sufficient conditions for solutions to exist to (2.1) (where the stochastic convolution is defined in the sense of Itô calculus) are given in [30]. IfX is a Brownian motion, we refer the reader to [25].

Example 2.4 (Operator Self-Decomposable Measures). Let µ be a Radon probability measure onEthat takes the form

µ=T µ∗ρ, (2.3)

whereT is a bounded linear operator onEandρis another Radon probability measure onE. Thenµ is operator self-decomposable (see [34]) andT is a skew operator with respect to the pair(µ, µ). There has been extensive work on such measures in the case where (2.3) holds withT =S(t)for allt≥0where(S(t), t≥0)is aC₀-semigroup onE (see e.g. [1, 18, 19]). Indeed such measuresµarise as the invariant measures of the Mehler semigroups of Example 2.3 (when these exist - see e.g. [7, 14]) and in the case of (2.2),ρis the law ofR∞

0 S(t−u)dL(u).

Definition 2.5. Letµbe a Radon probability measure onEsatisfyingµ(xb ^∗)6= 0for all x^∗∈E^∗. For each Borel functionφ:E→Rwe define the functionK_µ,φ:E→C^by

Kµ,φ(x) := exp(iφ(x)) µ(φ)b . We callK_µ,φanexponential martingale vector.

Proposition 2.6. Letµ1andµ2 be Radon probability measures onE1andE2, respec- tively, withµc₂(x^∗) 6= 0 for allx^∗ ∈ E₂^∗. LetT be a skew map with respect to the pair (µ₁, µ₂). Then for allx^∗∈E₂^∗we have

PTKµ₂,x^∗=Kµ₁,x^∗◦T. (2.4) Proof. Letρdenote the associated skew convolution factor. From the identity

µb2(x^∗) =T(µ\1)(x^∗)ρ(xb ^∗) =cµ1(x^∗◦T)ρ(xb ^∗) we deduce that for allx∈E

PTKµ₂,x^∗(x) = Z

E₂

Kµ₂,x^∗(T(x) +y)dρ(y)

= exp(ihT(x), x^∗i) µb₂(x^∗)

Z

E2

exp(ihy, x^∗i)dρ(y)

= exp(ihT(x), x^∗i)

µb2(x^∗) ρ(xb ^∗) =exp(i(x^∗◦T)(x))

µb1(x^∗◦T) =Kµ1,x^∗◦T(x).

Fix a Radon probability measureµonE and letEµdenote the linear span of the set of exponential martingale vectors{Kµ,x^∗;x^∗∈E^∗}. The proposition implies that, under the stated hypotheses onµ₁,µ₂, andT, the mapping

Kµ₂,x^∗ 7→Kµ₁,x^∗◦T

has a well-defined linear extension to a contraction from Eµ2 → Eµ1 (this extension being also denotedPT). Under suitable assumptions on the measures one may show that the functionsKµ,x^∗ are in fact linearly independent. This fact is of some interest by itself but is not needed here; therefore we have included it in an appendix at the end of this paper. Using the injectivity of the Fourier transform, a standard argument shows thatEµis dense inL^p(E, µ;C)for all1≤p <∞(see e.g. [2, Lemma 5.3.1]), and consequentlyP_T is the unique such extension.

3 Second quantisation: The Gaussian case

In this section we connect, in the Gaussian setting, the notions of skew operators with second quantisation. The presentation is slightly different from the usual one, in that we introduce a form of the chaos expansion that utilises iterated Malliavin deriva- tives that was introduced by Stroock [32]. This approach will bring out the analogies between the Gaussian and the Poisson case (which we present in the next section) very elegantly.

We begin by recalling some standard results from the theory of Gaussian measures.

Proofs and more details can be found in the monographs [5, 26, 35].

Let µ be a Gaussian measure on the real Banach space E, and let H denote its reproducing kernel Hilbert space, which is defined as follows. The covariance operator Qofµis given by

Qx^∗= Z

E

hx, x^∗ix µ(dx), x^∗∈E^∗.

