From <span class="mathjax-formula">$(n+1)$</span>-level atom chains to <span class="mathjax-formula">$n$</span>-dimensional noises

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From (n + 1)-level atom chains to n-dimensional noises

Stéphane Attal

, Yan Pautrat

aInstitute C. Jordan, UMR 5208, université de Lyon 1, bât. 101, 69622 Villeurbanne cedex, France bLaboratoire de Mathématiques, UMR 8628, Université Paris Sud, bât. 425, 91405 Orsay cedex, France

Received 29 January 2004; received in revised form 4 October 2004; accepted 19 October 2004 Available online 25 March 2005

This article is dedicated to the memory of Paul-André Meyer

Abstract

In quantum physics, the state space of a countable chain of(n+1)-level atoms becomes, in the continuous field limit, a Fock space with multiplicityn. In a more functional analytic language, the continuous tensor product space overR⁺of copies of the spaceCⁿ⁺¹is the symmetric Fock spaceΓs(L²(R⁺;Cⁿ)). In this article we focus on the probabilistic interpretations of these facts. We show that they correspond to the approximation of then-dimensional normal martingales by means of obtuse random walks, that is, extremal random walks inRⁿwhose jumps take exactlyn+1 different values. We show that these probabilistic approximations are carried by the convergence of the matrix basisa_jⁱ(p)of

NCⁿ⁺¹to the usual creation, annihilation and gauge processes on the Fock space.

2005 Elsevier SAS. All rights reserved.

Résumé

En physique quantique, l’espace d’état d’une chaîne dénombrable d’atomes à(n+1)niveaux devient, dans la limite continue, un espace de Fock de multiplicitén. De manière plus analytique, le produit tensoriel continu de copies deCⁿ⁺¹indexées par R⁺est l’espace de Fock symétriqueΓs(L²(R⁺;Cⁿ)). Dans cet article, nous nous intéressons aux interprétations probabilistes de ces résultats. Nous montrons qu’ils correspondent à l’approximation de martingales normalesn-dimensionnelles par des marches aléatoires obtuses, c’est-à-dire des marches aléatoires extémales deRⁿ dont les sauts prennent exactementn+1 valeurs différentes. Nous montrons que ces approximations sont contenues dans la convergence de la base canoniquea_jⁱ(p)de l’espace des matrices sur

NCⁿ⁺¹vers les processus de création, annihilation et nombre de l’espace de Fock.

2005 Elsevier SAS. All rights reserved.

* Corresponding author.

E-mail address: [email protected] (S. Attal).

0246-0203/$ – see front matter 2005 Elsevier SAS. All rights reserved.

doi:10.1016/j.anihpb.2004.10.003

1. Introduction

In functional analysis, the tensor product of a family of Hilbert spaces indexed by a continuous set, is a well- understood notion (see the very complete book [6]) which leads to notions such as “Fock spaces” or “symmetric space associated to a measured space”.

A physical interpretation of those continuous tensor product spaces consists in considering them as the contin- uous field limit of a countable chain of quantum system state spaces (such as a spin chain, for example).

The interesting point in these constructions is that, for alln∈N, the continuous tensor product space

R⁺

Cⁿ⁺¹

is the symmetric Fock spaceΓ_s(L²(R⁺;Cⁿ)). In a more physical language, the continuous field limit of the state space of a countable chain of(n+1)-level atoms is a Fock space with multiplicityn. A rigourous setting in which such an approximation is made true is developed in [1].

Both spaces

NCⁿ⁺¹andΓ_s(L²(R⁺;Cⁿ))admit natural probabilistic interpretations. In particular, the Fock spaceΓ (L²(R⁺;Cⁿ))admits natural probabilistic interpretations in terms ofn-dimensional normal martingales, such asn-dimensional Brownian motion,n-dimensional Poisson process,n-dimensional Azéma martingales, etc.

(cf. [2] and [3]). The aim of this article is to understand how the approximation ofΓ (L²(R⁺;Cⁿ))by means of spaces

NCⁿ⁺¹can be interpreted in probabilistic terms.

The structure of the space

NC⁽ⁿ⁺¹⁾suggests that we are dealing with random walks whose jumps take(n+1) different values.

