WHITE NOISE LÉVYMEIXNER PROCESSES THROUGH A TRANSFER PRINCIPAL FROM ONE-MODE TO ONE-MODE TYPE...

June 9, 2010 17:50 FILE NAME abr-LMWNP-Revised-IDAQP09

Infinite Dimensional Analysis, Quantum Probability and Related Topicsc© World Scientific Publishing Company

White Noise Levy-Meixner Processes Through a Transfer PrincipalFrom one-Mode to one-Mode Type Interacting Fock Spaces

LUIGI ACCARDI

Centro Vito Volterra, Faculta di EconomiaUniversita di Tor Vergata

Via di Tor Vergata, 00133 Roma, [email protected]

ABDESSATAR BARHOUMI

Department of Mathematics, Higher School of Sciences and Technologiesof Hammam-Sousse, Sousse University, Sousse, Tunisia

[email protected]

ANIS RIAHI

Department of Mathematics, Higher Institute of Applied Sciencesand Technologies of Gabes, University of Gabes, Tunisia

[email protected]

Received (January 30, 2009)Revised (August 18, 2009)

Accepted (September 06, 2009)

Consider the Levy-Meixner one-mode interacting Fock Space ΓLM , 〈 ·, · 〉LM. Inspiredfrom a derivative formula appearing in 〈 ·, · 〉LM , we define scalar products 〈 ·, · 〉LM,n insymmetric n-particle spaces. Then, we introduce a classe of one-mode type interactingFock spaces ΓLM (H) naturally associated to the one-dimensional infinitely divisible dis-tributions with Levy-Meixner type µr ; r > 0. The Fourier transform in generalizedjoint eigenvectors of a family Jφ ; φ ∈ E of Levy-Meixner Jacobi fields provides a wayto explicit a unitary isomorphism ULM between ΓLM (H) and the so-called Levy-Meixnerwhite noise space L2(E ′,B(E ′), ΛLM ). We derive a chaotic decomposition property of thequadratic integrable functionals of the Levy-Meixner white noise processes in terms ofan appropriate wick tensor product. For their stochastic regularity, we give explicit formand sharp estimate of the associated Donsker’s delta function.

Keywords: Chaotic representation, Donsker’s delta, Generating function, InteractingFock space, Jacobi field, Levy-Meixner white noise process, Spectral measure.

AMS Subject Classification: primary 60J65; secondary 60J45, 60J51, 60H40.

1. Introduction

The questions considered here are interrelated by two basic notions: the notion of in-teracting Fock space emerged from the stochastic limit of quantum electro dynamicswhich provides a natural frame to deal with non-Gaussian measures and interacting

1


2 Luigi Accardi, Abdessatar Barhoumi and Anis Riahi

fields (see Refs. 4,5,6). This first main concept provides a way to propound a whitenoise framework to study Levy-Meixner white noise functionals.

The foundation of white noise calculus initiated by Hida 23 is a Schwartz typedistribution theory on a Gaussian space (E ′, Λ); more precisely, it is based on aparticular choice of a Gel’fand triple 21 :

(E) ⊂ L2(E∗, Λ) ⊂ (E)∗,

where L2(E ′, Λ) is isomorphic to a Boson Fock space through the Weiner-Ito-Segalisomorphism. Investigations in this line are closely connected with regularity ofGaussian measure and the associated expansion into Hermite polynomials. Later on,an extension of white noise theory to non-Gaussian analysis was established (e.g.,the Poisson white noise analysis in Refs. 25,27, the gamma white noise calculus inRefs. 27,35,16 and general cases in Refs. 7,28,34,19). The main tools in this theory arethe Fock expansion of functionals in which the wick tensor powers and S-transformare crucial.

It is well-known that Boson Fock space is adequate for chaotic representationof the Gaussian and Poisson white noise spaces; however, this fails in general Levywhite noise spaces. The natural question arises: which Fock space structure servesfor the analogs of the Weiner-Ito-Segal isomorphism in non Gaussian-Poisson case.This question is the main motivation of the present paper. In an attempt to find amathematically rigorous answer for this problem, some analogs of the Fock expan-sion involving the so-called extended Fock spaces were established, especially for thegamma processes (see Refs. 16,26,27). However, to our opinion, these developmentsare rather complicated and need to be expanded in a sample setting. In the presentwork, to overcome these difficulties, we propose a new and intrinsic method for con-structing a class of one-mode type interacting Fock spaces by a simple embeddingargument starting from the natural class of one-particle Levy-Meixner interact-ing Fock spaces. Our approach is motivated by the original machinery proposedby Gel’fand et al. 21 where it is suggested to construct a white noise measure ininfinite-dimensional space by lifting a one dimensional infinitely divisible distribu-tion. The construction presented here have the advantage to be directly connectedto the Levy-Meixner distributions which enables us to prove the above mentionedresults on the chaotic representation and to obtain some new prospects.

Related results to our theory, but in a different context and with another scope,have been established in previous papers 11,12. In the paper 2, as applications ofthe above mentioned development, we focus on the quantum nonlinear white noisecalculus based on Levy-Meixner processes. Representations of some ?-Lie algebras,such as the Virasoro-Zamolodchikov-w∞ ?-Lie algebras, in the basis of higher powersof Levy-Meixner white noise operators will be in our considerations.

The outline of our paper is the following: In Section 2, by considering a class ofone-mode interacting Fock space ΓLM , 〈 ·, · 〉LM of Levy-Meixner type, we extend aderivative formula appearing in 〈 ·, · 〉LM to the symmetric n-particle spaces. Then,we introduce a class of one-mode type interacting Fock spaces ΓLM(H) naturally


White Noise Levy-Meixner Processes on one-Mode Type IFS 3

associated to the one-dimensional infinitely divisible distributions µr. In Section 3,for the sake of completeness, we discuss a family of Levy-Meixner field operatorsand we construct an associated Fourier transform in generalized joint eigenvectors.Then, we give a description of the elements of square integrable Levy-Meixner whitenoise functionals. For more details and other related results, we refer the interestedreader to Ref. 26 and references cited therein. In Section 4, we give an essence to ourapproach for defining the Levy-Meixner white noise functionals by expanding morefeatures of the white noise Levy-Meixner processes. Finally in Section 5, we con-struct the spaces of test and generalized functions and we give the interacting Fockexpansion of the white noise Levy-Meixner processes as well as there S-transforms.As application, for stochastic regularity, we give explicit form of the chaotic repre-sentations and a sharp estimate of the Donsker’s delta function for Levy-Meixnerprocesses.