This integral is known to be absolutely convergent inEand defines a bounded operator Q ∈ L(E^∗, E) which is positive in the sense that hQx^∗, x^∗i ≥ 0 for all x^∗ ∈ E^∗ and symmetric in the sense that hQx^∗, y^∗i = hQy^∗, x^∗i for all x^∗, y^∗ ∈ E^∗. The mapping (Qx^∗, Qy^∗) 7→ hQx^∗, y^∗idefines an inner product on the range of Q. The real Hilbert space H is defined to be the completion of the range of Q with respect to this inner product. The identity mapping Qx^∗ 7→ Qx^∗ extends to a bounded injective operator j:H →E, and we have the factorisationQ=j◦j^∗. Here we have identifiedH and its dual via the Riesz representation theorem.

Each elementh∈Hof the formh=j^∗x^∗defines a real-valued functionφh∈L²(E, µ) byφh(x) :=hx, x^∗i, and we have

kφhk²_L2(E,µ)= Z

E

hx, x^∗i²dµ(x) =kj^∗x^∗k²_H=khk²_H.

Sincej^∗ has dense range in H, the mappingh 7→φ_h uniquely extends to an isometry fromH intoL²(E, µ).

Suppose now that µ1 and µ2 are Gaussian Radon measures on Banach spaces E1

andE2, with reproducing kernel Hilbert spacesH1andH2respectively. In the next two Propositions 3.1 and 3.2 we shall investigate the relationship between linear skew maps fromE1toE2with respect to the pair(µ1, µ2)and linear contractions fromH1toH2.

We begin by proving that if T is a skew operator with respect to the pair (µ₁, µ₂), thenT restricts to a contraction between the reproducing kernel Hilbert spaces. This result and its proof extend a similar result for semigroup operators in [8, 23].

Proposition 3.1. If T is a bounded linear operator from E1 to E2 which is a skew operator with respect to the pair (µ1, µ2) of Gaussian measures, thenT restricts to a contraction fromH₁toH₂.

Proof. By assumption we haveT µ1∗ρ=µ2 for some Radon probability measureρ. We claim thatρis Gaussian. Indeed, using the fact thatT µ1has mean zero, we have

Z

E

hx, x^∗i²µ2(dx) = Z

E

Z

E

hT x+y, x^∗i²µ1(dx)ρ(dy)

= Z

E

hx, x^∗i²T µ1(dx) + Z

E

hy, x^∗i²ρ(dy).

Hence, denoting the covariances ofµ1andµ2 byQ1andQ2(respectively), we see that the operatorR:=Q2−T Q1T^∗is positive and symmetric as an operator fromE₂^∗toE2. SinceR ≤ Q2, a well-known tightness result for Gaussian measures implies that R is the covariance of a Gaussian Radon measureρ˜onE₂. The identity T Q₁T^∗+R = Q₂ impliesT µ₁∗ρ˜=µ₂. Sinceµ₂is a Gaussian measure, its characteristic function vanishes nowhere and hence, by the observation following Definition 2.1,ρ= ˜ρ. This proves the claim.

Recall thatQ1=j1◦j₁^∗, wherej1:H1,→E is the canonical inclusion mapping, and likewise we haveQ₂=j₂◦j₂^∗andR=j_R◦j^∗_R. For allx^∗∈E^∗we have

kj₁^∗T^∗x^∗k²_H

1 =hT Q₁T^∗x^∗, x^∗i

=hQ2x^∗, x^∗i − hR^∗x^∗, x^∗i ≤ hQ^∗₂x^∗, x^∗i=kj^∗₂x^∗k²_H2. ^(3.1)

Hence,

|hQ₁T^∗x^∗, y^∗i|=|[j₁^∗T^∗x^∗, j₁^∗y^∗]_H1| ≤ kj₂^∗x^∗k_H2kj^∗₁y^∗k_H1. (3.2) Define a linear functionalψy^∗on the range ofj₂^∗by

ψ_y^∗(j₂^∗x^∗) :=hQ₁T^∗x^∗, y^∗i.