In this article we show that the key point of this approximation is the notion of obtuse random walks, developed in [4]. They are the centered and normalized random variables inRⁿwhich take exactly(n+1)different values.

These obtuse random variables are naturally associated to an algebraic object called sesqui-symmetric 3-tensor and the associated random walk satisfies a discrete-time structure equation. This structure equation allows us to represent the multiplication operators by this random walk in terms of some basic operators of

NCⁿ⁺¹. Considering the approximation of the Fock spaceΓs(L²(R⁺;Cⁿ))by means of spaces

NCⁿ⁺¹, we obtain the approximation of a continuous-time normal martingale. The sesqui-symmetric 3-tensorΦthen converges to a so- called doubly-symmetric 3-tensor which is the key of the structure equation describing the probabilistic behaviour of that normal martingale (jumps, continuous and purely discontinuous parts . . . ).

This article is organized in the following way. In Section 2 we introduce the state space of the atom chain and the associated operators. In Section 3, we describe obtuse random walks inRⁿ, their structure equations and their representations as operators on the state space of the atom chains. In Section 4 we introduce Fock space and its quantum stochastic calculus, and the relation of these objects with the atom chains. In Section 5 we describe structure equations for normal martingales and the information given by these equations in a special case. In Section 6 we put together all of our tools and prove convergence in law of random walks to well-identified normal martingales. In Section 7 we review some explicit and illustrative examples.

2. The structure of the atom chain

We here introduce the mathematical structure and notations associated to the space

NCⁿ⁺¹. As the reader will easily see, this only means choosing a particular basis for the vectors and for the operators on that space. The physical-like terminology that we use from time to time is not necessary for the sequel, but is relevant (and used in references such as [5]).

Consider the spaceCⁿ⁺¹in which we choose an orthonormal basis denoted by{Ω, X¹, . . . , Xⁿ}. This space and this particular choice of an orthonormal basis physically represent either a particle withnexcited statesXⁱ and a

ground stateΩ, or a site which is either empty (Ω) or occupied by a typeiparticle (Xⁱ). We often writeX⁰forΩ when we need unified notations, but it is important in the sequel to distinguish one of the basis states.

Together with this basis ofCⁿ⁺¹we consider the following natural basis ofL(Cⁿ⁺¹)=Mn+1(C):

a_jⁱX^k=δ_kiX^j,

for alli, j, k=0, . . . , n. The operatora_jⁱ puts ani-level state into aj-level state.

We now consider a chain of copies of this system, like a chain of(n+1)-level atoms. That is, we consider the Hilbert space

TΦ=

i∈N

Cⁿ⁺¹,

the countable tensor product, indexed byN, of copies ofCⁿ⁺¹with respect to the stabilizing sequence of vectorsΩ.

By this we mean that a natural orthonormal basis of TΦis described by the family {X_A;A∈P}

where

– P is the set of finite subsetsA= {(n₁, i₁), . . . , (nk, ik)}ofN× {1, . . . , n}such that theni’s are two by two different. Another way to describe the setP is to identify it to the set of sequences(Ak)_k∈N with values in {0, . . . , n}, but taking only finitely many times a value different from 0.

– XAdenotes the vector

Ω⊗ · · · ⊗Ω⊗Xⁱ¹⊗Ω⊗ · · · ⊗Ω⊗Xⁱ²⊗ · · ·

of TΦ, where Xⁱ¹ appears in the copy numbern₁, Xⁱ² appears in the copy n₂, . . .. When A is seen as a sequence(A_k)_k∈Nas above, thenX_Ais advantageously written

kX_Ak.

The physical meaning of this basis is easy to understand: we have a chain of sites, indexed byN; on each site there is an atom in the ground state or an atom at energy level 1. . . . The above basis vectorX_Aspecifies that there is an atom at leveli₁in the siten₁, an atom at leveli₂in the siten₂, . . ., all the other sites being at the ground state.

The space TΦis what we shall call the(n+1)-level atom chain.

We denote bya_jⁱ(k)the natural ampliation of the operatora_jⁱ to TΦ which acts asaⁱ_j on the copy number k ofCⁿ⁺¹and as the identity on the other copies. These operators relate naturally to the creation and annihilation operators of the Fermionic Fock space overCⁿ.