2. A transfer principle from one-mode to one-mode type IFS

Let µr be an infinitely divisible distribution such that its Laplace transform is givenby

Lµr (x) =∫

Rextµr(dt) = erf(x), r > 0, (2.1)

where f is analytic in some neighborhood of zero with f(0) = 0. Suppose that thegenerating function of the orthogonal polynomials

(Pr,n

)∞n=0

with respect to µr hasthe following form :

Gµr (x, t) :=∞∑

n=0

tn

n!Pr,n(x) = exp

xρ(t)− rf(ρ(t))

, (2.2)

where ρ is analytic in some neighborhood of zero with ρ(0) = 0 and ρ′(0) = 1 andf is a Laplace exponent satisfying Eq. (2.1). By a multiplicative renormalizationprocedure (as in Ref. 9), a function Gµr (x, t) in the form (2.2) is a generatingfunction of orthogonal polynomials if and only if there exist a function Ψr withΨr(0) = 0 which is analytic in some neighborhood of zero, such that

rf(ρ(t) + ρ(s)

)− rf(ρ(t)

)− rf(ρ(s)

)= Ψr(st). (2.3)

We call the function Ψr interaction exponent of µr. In this case, there exist twoconstants β, γ ∈ R+ such that the family of polynomials (Pr,n)n≥0 satisfies therecurrence formula (see Ref. 24)

xPr,n(x) = Pr,n+1(x) + αr,nPr,n(x) + ωr,nPr,n−1(x), (2.4)

with

αr,n = rc1 + γn , ωr,n = n(rc2 + β(n− 1)), (2.5)

where c1 = f ′(0) and c2 = f ′′(0). The measure µr satisfying condition (2.3) willbe referred to as the Levy-Meixner distribution with parameter r. In general, thefunction Ψr only has two forms:



(1) Ψr = rc2t, for β = 0;

(2) Ψr = −rc2

βln(1− βt), for β > 0,

where, for r = 1, (1) includes the distributions of Gaussian and Poisson, and (2)is in the case of distributions of Meixner’s type including the gamma, Pascal andMeixner ones. For more details see Ref. 24.

By using Eqs. (2.1), (2.2) and (2.3), for t, s ≥ 0, we have

⟨Gµr (·, t), Gµr (·, s)

⟩L2(µr)

=∞∑

m=0

∞∑n=0

smtn

m!n!〈Pr,m, Pr,n〉L2(µr)

= exp− rf(ρ(t))− rf(ρ(s))

∫

Rex(ρ(t)+ρ(s))µr(dx)

= exp

rf(ρ(t) + ρ(s))− rf(ρ(t))− rf(ρ(s))

= expΨr(st)

=∞∑

n=0

(st)n

n!dn

dtn

∣∣∣t=0

exp(Ψr(t))

.

By comparing the coefficients of smtn, we obtain

〈Pr,m, Pr,n〉L2(µr) = n!dn

dtn

∣∣∣t=0exp(Ψr(t))δn,m =: λr,nδn,m. (2.6)

For such a sequence λr = λr,nn∈N we define a Hilbert space ΓLM by

ΓLM =−→z = (zn)n≥0 ; zn ∈ C,

∞∑n=0

λr,n|zn|2 < ∞

with norm ‖ · ‖r given by

‖−→z ‖2r =∞∑

n=0

n!dn

dtn

∣∣∣t=0exp(Ψr(t))|zn|2.

If −→z = (zn)n∈N and −→ω = (ωn)n∈N the scalar product on ΓLM is given by

〈−→z ,−→ω 〉LM =∞∑

n=0

n!dn

dtn

∣∣∣t=0exp(Ψr(tzω)). (2.7)

The space ΓLM is nothing than the one-mode interacting Fock space associated withthe sequence λr. ΓLM will be referred to as the Levy-Meixner one-mode interactingFock space. It is well-known that ΓLM and L2(R, µr) are unitary isomorphics throughAccardi-Bozejko isomorphism (see Ref. 3). For later use we recall this formalism interms of Jacobi matrix and spectral measure. In the next section we shall study aninfinite-dimensional analogous of this isomorphism.



Define the infinite matrix J (r) = (J (r)n,m)∞n,m=0 with

J (r)n,n = αr,n, J

(r)n,n−1 = J

(r)n−1,n =

√wr,n, n ∈ N,

and J(r)n,m = 0 for |n−m| > 1, where αr,n, wr,n are Szego-Jacobi parameters asso-

ciated to µr (see Eq. (2.5)). We denote by ΓLM,fin the dense subset of ΓLM consistingof all finite vectors, i.e.,

ΓLM,fin :=

(zn)n∈N : ∃m ∈ N such that zn = 0 for all n ≥ m

.

Each Jacobi matrix J (r) determines a linear symmetric operator in ΓLM with domainΓLM,fin by the following formula

J (r)en =√

wr,n+1en+1 + αr,nen +√

wr,nen−1, n ∈ N, (2.8)

where en = (0, · · · , 0, 1, 0, · · · ) with 1 in the (n + 1)-th entry and e−1 = 0. Wedenote by J (r) the closure of J (r), which evidently exists since the operator J (r) issymmetric. Then, there exists a unique unitary operator

U (r)LM

: ΓLM −→ L2(R, µr) (2.9)

such that U (r)LM

e0 = 1 and, under U (r)LM

, the operator J (r) corresponds to the multi-plication operator by x on L2(R, µr), i.e.,

(U (r)

LMJ (r) (U (r)

LM)−1h

)(x) = xh(x),

h ∈ U (r)LM

(Dom(J (r))) =

g ∈ L2(R, µr) ;∫

Rx2g2(x)µr(dx) < ∞

.

Here the spectral measure of the Jacobi matrix J (r) is the measure µr.From now on, we suppose that β > 0. Then, from Eq. (2.4), one can easily

obtain

‖Pr,n‖2L2(µr) = βnn!(

rc2

β

)

n

,

where, for a real number $, the Bochhammer symbol ($)n is defined by

($)0 = 1 and ($)n = $($ + 1) · · · ($ + n− 1) =Γ($ + n)

Γ($), n ≥ 1.

Set

Qr,n(x) =[βnn!

(rc2

β

)

n

]−1/2

Pr,n(x), (2.10)

then we get an orthonormal basis Qr,n; n = 0, 1, 2, · · · for H := L2(R, µr).It is well-known (see Ref. 10) that the family of polynomials Pr,n∞n=0 satisfies

the second order difference equation

(γ − 1)x∆+∆−Pr,n(x) + 2(x− rc1)∆+Pr,n(x)− 2nPr,n(x) = 0,



where ∆+ and ∆− are, respectively, the forward difference and the backward dif-ference of h at x :

∆+h(x) = h(x + 1)− h(x) , ∆−h(x) = h(x)− h(x− 1).

This motivate the introduction of the operator Ar defined by

Ar = 1 + (γ − 1)x∆+∆− + 2(x− rc1)∆+

and diagonalized by Qr,n :

ArQr,n = υnQr,n, with υn = 2n + 1, n = 0, 1, 2, · · · (2.11)

Now, for each p ∈ R, define a norm | · |p on H by

|h|p = |Apr h|0 =

( ∞∑n=0

υ2pn 〈h,Qr,n〉2

)1/2

, h ∈ H,

where | · |0 and 〈 · , · 〉 are , respectively, the norm and the inner product of H. Forp ≥ 0, let Ep be the Hilbert space consisting of all h ∈ H with |h|p < ∞, and let E−p

be the completion of H with respect to | · |−p. Since A−1r is of Hilbert-Schmidt type,

identifying H with its dual space we come to the real standard nuclear Gel’fandtriple (see Ref. 38)

E :=⋂

p≥0

Ep ⊂ H ⊂⋃

p≥0

E−p =: E ′.

Finally, by taking complexification, we get a complex Gel’fand triple:

EC ⊂ H ⊂ E ′C,where EC, H and E ′C are the complexifications of E , H and E ′, respectively. Thecanonical C-bilinear form on E ′C×EC which is compatible with the inner product ofH is denoted by 〈 · , · 〉 again.

Denote by Γ (n)LM

(H) the n-th symmetric tensor power of H, i.e.,

Γ (0)LM

(H) := C and Γ (n)LM

(H) := Hb⊗n, n ≥ 1.

From 17 ( see also 22), for smooth function h, we have

dn

dtneh(t) =

∑

i1+2i2+···+kik=n

n!i1! · · · ik!

(h(1)(t)1!

)i1 · · ·(h(k)(t)

k!