Ifj₂^∗x^∗ = 0, thenj₁^∗T^∗x^∗ = 0by (3.1), soψy^∗ is well-defined. By (3.2),ψy^∗ extends to a bounded linear functional onH2of norm≤ kj₁^∗y^∗kH1. Identifyingψy^∗ with an element ofH₂, for allx^∗∈E^∗we have

hj2ψy^∗, x^∗i= [j₂^∗x^∗, ψy^∗]H₂ =hQ1T^∗x^∗, y^∗i=hT Q^∗₁y^∗, x^∗i.

Hence, T Q₁y^∗ = j₂ψ_y^∗ and kT Q₁y^∗k_H2 ≤ kj₁^∗y^∗k_H1. Writing Q₁ = j₁j^∗₁ we see that the restriction of T|H₁ toH1 maps j₁^∗y^∗ to the element j2ψy^∗ of H1, and that T|H is contractive on the dense range ofj₁^∗inH1. This gives the result.

In the converse direction we have the following result.

Proposition 3.2. SupposeT :H1→H2is a linear contraction. ThenT admits a linear Borel measurable extensionT¯:E1→E2with the following properties:

1. the image measureT¯(µ1)is a Gaussian Radon measure;

2. there exists a Gaussian Radon measureρonE2such thatT¯(µ1)∗ρ=µ2. In particular,T¯is a linear skew map for the pair(µ1, µ2).

Proof. The following facts follows from the general theory of Gaussian measures (see, e.g., [5, 13]):

1. the mappingT :H₁→H₂admits an extension to a linear Borel mappingT¯:E₁→ E2;

2. the operatorQ=j2T T^∗j₂^∗is the covariance of a Gaussian measureµonE2; 3. µcoincides with the image measureT¯(µ₁).

In terms of the covariance operatorsQandQ2ofµandµ2we have hQx^∗, x^∗i=kT^∗j₂^∗x^∗k²_H1 ≤ kj₂^∗x^∗k²_H2 =hQ2x^∗, x^∗i.

Hence the positive symmetric operator R := Q2−Qis the covariance of a Gaussian measureρfor which we haveT¯(µ1)∗ρ=µ∗ρ=µ2.

Our next objective is to relate the abstract second quantisation procedure of the previous section to the Wiener-Itô decompositions ofL²(E1, µ2)andL²(E2, µ2).

Following the presentation in [26], for each n ≥ 1 we define Hn to be the closed linear subspace ofL²(E, µ)spanned by the functions Hn(φh), where h∈ H has norm one andHnis then-th Hermite polynomial given by the generating function expansion

exp tx−1 2t²

=

∞

X

n=0

tⁿ n!H_n(x).

The Wiener-Itô decomposition theorem asserts that we have an orthogonal direct sum decomposition

L²(E, µ) =M

n≥0

Hn.

LetS_nbe the permutation group onnelements. The range of the symmetrising projec- tionΣ_n :H^⊗n→H^⊗n defined by

Σn(h1⊗. . .⊗hn) := X

σ∈Sn

(h_σ(1)⊗. . .⊗h_σ(n))

is denoted byH^snand is called then-fold symmetric tensor productofH. Let(hn)_n≥1 be an orthonormal basis ofH (the Hilbert spaceH, being a reproducing kernel Hilbert space of a Gaussian Radon measure, is separable (see e.g. [5])).