Note, for information only, that the operatorsa_jⁱ(k)form a basis of the algebraB(TΦ)of bounded operators on TΦ. That is, the von Neumann algebra generated by thea_jⁱ(k),i, j=0, . . . , n,k∈N, is the whole ofB(TΦ)(for TΦadmits no subspace which is non-trivial and invariant under this algebra).

3. Obtuse random walks inRⁿ

We now abandon for a while this structure in order to concentrate on the probabilistic and algebraic structure of the obtuse random variables. The space TΦwill return naturally when describing the obtuse random walks.

LetX be a random variable in Rⁿ which takes exactly n+1 different values v₁, . . . , v_n+1 with respective probabilityα₁, . . . , α_n+1(all different from 0 by hypothesis). We assume, for simplicity, thatXis defined on its canonical space(A,A, P ), that is,A= {1, . . . , n+1},Ais theσ-field of subsets ofA, the probability measureP is given byP ({i})=αi andXis given byX(i)=vi, for alli=1, . . . , n+1.

Such a random variableXis called centered and normalized ifE[X] =0 and Cov(X)=I.

A family of elementsv₁, . . . , v_n+1ofRⁿis called an obtuse system if vi, vj = −1

for alli=j.

We consider the coordinatesX¹, . . . , XⁿofX in the canonical basis ofRⁿ, together with the random variable Ωon(A,A, P )which is deterministic and always equal to 1.

We putXⁱ to be the random variableXⁱ(j )= √α_jXⁱ(j )andΩ(j ) = √α_j. For any elementv=(a₁, . . . , a_n) ofRⁿwe putvˆ=(1, a₁, . . . , a_n)∈Rⁿ⁺¹.

The following proposition is rather straightforward and left to the reader.

Proposition 1. The following assertions are equivalent.

(i) Xis centered and normalized.

(ii) The(n+1)×(n+1)-matrix(Ω, X¹, . . . ,Xⁿ)is unitary.

(iii) The(n+1)×(n+1)-matrix(√

α₁vˆ₁, . . . ,√α_n+1vˆ_n+1)is unitary.

(iv) The familyv₁, . . . , v_n+1is an obtuse system ofR^αand α_i= 1

1+ v_i².

LetT be a 3-tensor inRⁿ, that is, a linear mapping fromRⁿtoMn(R). We writeT_k^ij for the coefficients ofT in the canonical basis ofRⁿ, that is,

T (x)

i,j= n

k=1

T_k^ijx_k.

Such a 3-tensorT is called sesqui-symmetric if (i) (i, j, k) →T_k^ij is symmetric and

(ii) (i, j, l, m) →

kT_k^ijT_k^lm+δijδlmis symmetric.

Theorem 2. IfXis a centered and normalized random variable inRⁿ, taking exactlyn+1 values, then there exists a sesqui-symmetric 3-tensorT such that

X⊗X=I+T (X). (1)

Proof. By Proposition 1, the matrix(√

α₁vˆ₁, . . . ,√α_n+1vˆ_n+1)is unitary. In particular the matrix(vˆ₁, . . . ,vˆ_n+1) is invertible and so is its adjoint matrix. But the latter is the matrix whose columns are the values of the random variablesΩ, X₁, . . . , Xn. As a consequence, thesen+1 random variables are linearly independent. They thus form a basis ofL²(A,A, P )for it is an+1 dimensional space.

The random variableXⁱX^j belongs toL²(A,A, P )and can thus be written as XⁱX^j=

n

k=0

T_k^ijX^k,

for some real coefficientsT_k^ij,k=0, . . . , n,i, j =1, . . . , n, where X⁰ denotesΩ. The fact that E[X^k] =0 and E[XⁱX^j] =δ_ij impliesT₀^ij=δ_ij. This gives the representation (1). ThatT is sesqui-symmetric is obtained from the relations

T_k^ij =E[XⁱX^j]

and

k

T_k^ijT_k^lm+δ_ijδ_lm=E[XⁱX^jX^lX^m]. 2

There is actually a natural bijection between the set of sesqui-symmetric 3-tensors and the set of obtuse random variables. The following result was obtained in [4], Theorem 2, pp. 268–272, which is far from obvious but which we shall not really need here.