)ik

eh(t). (2.12)

Put h(t) = Ψr(t) in (2.12), direct computation gives :

dn

dtn

∣∣∣t=0

exp

(〈Ψr(tφψ)〉r

)

=∑

i1+2i2+···+kik=n

n!βn(rc2)i1+···+ik〈φ, ψ〉i1 · · · 〈φk, ψk〉ik

i1! · · · ik!βi1(2β)i2 · · · (kβ)ik, φ, ψ ∈ H, (2.13)



where we used the notation 〈φ〉r :=∫

Rφ(x)µr(dx). Hence, by a simple embedding

argument, the sesquilinear form in the right hand side of (2.13), can be showed tobe positive-definite. Motivated by (2.7), where a scalar product is proposed for theLevy-Meixner one-particle space, we introduce in the following its natural n-particlecounterpart.

Definition 2.1. The Levy-Meixner n-particle scalar product on Γ (n)LM

(H) is definedby⟨φ⊗n, ψ⊗n

⟩LM,n

=∑

i1+2i2+···+kik=n

n!βn(rc2)i1+···+ik〈φ, ψ〉i1 · · · 〈φk, ψk〉ik

i1! · · · ik!βi1(2β)i2 · · · (kβ)ik. (2.14)

The following recurrence identity is important for the proofs of technical lemmasbelow.

Theorem 2.1. The family of scalar products〈 ·, · 〉LM,n

n∈N in (2.14) satisfies the

recurrence relation:

〈φ⊗n, ψ⊗n〉LM,n =n−1∑

k=0

(n− 1

k

)〈φ⊗k, ψ⊗k〉

LM,kΘ(n−k)

r (0), (2.15)

with

Θr(t) = 〈Ψr(tφψ)〉r, φ, ψ ∈ H.

Proof. From (2.13) and (2.14), we have

Fr(t) := exp (Θr(t)) =∞∑

n=0

tn

n!〈φ⊗n, ψ⊗n〉LM,n .

On the other hand

F ′r(t) =∞∑

n=0

tn−1

(n− 1)!〈φ⊗n, ψ⊗n〉LM,n

and

F ′r(t) =

( ∞∑n=0

tn

n!〈φ⊗n, ψ⊗n〉LM,n

)( ∞∑n=0

tn

n!Θ(n+1)

r (0)

)

=∞∑

n=0

n∑

k=0

(n

k

)Θ(n−k+1)

r (0)〈φ⊗k, ψ⊗k〉LM,k

tn

n!

=∞∑

n=1

n−1∑

k=0

(n− 1

k

)Θ(n−k)

r (0)〈φ⊗k, ψ⊗k〉LM,k

tn−1

(n− 1)!.

Then we deduce

〈φ⊗n, ψ⊗n〉LM,n =n−1∑

k=0

(n− 1

k

)〈φ⊗k, ψ⊗k〉LM,k

Θ(n−k)r (0),



as desired. ¤

Let be given w ∈ E ′. The k-linear function

(φ1, φ2, · · · , φk) 7−→ 〈w, φ1φ2 · · ·φk〉is continuous on Ek. Then, by the kernel theorem there exists a unique elementτk(w) ∈ E ′b⊗k such that

〈τk(w), φ1⊗φ2⊗ · · · ⊗φk〉 = 〈w, φ1φ2 · · ·φk〉.τk(w) is called the trace of order k with kernel w. In the scalar case w = 1, we writeτk = τk(1) for short.

Definition 2.2. The generalized Levy-Meixner trace of order n, Υn ∈ E ′b⊗n, isdefined recursively by

Υ0 = 1, Υn =n−1∑

k=0

(n− 1

k

)Υk⊗Ψ(n−k)

r (0)τn−k, n = 1, 2, · · · (2.16)

In view of (2.15) and (2.16), we have

〈φ⊗n, ψ⊗n〉LM,n = 〈Υn, (φψ)⊗n〉.Then, by a density imbedding arguments, for arbitrary φ(n), ψ(n) ∈ Γ (n)

LM(H), the

inner product 〈 · , · 〉LM,n is given by

〈φ(n), ψ(n)〉LM,n = 〈Υn, φ(n)ψ(n)〉. (2.17)

Definition 2.3. The Levy-Meixner one-mode type interacting Fock space ΓLM(H)is defined as the direct orthogonal Hilbert sum of

Γ (n)

LM(H), 〈·, ·〉LM,n

n∈N:

ΓLM(H) :=+∞⊕n=0

n! Γ (n)LM

(H). (2.18)

Let Γfin(E) be the algebraic direct sum of the spaces E b⊗nC . Observe that Γfin(E)

is dense in ΓLM(H). Moreover, if Γ∗fin(E) denote the dual of Γfin(E) with respect tothe zero space ΓLM(H), we have the Gel’fand triple

Γfin(E) ⊂ ΓLM(H) ⊂ Γ∗fin(E).

The space Γ∗fin(E) consists of infinite sequences Φ = (Φ(n))∞n=0, where Φ(n) ∈ E ′b⊗n

and its action on ϕ = (ϕ(n))∞n=0 ∈ Γfin(E) is given by

〈〈Φ, ϕ〉〉 =∞∑

n=0

n!⟨Φ(n), ϕ(n)

⟩,

where 〈 · , · 〉 is the canonical C-bilinear form on E ′b⊗nC × E b⊗n

C .



3. Levy-Meixner White Noise Functionals

In this Section, we will define a family of Levy-Meixner field operators and weconstruct an associated Fourier transform in generalized joint eigenvectors. Forself-completeness, results are given with there proofs.

For each φ ∈ E , let a†φ denote the standard creation operator acting on Γfin(E)by

a†φϕ(n) := φ⊗ϕ(n) , ϕ(n) ∈ E b⊗n , n ∈ N. (3.1)

From the obvious inequality∣∣∣φ⊗ϕ(n)

∣∣∣p≤ |φ|p

∣∣∣ϕ(n)∣∣∣p

,

a†φ is a densely defined continuous linear operator on ΓLM(H).

Lemma 3.1. The adjoint operator, aφ, of a†φ in ΓLM(H) is given by

aφϕ⊗n = nc2〈φ, ϕ〉ϕ⊗(n−1) + n(n− 1)β(φϕ2)⊗ϕ⊗(n−2).

Proof. First, we have the following much simpler recursion formulas

Υn+1 = c2Υn⊗τ1 + nβΥn⊗1τ2, n = 1, 2, · · · , (3.2)

where Υn⊗1τ2 ∈ E ′b⊗(n+1) defined by

〈Υn⊗1τ2, ξ⊗(n+1)〉 = 〈Υn, ξ2⊗ξ⊗(n−1)〉, ξ ∈ E .

By virtues of (2.18), (3.1) and (3.2), we have for arbitrary φ, ϕ, ψ ∈ E ,⟨a†φϕ⊗n, ψ⊗(n+1)

⟩ΓLM (H)

= (n + 1)!⟨Υn+1, (φψ)⊗(ϕψ)⊗n

⟩

= (n + 1)!⟨c2Υn⊗τ1 + nβΥn⊗1τ2, (φψ)⊗(ϕψ)⊗n

⟩

= (n + 1)![c2〈φ, ψ〉〈Υn, (ϕψ)⊗n〉+ nβ〈Υn, (φϕψ2)⊗(ϕψ)⊗(n−1)〉

]

=⟨ϕ⊗n, (n + 1)c2 〈φ, ψ〉ψ⊗n + n(n + 1)β(φψ2)⊗ψ⊗(n−1)

⟩ΓLM (H)

.