Consider then-fold stochastic integralI_n :H^sn→Hn, defined by In Σn(h^⊗k_j ¹

1 ⊗. . .⊗h^⊗k_j ^m

m ) :=

m

Y

l=1

Hk_l(φh_jl) withj₁<· · ·< j_mandk₁+· · ·+k_m=n. Then ^√1

n!I_nsets up an isometric isomorphism H^sn ' Hn. Stated differently, the mapping I = L∞

n=0

√1

n!In defines an isometric isomorphism

L²(E, µ)'Γ(H), where

Γ(H) :=

∞

M

n=0

H^sn with normk(hn)^∞_n=0k²_Γ(H)=P∞

n=0khnk²_Hns is thesymmetric Fock spaceoverH. For a functionf :E→Rof the form

f =g(φh₁, . . . , φh_n)

withh1, . . . , hn orthonormal inH and g : Rⁿ → C^{of class}C¹, we define theMalliavin derivative in the direction ofH as the functionDf :E→H given by

Df =

n

X

j=1

∂jg(φh₁, . . . , φh_n)⊗hj.

As is well known (see e.g. [26]), for all1≤p <∞the linear operatorDis closable and densely defined fromL^p(E, µ)toL^p(E, µ;H). From now on we will denote its closure by Das well, and denote the domain of its closure byW^1,p(E, µ). The higher order deriva- tivesD^kf :E→H^⊗k are defined recursively byD^kf :=D(D^k−1f). These operators are closable as well and the domains of their closures will be denoted byW^k,p(E, µ). We define the spacesW^∞,p(E, µ) :=T

k∈NW^k,p(E, µ).

The next proposition is due to Stroock [32] in the context of an abstract Wiener space. We give a different proof for Gaussian measures on Banach spaces. We write E^µf =R

Ef dµ.

Proposition 3.3. The spaceW^∞,2(E, µ)is dense inL²(E, µ)and for allf ∈W^∞,2(E, µ) we have

f =

∞

X

n=0

1

n!In(E^µDⁿf).

Proof. For eachh∈H, the functione_h:E →Ris defined bye_h:= exp(φ_h−¹₂khk²_H). It is well known that the linear span of{e_h, h∈E}is dense inL²(E, µ)(see e.g. [26] for a proof). Since

Dⁿeh=eh⊗(h⊗. . .⊗h

| {z }

ntimes

)

for alln∈N, the first assertion follows. We clearly have

E^µDⁿeh=E^µeh⊗(h⊗. . .⊗h) =h⊗. . .⊗h.

Applying the n-fold stochastic integral and using the generating function identity for the Hermite polynomials we obtain

∞

X

n=0

1

n!I_n(EµDⁿe_h) =

∞

X

n=0

1

n!I_n(h⊗. . .⊗h

| {z }

ntimes

)

=

∞

X

n=0

1

n!Hn(φⁿ_h) = exp φh−1 2khk²_H

=eh, and the required result follows by density.

Let us now return to the setting whereµ1andµ2are Gaussian measures onE1and E2, having reproducing kernel Hilbert spacesH1andH2, respectively. In order to avoid unnecessary notational complexity we will use the notationDfor Malliavin derivatives acting on bothL²(E₁, µ₁)andL²(E₂, µ₂), and define

Dhf := [Df, h].

Lemma 3.4. Let T : H₁ → H₂ be a linear contraction. Then for all f ∈ W^n,2(E₂, µ₂) andh₁, . . . h_n∈H₁,

E^µ1D_hⁿ_1,...,hnPTf =E^µ2Dⁿ_{T h1,...,T hn}f.

Proof. Let us check this first for n = 1. By an easy computation (see [22]), for f ∈ W^1,2(E₂, µ₂)we haveP_Tf ∈W^1,2(E₂, µ₂)and

DPTf = (PT ⊗T^∗)Df.

Consequently,

Eµ1D_hP_Tf =Eµ1[(P_T ⊗T^∗)Df, h]

= Z

E

Z

E

[Df(T x+y), T h]dρ(y)dµ₁(x)

= Z

E

[Df(z), T h]dµ₂(z) =Eµ2D_{T h}f.

(3.3)

Here,ρis the measure constructed in Proposition 3.2. The higher order case is proved along similar lines.