Theorem 3. The formulas S=

x∈Rⁿ; x⊗x=I+T (x) and

T (x)=

y∈S

pyy, xy⊗y,

wherep_x=1/(1+ x²), define a bijection between the set of sesqui-symmetric 3-tensorsT onRⁿand the set of obtuse systemsSinRⁿ.

Now we wish to consider the random walks which are induced by obtuse systems. That is, on the probability space(A^N,A^⊗N, P^⊗N), we consider a sequence(X(p))_p∈Nof independent random variables with the same law as a given centered normalized random variableX.

Recalling the notations of Section 2, for anyA∈P, we define the random variable

X_A=

(p,i)∈A

Xⁱ(p) with the convention

X_∅=1.

Proposition 4. The family{X_A; A∈P}forms an orthonormal basis of the spaceL²(A^N,A^⊗N, P^⊗N).

Proof. For anyA, B∈Pwe have

X_A, X_B =E[X_AX_B] =E[X_AB]E[X²_A∩B]

by the independence of theX(p). For the same reason, the first termE[X_AB]gives 0 unlessAB= ∅, that is A=B. The second termE[X²_A∩B]is then equal to

(p,i)∈AE[Xⁱ(p)²] =1. This proves the orthonormal character of the family{X_A; A∈P}.

Let us now prove that it generates a dense subspace of L²(A^N,A^⊗N, P^⊗N). Had we considered random walks indexed by{0, . . . , N}instead ofN, theX_A,A⊂ {0, . . . , N}would have formed an orthonormal basis of L²(A^N,A^⊗N, P^⊗N), for their dimensions are equal. Now any elementf ofL²(A^N,A^⊗N, P^⊗N)can be approxi- mated by a sequence(f_N)_N such thatf_N∈L²(A^N,A^⊗N, P^⊗N), for allN, by taking conditional expectations on the trajectories ofXup to timeN. 2

For every obtuse random variableX, we thus obtain a Hilbert space TΦ(X)=L²(A^N,A^⊗N, P^⊗N),

with a natural orthonormal basis{XA; A∈P}which emphasizes the independence of theX(p)’s. In particular there is a natural isomorphism between all the spaces TΦ(X)which consists in identifying the associated bases.

In the same way, all these canonical spaces TΦ(X)of obtuse random walks are naturally isomorphic to the atom chain TΦof previous section (again by identifying their natural orthonormal bases).

Of course this identification of Hilbert spaces does not mean much for the moment: in particular, it loses all the probabilistic properties of the random variablesXⁱ(p), be it individual (the law) or collective (probabilistic independence) properties.

The only way to recover the full probabilistic information onXⁱ(p)in the Hilbert space formalism associated to TΦis to consider the multiplication operator byXⁱ(p)instead of the Hilbert space elementXⁱ(p). Indeed, if we know the representation in TΦof the operatorMXⁱ(p)of multiplication byXⁱ(p)on TΦ(X), we know everything about the random variable Xⁱ(p) and its relation with other random variables. The above idea is what makes quantum probabilistic tools relevant for the study of classical probability; following this idea, the next theorem is one of the keys of this article. It is what allows us to translate probabilistic properties into operator-theoretic language, showing that all the obtuse random walks inRⁿ can be represented in a single space TΦ with very economical means: linear combinations of the operatorsaⁱ_j(p).

Theorem 5. LetXbe an obtuse random variable, let(X(p))_p∈Nbe the associated random walk on the canonical space TΦ(X). LetT be the sesqui-symmetric 3-tensor associated toX. LetUbe the natural unitary isomorphism from TΦ(X)to TΦ; then, for allp∈N, i= {1, . . . , n}, we have

UMX_i(p)U^∗=a_i⁰(p)+a₀ⁱ(p)+ n

j,l=1

T_i^{j l}a^j_l(p).