Thus aφ is defined on Γfin(E) by

aφϕ⊗n = nc2〈φ, ϕ〉ϕ⊗(n−1) + n(n− 1)β(φϕ2)⊗ϕ⊗(n−2).

¤

Finally, we define on Γfin(E) the preservation operator nφ at φ ∈ E by

nφϕ⊗n = n(φϕ)⊗ϕ⊗(n−1) , ϕ ∈ E .

Now, we are ready to define the Levy-Meixner field operators, Jφ, on Γfin(E) :

Jφ := a†φ + aφ + γnφ + c1〈φ〉r I,



where I denote the identity operator.Note that Jφ, φ ∈ E , are symmetric operators. Furthermore, if we denote Jφ

the closure of Jφ, one can show that Jφ, φ ∈ E are self-adjoint operators 26. Thus,by construction, the family of operators Jφ; φ ∈ E has a Jacobi field structure inthe interacting Fock space ΓLM(H). This family will be called Levy-Meixner-Jacobifield. For more details on the notion of Jacobi field, see 15,18,14,13.

Lemma 3.2. Jφ; φ ∈ E is a family of commuting operators in ΓLM

(H).

Proof. Any operators a†φ1and a†φ2

evidently commute. Hence, their adjoints aφ1

and aφ2 also commute. Next, the preservation operators nφ1 and nφ2 commute. Onthe other hand we have

aφ

(ϕ1⊗ϕ⊗n

2

)= 2nβ(φϕ1ϕ2)⊗ϕ

⊗(n−1)2 + n(n− 1)βϕ1⊗(φϕ2

2)⊗ϕ⊗(n−2)2

+c2〈φ, ϕ1〉ϕ⊗n2 + nc2〈φ, ϕ2〉ϕ1⊗ϕ

⊗(n−1)2

and

a†φ(ϕ1⊗ϕ⊗n

2

)= φ⊗ϕ1⊗ϕ⊗n

2 .

Then we obtain

(a†φ1

aφ2 + aφ1a†φ2

)ϕ⊗n

= nc2〈φ2, ϕ〉φ1⊗ϕ⊗(n−1) + n(n− 1)βφ1⊗(φ2ϕ2)⊗ϕ⊗(n−2)

+ c2〈φ1, φ2〉ϕ⊗n + nc2〈φ1, ϕ〉φ2⊗ϕ⊗(n−1)

+ 2nβ(φ1φ2ϕ)⊗ϕ⊗(n−1) + n(n− 1)βφ2⊗(φ1ϕ2)⊗ϕ⊗(n−2) (3.3)

which is symmetric in φ1 and φ2. Thus, arbitrary Jφ1 and Jφ2 commute as desired.¤

As it is expected in the transfer principle in the construction of the one-modetype interacting Fock space ΓLM(H) from the one-mode interacting Fock space ΓLM

(see section 2), we shall construct a white noise L2-space unitary isomorphic toΓLM(H) in term of an infinite-dimensional analog of Accardi-Bozejko isomorphismas in (2.9).

Theorem 3.1. (Levy-Meixner isometry)There exist a unique probability measure ΛLM on (E ′,B(E ′)) and a unique unitaryisomorphism

ULM : ΓLM(H) −→ L2(E ′,B(E ′), ΛLM)



defined on the dense set Γfin(E) as follows

Γfin(E) 3 ϕ =(ϕ(n)

)∞n=0

7−→ ULM ϕ = (ULM ϕ) (z) =+∞∑n=0

⟨: z⊗n :LM , ϕ(n)

⟩, (3.4)

where, for z ∈ E ′, : z⊗n :LM

is the continuous linear functional in E ′b⊗n, given recur-sively by

⟨: z⊗(n+1) :

LM, φ⊗(n+1)

⟩:= 〈z, φ〉〈: z⊗n :

LM, φ⊗n〉

− ⟨: z⊗n :

LM, γn(φ2)⊗φ⊗(n−1) + c1 〈φ〉φ⊗n

⟩

− ⟨: z⊗(n−1) :LM , nc2

⟨φ2

⟩φ⊗(n−1) + n(n− 1)β(φ3)⊗φ⊗(n−2)

⟩ (3.5)

with the conventions

: z⊗(−1) :LM= 0 , : z⊗0 :LM= 1.

Under the isomorphism ULM , each operator Jφ, φ ∈ E, goes over into the operatorof multiplication by the monomial 〈 · , φ〉 in L2(E ′,B(E ′),ΛLM), i.e.,

ULM Jφ U−1LM

= M〈 · ,φ〉.

The space L2(E ′,B(E ′), ΛLM) will be denoted (L2)LM for short.

Proof. As easily seen, for any φ ∈ E and n ∈ N, the operators a†φ, nφ and aφ act

continuously from E b⊗nC into E b⊗(n+1)

C , E b⊗nC and E b⊗(n−1)

C , respectively. Therefore, forany φ ∈ E , Jφ acts continuously on Γfin(E). Furthermore, for any fixed ϕ ∈ Γfin(E),the mapping E 3 φ 7−→ Jφϕ ∈ Γfin(E) is linear and continuous. Finally, the vacuumΩ is evidently a cyclic vector for the operators a†φ, φ ∈ E in Γfin(E), and hencein ΓLM(H). Then, by using the Jacobi field structure of Jφ, we can easily showthat Ω is a cyclic vector for Jφ, φ ∈ E , in ΓLM(H). Thus, we deduce the existenceof a unique probability measure ΛLM on (E ′,B(E ′)) and a unique unitary operatorULM : ΓLM(H) −→ (L2)LM such that, for each φ ∈ E , under ULM , Jφ goes over intothe operator of multiplication by the function 〈 · , φ〉 and ULM Ω = 1.

Now, for ΛLM-a.e. z ∈ E ′, there exists a generalized joint vector

Φz = (Φ(n)z )

n∈N ∈ Γ∗fin(E)

of the family Jφ; φ ∈ E :

〈〈Φz,Jφϕ〉〉 = 〈z, φ〉〈〈Φz, ϕ〉〉 , ϕ ∈ Γfin(E), (3.6)

and, for each ϕ = (ϕ(n))n∈N ∈ Γfin(E), the action of ULM

on ϕ is given by

(ULM

ϕ)(z) = 〈〈Φz, ϕ〉〉 =∞∑

n=0

n!⟨Φ(n)

z , ϕ(n)⟩

, z ∈ E ′. (3.7)

For ΛLM -a.e. z ∈ E ′, put

Ξn(z) := n!Φ(n)z ,



which is an element of E ′b⊗n. By (3.6) and (3.7), for ϕ = (0, · · · , 0, φ⊗n, 0, · · · ) ∈Γfin(E), we have

〈z, φ〉〈Ξn(z), φ⊗n〉 =⟨Ξn+1(z), φ⊗(n+1)

⟩

+⟨Ξn(z), γn(φ2)⊗φ⊗(n−1) + c1 〈φ〉φ⊗n

⟩

+⟨Ξn−1, nc2

⟨φ2

⟩φ⊗(n−1) + n(n− 1)β(φ3)⊗φ⊗(n−2)

⟩.

Hence, the Ξn(z)’s are given by the recurrence relation (3.5). Then, we concludethat, for any n ∈ N, Ξn(z) = : z⊗n :LM and (3.4) follows. This complete the proof.¤

Definition 3.1. The probability measure ΛLM on E ′, highlighted in the abovetheorem is called the Levy-Meixner white noise measure. The probability space(E ′,B(E ′),ΛLM) is called the Levy-Meixner white noise space.