In terms of the global derivative, the computation (3.3) shows that E^µ1DPTf = T^∗E^µ2Df and more generally we have

E^µ1DⁿPTf = (T^∗)^snE^µ2Dⁿf. (3.4) ApplyingInto both sides of (3.4) and using Proposition 3.3 together with the density ofW^∞,2(E, µ)inL²(E, µ)we have proved:

Theorem 3.5. The following diagram commutes:

L²(E2, µ2) −−−−→^PT L²(E1, µ1)

L∞ n=0

√1 n!I_n

x





x





L∞ n=0

√1 n!I_n

Γ(H2)

L∞

n=0(T^∗)^ns

−−−−−−−−−→ Γ(H1)

The operatorΓ(T^∗) :=L∞

n=0(T^∗)^sn is usually called thesymmetric second quanti- sationof the operatorT^∗.

Remark 3.6. The operatorL∞ n=0

√1

n!I_nis inverse toL∞ n=0

√1

n!EµDⁿby Proposition 3.3, so we may rewrite the commutative diagram in the following equivalent form:

L²(E₂, µ₂) −−−−→^PT L²(E₁, µ₁)

L∞ n=0√1

n!Eµ2Dⁿ



 y



 y

L∞ n=0√1

n!Eµ1Dⁿ

Γ(H2)

L∞

n=0(T^∗)^ns

−−−−−−−−−→ Γ(H1)

This diagram should be compared with the one in the next section.

Let us finally return to the setting of the previous section and derive the identity (2.4) by the methods of the present section. Fix x^∗ ∈ E₂^∗ and let h := j₂^∗x^∗. Then Kµ₂,x^∗= exp(iφh− khk²_H

2)and therefore by Lemma 3.4 (which we apply to the real and imaginary parts ofKµ₂,x^∗), for allg∈H1we have

E^µ1DgPTKµ₂,x^∗=E^µ2DT gKµ₂.x^∗=i[h, T g]E^µ2Kµ₂,x^∗=i[T^∗h, g], soE^µ1DPTKµ₂,x^∗=iT^∗h.Likewise we have

E^µ1DⁿPTKµ₂,x^∗=iⁿ⊗(T^∗h⊗ · · · ⊗T^∗h

| {z }

ntimes

).

Applying the n-fold stochastic integral, using Proposition 3.3 (again considering real and imaginary parts separately), and using the (analytic extension of the) generating function identity for the Hermite polynomials, we obtain

PTKµ₂,x^∗=

∞

X

n=0

1

n!In(E^µDⁿPTKµ₂,x^∗)

=

∞

X

n=0

iⁿ

n!I_n(T^∗h⊗. . .⊗T^∗h

| {z }

ntimes

)

=

∞

X

n=0

iⁿ

n!Hn(φⁿ_T∗h)

= exp iφT^∗h−1

2kT^∗hk²_H

=Kµ₁,T^∗x^∗,

where the last identity used thatT◦j2=j2◦T impliesT^∗h=T^∗j₂^∗x^∗=j₂^∗T^∗x^∗.

4 Second quantisation: the Poisson random measure case

We proceed with a similar result in the case whereµ1andµ2 are infinitely divisible measures of pure jump type. For this we need do delve a bit deeper into the structure of such measures and develop their connection with Poisson random measures.

Let(Y,Y, ν)be aσ-finite measure space and letN(Y)denote the set of allN^-valued measures onY. We endow this space with theσ-algebraσ(Y)generated byY^{, that is,} the smallestσ-algebra which renders the mappingsξ7→ξ(B)measurable for allB ∈Y^. Let (Ω,F,P) be a probability space and Π be a Poisson random measure having intensity measureν. We denote byP^Πthe distribution ofΠ, that is,P^Πis the probability measure on(N(Y), σ(Y))given by

P^Π(B) =P(Π∈B)

for measurableB ⊆N(Y).

Following Last and Penrose [20], for a measurable functionf :N(Y)→R^andy∈Y we define the measurable functionDyf :N(Y)→R^by

Dyf(η) :=f(η+δy)−f(η).