Proof. It suffices to compute the action ofMX_i(p)on the basis elementsX_A,A∈P. Denote by “(p,·) /∈A” the claim “for noidoes(p, i)belong toA”. Then, by Theorem 1, there exists a sesquisymmetric tensorT onRⁿsuch that

X_i(p)X_A=1(p,·) /∈AX_i(p)X_A+ n

j=1

1(p,j )∈AX_i(p)X_A

=1(p,·) /∈AX_A∪{(p,i)}+ n

j=1

1(p,j )∈AX_i(p)X_j(p)X_{A\{(p,j )}}

=1(p,·) /∈AXA∪{(p,i)}+ n

j=1

1(p,j )∈A

δij+

l

T_l^ijXl(p)

XA\{(p,j )}

=1(p,·) /∈AX_A∪{(p,i)}+1(p,i)∈AX_A\(p,i)+ n

j=1

n

l=1

1(p,j )∈AT_l^ijX_{A\{(p,j )}∪{(p,i)}} and we recognize the formula for

a_i⁰(p)XA+a₀ⁱ(p)XA+

p,l

T_l^ija_l^j(p)XA. 2

Let us now return to quantum probabilistic structures and describe the Fock space and its approximation by the atom chain.

4. Approximation of the Fock space by atom chains

We recall the structure of the bosonic Fock spaceΦand its basic objects (see e.g. [3] or [7] for details).

LetΦ=Γ_s(L²(R⁺;Cⁿ))be the symmetric (bosonic) Fock space over the spaceL²(R⁺;Cⁿ). We shall here give a very efficient presentation of that space, the so-called Guichardet interpretation of the Fock space.

LetI= {1, . . . , n}and let P be the set of finite subsets {(s₁, i₁), . . . , (s_k, i_k)}of R⁺×I such that thes_i are mutually distinct (we use the same symbol as in the discrete case; the context will always prevent confusion). Then P=

kP(k)whereP(k)is the set ofk-elements subsets ofR⁺×I. By ordering theR⁺-part of the elements of σ∈P(k), the setP(k)can be identified to the increasing simplexΣ_k= {0< t₁<· · ·< t_k} ×I ofR^k×I. Thus P(k)inherits a measured space structure from the product of Lebesgue measure onR^k and the counting measure onI. This also gives a measure structure onP if we specify that onP(0)= {∅}we put the measureδ_∅. Elements ofPare usually denoted byσ, the measure onPis denoted by the infinitesimaldσ. Theσ-field obtained this way onPis denotedF.

We identify any elementσ ∈P with a family{σ₁, . . . , σ_n}of (two by two disjoint) subsets ofR⁺where σ_i=

s∈R⁺;(s, i)∈σ .

For as∈R⁺we denote by{s}i the elementσ = {∅, . . . ,∅,{s},∅, . . . ,∅}ofPwhere{s}is at thei-th position.

The Fock spaceΦis the spaceL²(P,F,dσ ). An elementf ofΦis thus a measurable functionf:P→Csuch that

f²=

P

f (σ )²dσ <∞.

One can define, in the same way,P[a,b]andΦ_[a,b]by replacingR⁺with[a, b] ⊂R⁺. There is a natural isomor- phism betweenΦ_[0,t]⊗Φ_[t,+∞[andΦ given byh⊗g →f wheref (σ )=h(σ∩ [0, t]) g(σ∩(t,+∞[). Define also1to be the vacuum vector, that is,1(σ )=δ_∅(σ ).

Defineχ_tⁱ∈Φ by

χ_t(σ )=1_[0,t](s) ifσ= {s}i,

0 otherwise.

Thenχ_t belongs to Φ_[0,t]. We even haveχ_tⁱ −χ_sⁱ ∈Φ_[s,t] for allst. This last property allows to define a so-called Itô integral onΦ. Indeed, let(g_tⁱ)t0be families inΦ, fori=1, . . . , n, such that

(i) t → g_tⁱis measurable, (ii) gⁱ_t∈Φ_[0,t]for allt, (iii) _∞

0 g_tⁱ²dt <∞, then one defines

i

_∞

0 g_tⁱdχ_tⁱto be the limit inΦof n

i=1

∞ j=0

1 tj+1−tj

tj+1

t_j

P_tjgⁱ_sds⊗(χ_tⁱ

j+1−χ_tⁱ

j) (2)

whereP_t is the orthogonal projection ontoΦ_[0,t]and{t_j, j∈N}is a partition ofR⁺, and the limit is taken along a sequence of refining partitions with mesh size going to zero. Note that _t ¹

j+1−tj

_tj+1

tj P_tjg_sdsbelongs toΦ_[0,tj], which explains the tensor product symbol in (2).