4. The White Noise Levy-Meixner Process

In the present Section, our aim is to make the Levy-Meixner white noise measure ΛLM

more useful by giving explicitly its Fourier transform and its generating function.The major step in the proof of these facts is the following lemma.

Lemma 4.1. Let Bb(R) denote the set of all bounded Borel sets on R, and for D ∈Bb(R) let χ

Ddenote the indicator of D. Then the distribution of the random variable

〈 · , χD〉 is the distribution µ|D|r with parameter |D|r := µr(D) =

∫R χ

D(x)µr(dx).

Proof. Let JχD

be the operator in ΓLM(H) which, under the unitary ULM , goes overinto the operator of multiplication by the function

〈z, χD〉 = 〈: z⊗1 :LM , χ

D〉+ c1|D|r = ULM (χ

D)(z) + c1|D|r .

For each n ∈ N, the vectors χ⊗n

D, with χ⊗0

D:= Ω, belongs to ΓLM(H). Let XD stand

for the subset of ΓLM(H) spanned by these vectors. We consider an approximatingsequence ϕn ∈ E ; n ∈ N of χ

D, in such away ϕn(x) → χ

D(x) as n → ∞ in

L2(R, µr). By using the definition of ΓLM(H), Jφ and the dominated convergencetheorem, we get

JχD

χ⊗n

D= χ⊗(n+1)

D+ (γn + c1|D|r )χ⊗n

D+ n (β(n− 1) + c2|D|r ) χ⊗(n−1)

D. (4.1)

Therefore, XD

is an invariant subspace for the operator JχD

. Let JD

stand for

the restriction of JχD

to XD

. Then JχD

is a densely defined operator in XD

. Letηn,D = ‖χ⊗n

D‖ΓLM (H), then the vectors

(ε(n)

D

)n∈N, with ε(n)

D= η−1

n,Dχ⊗n

D, constitute



an orthonormal basis for XD

. (4.1) yields

JD

ε(n)D

=ηn+1,D

ηn,D

ε(n+1)D

+(nγ + c1|D|r

)ε(n)

D

+ n(β(n− 1) + c2|D|r

) ηn−1,D

ηn,D

ε(n−1)D

.

(4.2)

The matrix of the operator JD

in the orthonormal basis(ε(n)

D

)n∈N must be symmet-

ric. Thus, together with the formula (4.2), this implies that this matrix is a Jacobione with recurrence formula :

JD

ε(n)D

= αn+1,D

ε(n+1)D

+ wn,D

ε(n)D

+ wn,D

ε(n−1)D

. (4.3)

By comparing (4.2) and (4.3) we obtain the equalityη

n,D

ηn−1,D

= n(β(n− 1) + c2|D|r

) ηn−1,D

ηn,D

,

from which we haveη

n,D

ηn−1,D

=√

n(β(n− 1) + c2|D|r ). (4.4)

Substituting (4.4) into (4.2), we conclude that the coefficients wn,D and αn,D in(4.3) can be rewritten as

wn,D

=√

n(β(n− 1) + c2|D|r

), α

n,D= γn + c1|D|r .

Then there exist a unique probability measure µ|D|r on R such that the system of

polynomials(P|D|r ,n

)n∈N

, defined by the recurrence formula

P|D|r ,n+1(x) = (x− αn,D )P|D|r ,n(x)− w2

n,DP|D|r ,n−1(x), (4.5)

forms an orthonormal basis for L2(R, µ|D|r ). Denote JD the closure in XD of theessentially self-adjoint operator JD . Then, under the unitary operator UD : XD −→L2(R, µ|D|r ) given by U

Dε(n)

D:= P|D|r ,n

, JD

goes over into the operator of multi-

plication by x. On the other hand, we know from theorem 3.1 that ULM JχD

ULM−1

coincides with the operator of multiplication by 〈z, χD〉. Hence, (4.3) and (4.5)

imply that

(ULM ε(n)D

)(z) = P|D|r ,n

(〈z, χ

D〉)

.

Therefore, µ|D|r coincides with the distribution of 〈 · , χD〉 regarded as a random

variable in the probability space (E ′,B(E ′), ΛLM). ¤

Theorem 4.1. The Laplace transform of ΛLM is given by the following formula∫

E′eu〈z,φ〉dΛLM(z) = expr〈f(uφ)〉r , φ ∈ E , u ∈ R. (4.6)



Proof. Let D ∈ Bb(R). For u ∈ R, by using lemma 4.1, we get∫

E′eu〈z,χ

D〉dΛLM(z) = exp |D|rf(u)

= exp∫

Rf(uχ

D(x))µr(dx)

.

Let be given an n disjoint sets D1, · · · , Dn ∈ Bb(R). Then, the random variables〈 · , χD1

〉, · · · , 〈 · , χDn〉 are independent. Hence, for any step function ϕ =

∑

i

uiχDi,

formula (4.6) holds. Now, for any ϕ ∈ E , choose any sequence of step functionsϕnn∈N such that ϕn converges uniformly to ϕ as n → ∞. Then, by the domi-nated convergence theorem, we deduce that the left-hand side of (4.6) converges tothe right-hand side of (4.6). On the other hand, we can easily show that 〈 · , ϕn〉converges to 〈 · , ϕ〉 in (L2)LM , and therefore also in probability. Hence, again by thedominated convergence theorem, the left-hand side of (4.6) with ϕ = ϕn convergesto the right-hand side of (4.6). This completes the proof of the theorem. ¤

The following is the Levy-Meixner analog of the bosonic wick tensor powers andHermite polynomials.

Corollary 4.1. For any D ∈ Bb(R), we have⟨: z⊗n :LM , χ⊗n

D

⟩= P|D|r ,n

(〈z, χ

D〉)

, ΛLM − a.e. z ∈ E ′ , (4.7)

where(P|D|r ,n

)∞n=0

are the polynomials on R defined by the recurrence relation

(4.5).

Proof. The statement follows from the proof of theorem 3.1. ¤

Remark 4.1. For t ∈ R∗+, setting in (4.6) ϕ = ` χ[0,t] and put µr([0, t]) =: |t|r , weobtain ∫

E′e〈z,ϕ〉dΛLM(z) = exp |t|rf(`) , ` ∈ R.

By Eq. (2.1), one can see that this function coincides with the Laplace transformof the measure µ|t|r . On the probability space (E ′,B(E ′), ΛLM), the random variableXt := 〈 · , χ[0,t]〉 has a Levy-Meixner distribution with parameter |t|r . So the familyof random variables

X = Xt ; t ∈ R, X0 = 0

is called a Levy-Meixner process. The image of the Levy-Meixner white noise mea-sure ΛLM under the random variable Xt(·) is the Levy-Meixner distribution µ|t|r .Then, for t > s > 0, the characteristic function E [expi`(Xt −Xs)] of Xt − Xs

has the following form :

E [expi`(Xt −Xs)] = exp [|t− s|rf(i`)] . (4.8)



We conclude that for s, t ∈ R the random variable Xt −Xs has the Levy-Meixnerdistribution with parameter |t− s|

r. Denote by 〈 · , φ 〉 the L2-limit of (〈 · , φn 〉)n∈N

with φn ∈ E . When φ = χ(s,t) (the indicator of (s, t)), the characteristic function of〈 · , φ 〉 is exactly the same as the one in (4.8). So the Levy-Meixner process X on(E ′,B(E ′), ΛLM) can be represented by

X(t; z) =

〈 z , χ(0,t) 〉 if t ≥ 0

−〈 z , χ(t,0) 〉 if t < 0 , z ∈ E ′

Taking the time derivative formally, we get

X(t; z) = z(t), z ∈ E ′.