The functionDⁿ_y1,...,ynf :N(Y)→Ris defined recursively by D_yⁿ

1,...,y_nf =D_ynDⁿ⁻¹_y

1,...,yn−1f,

fory1, . . . , yn ∈Y. This function is symmetric, i.e. it is invariant under any permutation of the variables.

We have a canonical isometryL²_s(Yⁿ) = (L²(Y))^sn, where the former denotes the closed subspace ofL²(Yⁿ)comprised of all symmetric functions. We set

Γ(L²_s(Y)) =

∞

M

n=0

L²_s(Yⁿ) with normk(fn)^∞_n=0k²_H =P∞

n=0kfnk²_L2

s(Yⁿ); forn= 0it is understood that(L²(Y))^s0= L²_s(Y⁰) := R^{. By} Iⁿ : L²_s(Yⁿ) → L²(Ω) we denote the n-fold stochastic integral as- sociated withΠas defined in [20]. We note that part (3) of the Last-Penrose theorem is essentially a Stroock formula for Poisson measures (cf. Proposition 3.3) and that a version of this result for a class of real-valued Lévy processes may be found in [12].

Theorem 4.1(Last-Penrose [20]). 1. For alln∈N^,y₁, . . . , y_n ∈ Y, andf ∈L²(PΠ) we haveτⁿf ∈L²_s(Yⁿ), where

τⁿf(y1, . . . , yn) :=EDⁿ_y1,...,ynf(Π).

2. The mappingτ:=L∞ n=0

√1

n!τⁿis a surjective isometry fromL²(PΠ)ontoΓ(L²_s(Y)). 3. For allf ∈L²(PΠ)we have

f(Π) =

∞

X

n=0

1

n!In(EDⁿf(Π)).

From this point on, we shall consider the special caseY =E, whereEis a separable real Banach space. We use the shorthand notation

Π(dx) =¯ 1{0<kxk≤1}Π(dx) +b 1_{kxk>1}Π(dx), whereΠb is the compensated Poisson random measure,

Π(B) = Π(B)b −ν(B),

andν is now assumed to be a Lévy measure onE (see e.g. [17, 21] for the definition).

We will need to use the Lévy-Itô decomposition for Banach space-valued Lévy processes, as established in [30], and the next lemma is key in that regard.

Lemma 4.2. The functionx7→xis Pettis integrable with respect toΠ¯.

Proof. LetNbe a Poisson random measure on[0,∞)×Ewith intensity measuredt×ν. By a theorem of Riedle and Van Gaans [30],x7→ xis Pettis integrable with respect to N¯. It follows thatx7→xis Pettis integrable with respect toM¯, whereM is the Poisson random measure onE given byM(B) = N([0,1]×B). SinceM andΠ are identically distributed (both being Poisson random measures with intensity measureν), this proves the lemma.

We will be interested in Borel probability measuresµonEwhich arise as the distri- bution ofE-valued random variablesX of the form

X =ξ+ Z

E

xΠ(dx)¯ (4.1)

whereΠis a Poisson random measure onEwhose intensity measureν is a Lévy mea- sure andξ ∈ E is a given vector. The interest of such random variables comes from the Lévy-Itô decomposition forE-valued Lévy processes, which asserts that if(L(t))_t≥0 is a Lévy process without Gaussian part, thenL(1)is precisely of this form (see [30]).

Note that µ is a Radon measure (since every Borel measure on a separable Banach space is Radon) and infinitely divisible. In particular, its characteristic function van- ishes nowhere.

It will be convenient to define, forf ∈L²(E, µ),

D˜yf(x) :=f(x+y)−f(x) (4.2) The higher order derivatives are defined recursively byD˜_yⁿ_1,...,yn= ˜Dy_nD˜ⁿ⁻¹_{y1,...,yn−1}.