We get that

i

_∞

0 gⁱ_tdχ_tⁱ is an element ofΦwith

i

∞

0

gtdχt

2

=

i

∞

0

g_tⁱ²dt. (3)

Letf∈L²(P); one can easily define the iterated Itô integral onΦ. I_n(f )=

P

f (σ )dχ_tⁱ¹

1 · · ·dχ_tⁱⁿ

n

by iterating the definition of the Itô integral. We use the following notation:

I_n(f )=

P

f (σ )dχ_σ

which we extend, in an obvious way, to anyf ∈Φ. We then have the following important representation.

Theorem 6. Any elementf ofΦadmits an abstract chaotic representation f=

P

f (σ )dχ_σ with

f²=

P

f (σ )²dσ

and an abstract predictable representation

f=f (∅)1+

i

∞

0

Dⁱ_tfdχ_tⁱ with

f²=f (∅)²+

i

∞

0

D_sⁱf²ds

where[Dⁱ_sf](σ )=f (σ∪ {s}i)1σ⊂[0,s[.

Let us now recall the definitions of the basic noise operatorsa_jⁱ(t ),i, j=0, . . . , n, onΦ. They are respectively defined by

a_i⁰(t )f

(σ )=

s∈σ_i∩[0,t]

f σ\ {s}i

,

[a₀ⁱf](σ )= t

0

f σ∪ {s}i

ds,

[a_jⁱf](σ )=

s∈σi∩[0,t]

f

σ\ {s}i∪ {s}j

fori, j=0 and a₀⁰(t )=t I.

There is a good common domain to all these operators, namely D=

f∈Φ;

P

|σ|f (σ )²dσ <∞

.

LetS= {0=t₀< t₁<· · ·< t_p<· · ·}be a partition ofR⁺ andδ(S)=sup_i|t_i+1−t_i|be the diameter ofS. For fixedS, defineΦ_p=Φ_[tp,tp+1],i∈N. We then haveΦ

p∈NΦ_p(with respect to the stabilizing sequence (1)p∈N).

For allp∈N, define fori, j=0 Xⁱ(p)= χ_tⁱ

p+1−χ_tⁱ

√ p

tp+1−tp ∈Φ_p, a₀ⁱ(p)=a₀ⁱ(t_p+1)−aⁱ₀(t_p)

√t_p+1−t_p P1], a_jⁱ(p)=P_1]

aⁱ_j(t_p+1)−aⁱ_j(t_p) P_1], a_j⁰(p)=P_1]

a⁰_j(t_p+1)−a⁰_j(t_p)

√t_p+1−t_p ,

whereP_1]is the orthogonal projection ontoL²(P(1))and where the above definition ofa_i⁰(p)is understood to be valid onΦ_p only, witha_i⁰(p)being the identity operatorI on the othersΦ_q’s (the same is automatically true for a₀ⁱ,a_jⁱ). We puta₀⁰(p)=I.

Proposition 7. We have a₀ⁱ(p)X^j(p)=δ_ij1,

a₀ⁱ1=0,

a_jⁱ(p)X^k(p)=δ_ikX^j(p), a_jⁱ1=0,

a_j⁰(p)Xⁱ(p)=0, a_j⁰(p)1=X^j(p).

Thus the action of the operatorsa_jⁱ on theXⁱ(p)is similar to the action of the corresponding operators on the atom chain of section two. We are now going to construct the atom chain insideΦ.

We are still given a fixed partitionS. Define TΦ(S)to be the space of vectorsf ∈Φ which are of the form

f=

A∈PN

f (A)X_A (withf²=

A∈PN|f (A)|²<∞).

The space TΦ(S)is thus clearly identifiable to the atom chain TΦ; the operatorsa_jⁱ(p)act on TΦ(S)exactly in the same way as the corresponding operators on TΦ. We have completely embedded the toy Fock space into the Fock space.

LetS= {0=t₀< t₁<· · ·< t_p<· · ·}be a fixed partition ofR⁺. The space TΦ(S)is a closed subspace ofΦ.

We denote byP_Sthe operator of orthogonal projection fromΦonto TΦ(S).