Thus, the elements of E ′ can be regarded as the sample paths of the Levy-Meixnerwhite noise, and members of (L2)LM are called Levy-Meixner white noise functionals.

We recall that in one-dimensional case 9, the generating function Gµ|D|rof the

sequence of polynomials(P|D|r ,n

)∞n=0

characterized by (4.5) is given by

Gµ|D|r(t; x) =

∞∑n=0

1n!

tnP|D|r ,n(x) = exp

xρ(t)− |D|rf(ρ(t))

. (4.9)

According to (4.7), as infinite dimensional analogue of (4.9), we shall define thegenerating function of the polynomials 〈: z⊗n :LM , φ⊗n〉 by

: e〈z,φ〉 :LM=∞∑

n=0

1n!〈: z⊗n :LM , φ⊗n〉 , z ∈ E ′, φ ∈ E .

Lemma 4.2. For any k1, · · · , km ∈ N such that k1 + · · ·+ km = n, any z ∈ E ′ andany disjoint D1, · · · , Dm ∈ Bb(R), we have

⟨: z⊗n :LM , χ⊗k1

D1⊗ · · · ⊗χ⊗km

Dm

⟩=

m∏

i=1

P|Di|r ,ki( 〈z, χDi

〉 ). (4.10)

Proof. We prove this lemma by induction in n ∈ N. For n = 1, formula (4.10) triv-ially holds. Now, suppose that (4.10) holds for any n ≤ N . Let k1, · · · , km ∈ N withk1 + · · ·+ km = N + 1. Applying the recurrence formula (3.5), for any 1 ≤ j ≤ m,direct calculus gives



⟨: z⊗(N+1) :LM , χ⊗k1

D1⊗ · · · ⊗χ⊗km

Dm

⟩

=〈z, χ

Dj〉 −

(γN + c1|Dj |r

)

×⟨: z⊗N :LM , χ⊗k1

D1⊗ · · · ⊗χ⊗(kj−1)

Dj⊗ · · · ⊗χ⊗km

Dm

⟩

− N(|Dj |r + β(N − 1)

)

×⟨: z⊗(N−1) :LM , χ⊗k1

D1⊗ · · · ⊗χ⊗(kj−2)

Dj⊗ · · · ⊗χ⊗km

Dm

⟩.

Applying formula (4.10) with n = N and n = N − 1 and then using the recurrencerelation (4.5) for the polynomials P|D|r ,n

, we conclude the statement. ¤

Theorem 4.2. For each φ ∈ E satisfying supx∈R

|φ(x)| = ‖φ‖∞ < 1, we have

: e〈z,φ〉 :LM= exp〈z, ρ(φ)〉 − r〈f(ρ(φ))〉r

. (4.11)

Proof. Fix D ∈ Bb(R), by using the equality (4.7) we obtain∞∑

n=0

1n!

⟨: z⊗n :LM , (uχ

D)⊗n

⟩=

∞∑n=0

un

n!P|D|r ,n

(〈z, χD〉).

Hence, it follows from (4.9) that (4.11) holds with φ = uχD

and |u| < 1 from aneighborhood of zero in R.

Now, for any disjoint D1, · · · , Dm ∈ Bb(R) and any u1, · · · , um ∈ R satisfying

|ui| < 1, i = 1, · · · ,m, put g :=m∑

i=1

uiχDi. By lemma 4.2, we get

∞∑n=0

1n!

⟨: z⊗n :LM , g⊗n

⟩=

m∏

i=1

( ∞∑n=0

uni

n!P|Di|r ,n

(〈z, χDi〉)

).

Hence, (4.11) holds with φ = g. Finally, fix any φ ∈ E . Choose any sequence ofstep functions φnn∈N such that φn converges uniformly in the L2-sense to φ asn →∞. Then, by the dominated convergence theorem, we conclude the statement.¤

Remark 4.2. In the one-dimensional setting, the Laplace transform

Lµ|D|r(u) :=

∫

Reuxdµ|D|r (x) = exp (|D|rf(u))

of the measure µ|D|r is well defined and holomorphic on a neighborhood of zero inR. Furthermore, for all x ∈ R and u from a neighborhood of zero, by using (4.9),we have

Gµ|D|r(x, u) =

exp(xρ(u))Lµ|D|r

(ρ(u)). (4.12)



Now for the Levy-Meixner white noise measure ΛLM , LΛLM(φ) =

∫E′ e

〈z,φ〉dΛLM(z) iswell defined and holomorphic on some neighborhood of zero O in EC. If we considera holomorphic EC-valued function ρ defined on O, we get the infinite dimensionalcounterpart of (4.12):

: e〈z,φ〉 :LM=exp (〈z, ρ(φ)〉)LΛLM

(ρ(φ)). (4.13)

Definition 4.1. The S-transform of F ∈ (L2)LM at φ ∈ EC is defined as the innerproduct

SF (φ) :=⟨F, : e〈·,φ〉 :LM

⟩(L2)LM

.

Let us remember, that a generalized hypergeometric series with k numeratorparameters a1, · · ·, ak and s denominator parameters b1, · · ·, bs is defined by

kFs(a1, · · ·, ak; b1, · · · , bs; z) =∞∑

n=0

(a1)n · · · (ak)n

(b1)n · · · (bs)n

zn

n!.

Theorem 4.3. The Levy-Meixner process (Xt) has the following chaotic represen-tation

Xt = |t|r (c1 + c2)+∞∑n=0

[βnn!

( |t|rc2

β

)n

]−1 ⟨: ·⊗n :LM , χ⊗n

[0,t]

⟩. (4.14)

Moreover, it’s S-transform is given by

S(Xt)(φ) = |t|r(c1 + c2) 1F2

(−;−,

|t|rc2

β;β−1〈φ〉r,t

),

where 〈φ〉r,t =∫ t

0

φ(s)µ|t|r (ds).

Proof . From Section 2 we recall that the family

ζ|t|r ,n =[βnn!

( |t|rc2

β

)n

]− 12

P|t|r ,n; n = 0, 1, 2, · · ·

forms an orthonormal basis for the Hilbert space L2(µ|t|r ). Then the expansion ofthe function x is given by

x =∞∑

n=0

〈x, ζ|t|r ,n〉ζ|t|r ,n(x)

=∞∑

n=0

[βnn!

( |t|rc2

β

)n

]−1 ⟨x, P|t|r ,n

⟩P|t|r ,n(x).

On the other hand, Eq. (2.4) gives :

P|t|r ,1(x) = x− |t|rc1,



then

⟨x, P|t|r ,n

⟩=

⟨P|t|r ,1 + |t|rc1, P|t|r ,n

⟩= |t|r (c1 + c2).

Thus

x = |t|r(c1 + c2)

∞∑n=0

[βnn!

( |t|rc2

β

)n

]−1

P|t|r ,n(x). (4.15)

From corollary 4.1, we have⟨: ·⊗n :LM , χ⊗n

[0,t]

⟩= P|t|r ,n(Xt). (4.16)

Thus, Eqs. (4.15) and (4.16) yield

Xt = |t|r (c1 + c2)+∞∑n=0

[βnn!

( |t|rc2

β

)n

]−1

P|t|r ,n

(〈·, χ[0,t]〉)

= |t|r (c1 + c2)+∞∑n=0

[βnn!

( |t|rc2

β

)n

]−1 ⟨: ·⊗n :LM , χ⊗n

[0,t]

⟩.