Suppose now thatµ1 and µ2 are two measures of the above form, associated with random variablesX1: Ω→E1andX2: Ω→E2which are given in terms of the vectors ξ1 ∈E1 and ξ2 ∈E2 and Poisson random measuresΠ1 and Π2 as in (4.1). Consider a linear skew mapT :E1→E2with respect to the pair(µ1, µ2), so that

T µ1∗ρ=µ2

for some unique Borel probability measureρ. We have the following analogue of Lemma 3.4:

Lemma 4.3. For allf ∈L²(E₂, µ₂)andy₁, . . . , y_n∈E₁, Eµ1D˜ⁿ_y

1,...,y_nP_Tf =Eµ2D˜ⁿ_{T y}

1,...,T y_nf. (4.3)

Proof. Suppose the random variableR: ˜Ω→E2, defined on an independent probabil- ity space( ˜Ω,F˜,P)˜ , has distribution ρ. Then using the fact that T X₁+R and X₂ are identically distributed,

Eµ1P_Tf =EE˜f(T X₁+R) =Ef(X₂) =Eµ2f and

E^µ1D˜yPTf =E^µ1PTf(·+y)−E^µ1PTf(·)

=EE˜f(T y+T X1+R)−f(T X1+R)

=Ef(T y+X₂)−f(X₂)

=E^µ2f(·+T y)−E^µ2f(·)

=E^µ2D˜T yf.

For the higher derivatives we use a straightforward inductive argument.

Below we will think of the left and right hand side of (4.3) as symmetric functions on Eⁿ. As such, the identity will be written as

E^µ1D˜ⁿPTf =E^µ1D˜ⁿf◦T^sn, where

(g◦T^sn)(y1, . . . , yn) :=g(T y1, . . . , T yn).

Define, fork= 1,2, the operatorsj_k:L²(E_k, µ_k)→L²(PΠk)by jkf(η) =f

ξ+ Z

E

xη(dx)¯

, η∈N(E).

The rigorous interpretation of this identity is provided by noting that kjkfk²_L2(PΠ)=E

f

ξ+ Z

E

xΠ(dx)¯

2

=kfk²_L2(E,µ_k),

which means thatjkf(η)is well-defined for P^Π-almost allη and that jk establishes an isometry fromL²(E_k, µ_k)intoL²(PΠk).Note that

jkf(Πk) =f(Xk) (4.4)

and

jk◦D˜ =D◦jk, and therefore, for allg∈L²(Ek, µk),

(τ_kⁿ◦j_k)g=EDⁿj_kg(Π_k) =Ej_kD˜ⁿg(Π_k) =ED˜ⁿg(X_k) =EµkD˜ⁿg.

Using this identity in combination with Lemma 4.3, for allf ∈L²(E2, µ2)we obtain (τ₁ⁿ◦j₁)P_Tf =Eµ1D˜ⁿP_Tf =Eµ1D˜ⁿf◦T^sn= (τ₁ⁿ◦j₁)f◦T^sn.

When combined with the contractivity ofPT and the surjectivity ofτ(see Theorem 4.1), this identity implies that the mapping f 7→ f ◦T^sn is a is a linear contraction from L²_s(E₂ⁿ, ν₂ⁿ)toL²_s(E₁ⁿ, ν₁ⁿ).

In summary we have proved the following theorem.

Theorem 4.4. Let T :E1 →E2 be a linear Borel mapping. Under the above assump- tions, the mapping(T^∗)^sn : f 7→ f ◦T^sn is a linear contraction from L²_s(E₂ⁿ, ν₂ⁿ) to L²_s(E₁ⁿ, ν₁ⁿ), and the following diagram commutes:

L²(E2, µ2) −−−−→^PT L²(E1, µ1)

L∞ n=0√1

n!Eµ2D˜ⁿ



 y



 y

L∞ n=0√1

n!Eµ1D˜ⁿ

Γ(L²(E2, ν2))

L∞

n=0(T^∗)^ns

−−−−−−−−−→ Γ(L²(E1, ν1))