We are now going to prove that the Fock spaceΦ and its basic operatorsa_jⁱ(t )can be approached by the toy Fock spaces TΦ(S)and their basic operatorsaⁱ_j(p).

We are given a sequence(Sp)p∈N of partitions which are getting finer and finer and whose diameter δ(Sp) tends to 0 whenptends to+∞. Let TΦ(p)=TΦ(Sp)and letPpbe the orthogonal projector onto TΦ(Sp), for allp∈N.

Theorem 8.

(i) For everyf ∈Φthere exists a sequence(fp)_p∈Nsuch thatfp∈TΦ(p), for allp∈N, and(fp)_p∈Nconverges tof inΦ.

(ii) For alli, jlet ε_ij=1

2(δ_0i+δ_0j).

IfSp= {0=t₀^p< t₁^p<· · ·< t_k^p<· · ·}, then for allt∈R⁺, the operators

k;t_k^pt

(t_k^p₊₁−t_k^p)^εija_jⁱ(k)

converge strongly onDtoaⁱ_j(t ).

Proof. (i) As theSp are refining then the(P_p)_p form an increasing family of orthogonal projections inΦ. Let P_∞= ∨pP_p. Clearly, for allst, alliwe have thatχ_tⁱ−χ_sⁱ belongs to RanP_∞. But by the construction of the Itô integral and by Theorem 5, we have that theχ_tⁱ −χ_sⁱ generateΦ. ThusP_∞=I. Consequently iff ∈Φ, the sequencef_p=P_pf satisfies the statements.

(ii) The convergence of

k,t_k^pt(t_k^p₊₁−t_k^p)^εija_jⁱ(k)toa_jⁱ(t )is easy from the definitions wheni=0. Let us check the case ofa⁰_i. We have, forf ∈D

k;t_k^pt

t_k^p₊₁−t_k^pa_i⁰(k)f

(σ )=

k;t_k^pt

1_|σ∩[t^p

k,t_k+1^p ]|=1

s∈σ∩[t_k^p,t_k^p₊₁]

f σ\ {s}

.

Putt^p=inf{t_k^p∈Sp; t_k^pt}. We have

k;t_k^pt

t_k^p₊₁−t_k^pa_i⁰(k)−a_i⁰(t )f ²

=

P

k;t_k^pt

1_|σ∩[t^p

k,t_k^p₊₁]|=1

s∈σ∩[t_k^p,t_k+1^p ]

f σ\ {s}

−

s∈σ∩[0,t]

f

σ\ {s}

2

dσ

2

P

s∈σ∩[t,t^p]

f

σ\ {s}

2

dσ+2

P

k;t_k^pt

1_|σ∩[t^p

k,t_k^p₊₁]|2×

s∈σ∩[t_k^p,t_k^p₊₁]

f

σ\ {s}

2

dσ.

For any fixedσ, the terms inside each of the integrals above converge to 0 whenptends to+∞. Furthermore we have, for large enoughp,

P

s∈σ∩[t,t^p]

f

σ\ {s}

2

dσ

P

|σ|

s∈σ st+1

f

σ\ {s}²dσ=

t+1

0

P

|σ| +1f (σ )²dσds (t+1)

P

|σ| +1f (σ )²dσ which is finite forf ∈D;

P

k;t_k^pt

1_|σ∩[t^p

k,t_k+1^p ]|2

s∈σ∩[t_k^p,t_k^p₊₁]

f

σ\ {s}

2

dσ

P k;t_k^pt

1_|σ∩[t^p

k,t_k^p₊₁]|2

s∈σ∩[t_k^p,t_k^p₊₁]

f

σ\ {s} ₂

dσ

P k;t_k^pt

s∈σ∩[t_k^p,t_k^p₊₁]

f

σ\ {s}₂ dσ

=

P s∈σ st^p

f

σ\ {s}2

dσ=

P

|σ|

s∈σ st^p

f

σ\ {s}²dσ(t+1)

P

|σ| +1f (σ )²dσ

in the same way as above. So we can apply Lebesgue’s theorem. This proves (ii). 2

5. Multidimensional structure equations

Let us recall some basic facts about normal martingales inRⁿ. Except for Theorem 13, all the statements in this section are taken from [4].