Therefore, one can see that the isometry ULM guaranteed that any F ∈ (L2)LM havethe following form

F =∞∑

n=0

〈: ·⊗n :LM , Fn〉, Fn ∈ Hb⊗n.

Then, we conclude that the S-transform of F can be given by

S(F )(φ) =∞∑

n=0

〈Fn, φ⊗n〉, φ ∈ EC. (4.17)

Finally, Eqs. (4.14) and (4.17) yield

S(Xt)(φ) = |t|r (c1 + c2)∞∑

n=0

[βnn!

( |t|rc2

β

)n

]−1

〈φ⊗n, χ⊗n[0,t]〉

= |t|r (c1 + c2)∞∑

n=0

( |t|rc2

β

)−1

n

(β−1〈φ〉r,t

)n

n!

= |t|r (c1 + c2) 1F2

(−;−,

|t|r c2β ; β−1〈φ〉r,t

).

¤



5. Donsker’s delta function for Levy-Meixner processes

A continuous, convex, increasing function θ : [0, +∞[−→ [0, +∞[ is called a Youngfunction if

θ(0) = 0 and limx→+∞

θ(x)x

= +∞.

Given a Young function θ, we define a weight sequences θn∞n=0 by

θn = infx>0

eθ(x)

xn, n = 0, 1, 2, · · · .

For simplicity of notation, we denote by

|fn|LM,p = |fn|Γ(n)LM (Ep)

, p ∈ Z.

Now, suppose a pair p ≥ 0,m > 0 is given. Then, for −→ϕ = (ϕn)∞n=0 with ϕn ∈ E b⊗np

we put

‖−→ϕ ‖2θ,p,m :=+∞∑n=0

θ−2n m−n

(rc2

β

)−1

n|ϕ(n)|2

LM,p.

Accordingly, we obtain a projective system of Hilbert spaces ΓLM(Ep,m, θ), p ≥0, m > 0, where

ΓLM(Ep,m, θ) =−→ϕ = (ϕn)∞n=0; ϕn ∈ E b⊗n

p , ‖−→ϕ ‖2θ,p,m < ∞

These are sometimes reffered to as weighted Fock space too. Finally, we define

ΓLM,θ(E) = proj lim

p→+∞;m↓0ΓLM(Ep,m, θ).

It is easily verified that ΓLM,θ(E) is a nuclear Frechet space.

From now on we assume that the Young function θ satisfies the following con-dition:

lim supx→∞

θ(x)x2

< ∞, (5.1)

or equivalently,

lim infx→∞

θ∗(x)x2

> 0, (5.2)

where θ∗ is the polar function of θ given by :

θ∗(x) = supt≥0

(tx− θ(t)), x > 0.

Proposition 5.1. If the Young function θ satisfies the condition (5.1), we haveΓ

LM,θ(E) ⊂ ΓLM(H), where the inclusion is continuous and has a dense image.

Proof. Let −→ϕ ∼ (ϕn)∞n=0 ∈ ΓLM,θ(E). By definition we have

‖−→ϕ ‖2ΓLM(H)

=∞∑

n=0

n!|ϕn|2Γ(n)LM (H)

=∞∑

n=0

(rc2

β

)nθ2

nmnn! θ−2n m−n

(rc2

β

)−1

n|ϕn|2

Γ(n)LM (H)

.(5.3)



On the other hand the condition (5.1) guarantees the existence of two constantnumbers a > 0 and b > 0 such that

θn ≤ a

(2be

n

)n/2

, (5.4)

then(rc2

β

)nθ2

nmnn! ≤ a2

(2be

n

)n [m

(rc2

β+ 1

)]n

n!nn = a2

[2bem

(rc2

β+ 1

)]n

n!.

Therefore, for m <

[2be

(rc2

β+ 1

)]−1

we have

K ≡ supn≥0

(rc2

β

)nθ2

nmnn! < ∞.

Thus (5.3) becomes

‖−→ϕ ‖2ΓLM(H)

≤ K

∞∑n=0

θ−2n m−n

(rc2

β

)−1

n|ϕn|2

Γ(n)LM (H)

= K‖−→ϕ ‖2θ,0,m ≤ K‖−→ϕ ‖2θ,p,m, p ≥ 0,

which means that ΓLM,θ(E) ⊂ ΓLM(H) and the inclusion is continuous. ¤

Now define FLM,θ(E) := ULM(ΓLM,θ

(E)). Then, in view of the Levy-Meixner isom-etry (see theorem 3.1), we have the following diagram :

ΓLM,θ(E) → ΓLM(H)↓ ↓

FLM,θ(E) → (L2)LM

(5.5)

The space of test functions FLM,θ(E) is the projective system of Hilbert space

FLM(Ep,m, θ); p ≥ 0, m > 0, where FLM(Ep, m, θ) is the set of function ϕ ofthe form

ϕ(w) =+∞∑n=0

〈: w⊗n :LM , ϕ(n)〉, −→ϕ = (ϕ(n)) ∈ ΓLM,θ(E)

such that

‖ϕ‖θ,p,m = ‖−→ϕ ‖θ,p,m < ∞.

Let FLM(E−p,m, θ) be the dual of FLM(Ep,m, θ) with respect to (L2)LM . One can seethat FLM(E−p,m, θ) is a Hilbert space with Hilbertian norm

‖Φ‖2θ,−p,m :=+∞∑n=0

(n!θn)2(rc2

β

)nmn|Φ(n)|2

LM,−p.

Then

F∗LM,θ

(E) ≡⋃

p≥0,m>0

FLM(E−p,m, θ),



endowed with the inductive limit topology. Next, the chaos decomposition of (L2)LM

induces the following natural decomposition for Φ ∈ F∗LM,θ

(E). For any Φ(n) ∈ E ′b⊗nC ,

there exists a unique distribution 〈: ·⊗n :LM , Φ(n)〉 in F∗LM,θ

(E), acting on smoothcontinuous polynomials ϕ = (ϕ(n))∞n=0 ∈ FLM,θ

(E) as⟨⟨〈: ·⊗n :LM ,Φ(n)〉, ϕ

⟩⟩:= n!〈Φ(n), ϕ(n)〉.

Any Φ ∈ F∗LM,θ

(E) then has a unique decomposition Φ =+∞∑n=0

〈: ·⊗n :LM , Φ(n)〉, where

the sum converges in F∗LM,θ

(E). It is verified that FLM,θ(E) and F∗

LM,θ(E) are mutually

dual spaces with respect to the canonical C-bilinear form on (L2)LM , and hence weobtain the nuclear triple :

FLM,θ(E) → (L2)LM → F∗

LM,θ(E). (5.6)

The C-bilinear form on F∗LM,θ

(E)×FLM,θ(E) is given by

〈〈Φ, ϕ〉〉 =+∞∑n=0

n!〈Φ(n), ϕ(n)〉, (5.7)

for any ϕ ∼ (ϕ(n))∞n=0 ∈ FLM,θ(E) and Φ ∼ (Φ(n))∞n=0 ∈ F∗LM,θ

(E). The space F∗LM,θ

(E)is referred to us the space of generalized functions.