To make the connection with the commuting diagram in the Gaussian case, which features then-fold stochastic integrals rather than n-fold derivatives, we note that by Theorem 4.1 the following diagram commutes as well fork= 1,2:

L²(E, µk) −−−−−−→^f7→f(Xk) L²(Ω)

L∞ n=0√1

n!E_µkD˜ⁿ



 y

x





L∞ n=0√1

n!In

Γ(L²(Ek, νk)) −−−−→⁼ Γ(L²(Ek, νk))

Theorem 4.4 is a generalisation of the result obtained by Peszat [28] in the case whereµ₁=µ₂is an invariant measure associated with a Mehler semigroup on a Hilbert spaceE1=E2.

As we did in the previous section, we wish to make the link with the results on skew operators. In principle we could repeat the Gaussian computation at the end of Section 3, but this requires the evaluation of a rather intractable Poisson stochastic integral.

There is, however, a simpler argument.

We start with some preparations. IfXandµare as in (4.1), thenK_µ,x^∗ = exp(ihx, x^∗i−

ζ(x^∗)), wherebµ(x^∗) = exp(ζ(x^∗))is the characteristic function ofµ(withζtheLévy sym- bolofµ; see [2, page 31]). Then, for ally∈Eand for allx^∗∈E^∗,

E^µD˜yKµ,x^∗=E^µKµ,x^∗(·+y)−E^µKµ,x^∗(·)

= exp(−ζ(x^∗))E^µexp(ih·, x^∗i)[exp(ihy, x^∗i)−1] = exp(ihy, x^∗i)−1.

Likewise, fory1, . . . , yn∈E,

E^µD˜ⁿ_y1,...,ynKµ,x^∗=

n

Y

j=1

[exp(ihyj, x^∗i)−1]. (4.5)

Now suppose thatT :E1→E2is a skew operator with respect to(µ1, µ2), where the measuresµk are the distributions of random variablesXk as in (4.1) fork= 1,2. Then, by (4.4), the Last-Penrose theorem, Lemma 4.3 and (4.5), for allx^∗∈E₂^∗we have

PTKµ2,x^∗(X1) =j1PTKµ2,x^∗(Π1) =

∞

X

n=0

1

n!In(EDⁿj1PTKµ2,x^∗(Π1))

=

∞

X

n=0

1

n!I_n(Eµ1D˜ⁿP_TK_µ2,x^∗)

=

∞

X

n=0

1

n!In(E^µ2D˜ⁿ_T·Kµ₂,x^∗)

=

∞

X

n=0

1

n!I_nYⁿ

j=1

[exp(ihT·j, x^∗i)−1]

and, by duality and then repeating the same computation backwards,

=

∞

X

n=0

1 n!In

Yⁿ

j=1

[exp(ih·j, T^∗x^∗i)−1]

=

∞

X

n=0

1

n!In(E^µ1D˜ⁿKµ₁,T^∗x^∗)

=

∞

X

n=0

1

n!In(EDⁿj1Kµ1,T^∗x^∗(Π1))

=j1Kµ₁,T^∗x^∗(Π1) =Kµ₁,T^∗x^∗(X1).

It follows thatP_TK_µ2,x^∗=K_µ1,x^∗_◦T, in agreement with (2.4).

Remark 4.5. The results of Sections 3 and 4 suggest the problem of extending the theory to that case where µ₁ and µ₂ are arbitrary infinitely divisible measures. We conjecture that Theorems 3.5 and 4.4 extend to this more general framework.

A Linear independence of the functions K

Thesupportof a Radon measureµonE is the complement of the union of all open µ-null sets in E. We denote the support of µ bysupp(µ)and its closed linear span by Eµ. We say thatµhaslinear supportifsupp(µ) =Eµ. The proof of the next result uses a variant of a standard technique of reduction to a system of linear equations that can be found in [16, pp. 20-21] or [27, pp. 126-7].