In the same way as the Fock spaceΦ=Γ (L²(R⁺;C)) admits probabilistic interpretations in terms of one- dimensional normal martingales (see [3]), the multiple Fock space Φ =Γ (L²(R⁺;Cⁿ)) admits probabilistic interpretations in terms of multidimensional normal martingales. The point here is that the extension of the notion of normal martingale, structure equation. . . to the multidimensional case is not so immediate. Some interesting algebraic structures appear.

A martingaleX=(X¹, . . . , Xⁿ)with values inRⁿis called normal ifX0=0 and if, for alliandj, the process X_tⁱX^j_t −δijtis a martingale. This is equivalent to saying that

Xⁱ, X^jt=δijt

for allt∈R⁺, or else this is equivalent to saying that the process [Xⁱ, X^j]t−δ_ijt

is a martingale.

A normal martingaleX=(X¹, . . . , Xⁿ)inRⁿis said to satisfy a structure equation if each of the martingales [Xⁱ, X^j]t−δ_ijtis a stochastic integral with respect toX:

[Xⁱ, X^j]t=δijt+ n

k=1

t

0

T_k^ij(s)dX^k_s where theT_k^ij are predictable processes.

Any family{A^ij_k; i, j, k∈ {1, . . . , n}}of real numbers is identified to a 3-tensor, that is, a linear mapAfromRⁿ toRⁿ⊗Rⁿby

(Ax)_ij= n

k=1

A^ij_kx_k.

Such a family is said to be diagonalizable in some orthonormal basis if there exists an orthonormal basis {e¹, . . . , eⁿ}ofRⁿfor which

Ae^k=λke^k⊗e^k

for allk=1, . . . , nand for some eigenvaluesλ₁, . . . , λn∈R. A family{A^ij_k; i, j, k∈ {1, . . . , n}}is called doubly symmetric if

(i) (i, j, k) →A^ij_k is symmetric on{1, . . . , n}³and (ii) (i, j, i, j) →_n

k=1A^ij_kAⁱ_k^jis symmetric on{1, . . . , n}⁴.

Theorem 9. For a family{A^ij_k; i, j, k∈ {1, . . . , n}}of real numbers, the following assertions are equivalent.

(i) Ais doubly symmetric.

(ii) Ais diagonalizable in some orthonormal basis.

This means that the condition of being doubly symmetric is the exact extension to 3-tensors of the symmetry property for matrices (2-tensors): it is the necessary and sufficient condition for being diagonalizable in some orthonormal basis.

A family{x¹, . . . , x^k}of elements ofRis called orthogonal family if thexⁱ are all different from 0 and are two by two orthogonal.

Theorem 10. There is a bijection between the doubly symmetric familiesAofRⁿ and the orthogonal familiesΣ which is given by

Af =

x∈Σ

1

x²x, fx⊗x and

Σ=

x∈Rⁿ\ {0};Ax=x⊗x .

These algebraic preliminaries are used to determine the behaviour of the multidimensional normal martingales.

Theorem 11. LetXbe a normal martingale inRⁿsatisfying a structure equation [Xⁱ, X^j]t=δ_ijt+

n

k=1

t

0

T_k^ij(s)dX^k_s.

Then for a.a.(t, ω)the family{T_k^ij(s, ω); i, j, k=1, . . . , n}is doubly symmetric. IfΣ_t(ω)is its associated orthog- onal system and ifπ_t(ω)denotes the orthogonal projection onto(Σ_t(ω))^⊥, then the continuous part ofXis given by

X^c,i_t = n

j=1

t

0

π_s^ijdX^j_s;

the jumps ofXhappen only at totally inaccessible times and they satisfy X_t(ω)∈Σ_t(ω).

We can now study a basic example. The simplest case occurs whenT is constant int. Contrarily to the unidi- mensional case, this situation is already rather rich.

Proposition 12. LetT be a doubly symmetric family onRⁿ. LetΣbe its associated orthogonal system. LetBbe a standard Brownian motion with values in the Euclidian spaceΣ^⊥. For eachx∈Σ, letN^xbe a Poisson process with intensityx⁻². We assumeBand all theN^xto be independent. Then the martingale

X_t=B_t+

x∈Σ

N_t^x− x⁻²t x