Lemma 5.1. For any p ≥ 0 and φ ∈ Ep,C, there exist a constant Cp such that

|φ⊗n|2LM,p

≤ Cnp βn

(rc2

β

)n|φ⊗n|2p (5.8)

Proof. It is easy to see that for any p ≥ 0, there exist Cp ≥ 0 such that

|φk|2LM,p

= 〈φk, φk〉Γ

(k)LM (Ep)

≤ Ckp |φ⊗k|2p, ∀k ∈ N. (5.9)

Then, Eqs. (2.14) and (5.9) gives

|φ⊗n|2LM,p

=∑

i1+2i2+···+kik=n

n!βn(rc2)i1+···+ik〈φ, φ〉i1 · · · 〈φk, φk〉ik

i1! · · · ik!βi1(2β)i2 · · · (kβ)ik

≤ Cnp 〈φ⊗n, φ⊗n〉

∑

i1+2i2+···+kik=n

n!βn(rc2)i1+···+ik

i1! · · · ik!βi1(2β)i2 · · · (kβ)ik.

On the other hand, by using Eqs. (2.6) and (2.14) we have∑

i1+2i2+···+kik=n

n!βn(rc2)i1+···+ik

i1! · · · ik!βi1(2β)i2 · · · (kβ)ik=

1n!‖Pr,n‖2 = βn

(rc2

β

)n.

This give the statement. ¤

Now we extend the S-transform to generalized functions. For φ ∈ EC, we have

‖ : e〈·,φ〉 :LM ‖2θ,p,m =+∞∑n=0

m−nθ−2n

(rc2

β

)−1

n(n!)−2|φ⊗n|2

LM,p.



Thus, by using Eq. (5.8) and by a same calculus as in lemma 2 in Ref. 20 , for allp ≥ 0 and m > 0, there exist c, m′ > 0 and q > p such that

‖ : e〈·,φ〉 :LM ‖2θ,p,m ≤+∞∑n=0

Cnp βnθ−2

n

mn(n!)2|φ|2n

p ≤ c2 exp

2θ∗( |φ|q

m′

). (5.10)

This shows that the Levy-Meixner exponential : e〈·,φ〉 :LM is a test function inFLM,θ

(E). Thus, we can define the S-transform of a distribution Φ ∈ F∗LM,θ

(E), atφ ∈ EC as follows

SΦ(φ) := 〈〈Φ, : e〈·,φ〉 :LM〉〉 =+∞∑n=0

⟨Φ(n), φ⊗n

⟩. (5.11)

Let δa be the Dirac delta function at a ∈ R. For the Brownian motion Bt itwas shown in Kuo 31 that δa(Bt) is a generalized function in Gaussian white noisetheory. More generally, it was shown in Kubo 29 that the composition G(Bt) is ageneralized function for any G ∈ E ′C. Related results to Donsker’s delta functionfor general Levy processes, but in a different context and with another scope, havebeen established in 33,37. In this paper, we will give a simple derivation of Kubo’stheorem for the Levy-Meixner process Xt and obtain a new generalized process inthe class F∗

LM,θ(E). The informal calculations presented here are partly based on the

development in Kuo’s book 32.

Theorem 5.1. Let G ∈ E ′C. Then G(Xt) is a generalized function in F∗LM,θ

(E) withthe following chaotic representation

G(Xt) =+∞∑n=0

[βnn!

( |t|rc2

β

)n

]−1∫

RG(y)P|t|r ,n(y)µ|t|r (dy)

⟨: ·⊗n :LM , χ⊗n

[0,t]

⟩. (5.12)


S(G(Xt))(φ) =Γ( |t|r c2

β )

2iπ

∫

R

∫ 0

−∞G(y)x−

|t|r c2β

× exp

x + yρ(φ(t)

βx

)− f ρ

(φ(t)βx

)dxµ|t|r (dy).

Proof. Let G ∈ E ′C, then we can express F as

G(x) =∞∑

n=0

〈G, ζ|t|r ,n〉ζ|t|r ,n(x),

where 〈 ·, · 〉 is the bilinear pairing with respect to the measure µ|t|r . The aboveequation can be rewritten as

G(x) =∞∑

n=0

[βnn!

( |t|rc2

β

)n

]−1

P|t|r ,n(x)∫

RG(y)P|t|r ,n(y)µ|t|r (dy). (5.13)



From Eq. (4.16), we get

G(Xt) =+∞∑n=0

[βnn!

( |t|rc2

β

)n

]−1∫

RG(y)P|t|r ,n(y)µ|t|r (dy)

⟨: ·⊗n :LM , χ⊗n

[0,t]

⟩

=+∞∑n=0

[βnn!

( |t|rc2

β

)n

]−1/2∫

RG(y)Q|t|r ,n(y)µ|t|r (dy)

⟨: ·⊗n :LM , χ⊗n

[0,t]

⟩(5.14)

Using Eq. (2.11), we have∣∣∣∣∫

RG(y)Q|t|r ,n(y)µ|t|r (dy)

∣∣∣∣ =∣∣∣∣⟨G, Q|t|r ,n

⟩L2(µ|t|r

)

∣∣∣∣

≤∣∣∣⟨A−p|t|r G,Ap

|t|r Q|t|r ,n

⟩∣∣∣ ≤∣∣∣A−p|t|r G

∣∣∣0

∣∣∣Ap|t|r Q|t|r ,n

∣∣∣0≤ (2n + 1)p|G|−p. (5.15)

Then from Eqs. (5.4), (5.13) and (5.15) we get

‖G(Xt)‖2θ,−p,m ≤∞∑

n=0

(n!θn)2mn(2n + 1)2p

βnn!(|t|r c2

β

)n

|G|2−p|χ⊗n[0,t]|2LM,−p

≤ c2|G|2−p

∞∑n=0

n!(2n + 1)2p

(|t|r c2

β

)n

(2mbe

nβ

)n

|χ⊗n[0,t]|2LM,−p

.

Thus one can see that the last series is convergent. This shows that G(Xt) ∈F∗

LM,θ(E). Moreover, for φ ∈ EC we have

S(G(Xt))(φ) =+∞∑n=0

[βnn!

( |t|rc2

β

)n

]−1

φn(t)∫

RG(y)P|t|r ,n(y)µ|t|r (dy). (5.16)

On the other hand, Eq. (2.2) yields

∞∑n=0

φn(t)

βnn!(|t|r c2

β

)n

P|t|r ,n(y) = Γ( |t|rc2

β

) ∞∑n=0

φn(t)P|t|r ,n(y)

βnn!Γ( |t|r c2β + n)

=Γ

(|t|r c2

β

)

2iπ

∫ 0

−∞exx−

|t|r c2β

∞∑n=0

P|t|r ,n(y)n!

(φ(t)βx

)n

dx

=Γ

(|t|r c2

β

)

2iπ

∫ 0

−∞x−

|t|r c2β exp

x + yρ

(φ(t)βx

)− f ρ

(φ(t)βx

)dx.(5.17)

The statement follows from Eqs. (5.16) and (5.17). ¤

Remark 5.1. For a ∈ R, put G = δa,then G(Xt) = δa(Xt) is the Donsker’s deltafunction for the Levy-Meixner process. By theorem 5.1, δa(Xt) is a generalized



function and it has the following chaotic decomposition

δa(Xt) =+∞∑n=0

[βnn!

( |t|rc2

β

)n

]−1

P|t|r ,n(a)⟨: ·⊗n :LM , χ⊗n

[0,t]

⟩.


S(δa(Xt))(φ) =Γ( |t|r c2

β )

2iπ

∫ 0

−∞x−

|t|r c2β exp

x + aρ

(φ(t)βx

)− f ρ

(φ(t)βx

)dx.

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WHITE NOISE LÉVYMEIXNER PROCESSES THROUGH A TRANSFER PRINCIPAL FROM ONE-MODE TO ONE-MODE TYPE...

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