Quantum stopping times and quasi-left continuity

Quantum stopping times and quasi-left continuity

Stéphane Attal

, Agnès Coquio

Université de Grenoble I, institut Fourier, UMR 5582 du CNRS et de l’UJF, UFR de mathématiques, B.P. 74, 38402 St Martin d’Hères cedex, France

Received 17 September 2003; received in revised form 3 October 2003; accepted 18 November 2003 Available online 20 May 2004

Abstract

We study the general properties of quantum stopping times on Hilbert spaces equipped with a filtration. We define and investigate notions such as the spaces of anterior events, the spaces of strictly anterior events and above all we define the propertyS < T for two stopping times together with the notion of predictable quantum stopping times. It is well-known that the natural filtration of any normal martingale with the predictable representation property is quasi-left continuous; with the help of our new notions we prove that this property is actually an intrinsic property of the symmetric Fock spaceΦoverL²(R⁺). We also apply these definitions to the case of a non commutative stochastic base. We show, in this context, that the fermionic Fock space overL²(R⁺), the quasi-free boson and fermion spaces are also quasi-left continuous.

2004 Elsevier SAS. All rights reserved.

Résumé

Sur les espaces de Fock munis d’une filtration, nous définissons la propriétéS < T pour 2 temps d’arrêt ainsi que la notion de temps d’arrêt prévisible. Il est bien connu que la filtration naturelle d’une martingale normale ayant la propriété de représentation prévisible est quasi continue à gauche. Grâce à ces nouvelles notions, nous prouvons que cette propriété est intrinsèque sur l’espace de Fock symétrique surL²(R⁺). Nous montrons aussi que l’espace de Fock fermionique, les espaces de Fock fermioniques et bosoniques quasi-libres sont quasi-continus à gauche.

2004 Elsevier SAS. All rights reserved.

MSC: 46L53; 60G40; 81S25

Keywords: Quantum stopping times; Fock space; Quantum stochastic calculus; Noncommutative stochastic base

1. Introduction

The fundamental importance of stopping times in the classical theory of stochastic processes does not need to be demonstrated any more. That is a reason why the difficulties to define a serious and efficient theory of stopping times in the framework of quantum processes can be felt as an obstacle to important developments.

* Corresponding author.

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

0246-0203/$ – see front matter 2004 Elsevier SAS. All rights reserved.

doi:10.1016/j.anihpb.2003.10.008

The theory of quantum processes and quantum noises has had an impressive development since the last 25 years and has found many deep applications in quantum physics. For example, the quantum statistical description of the dilation of the dynamic of a quantum open system, with the help of quantum noises, is one of the most remarkable application of quantum probability theory [11].

The theory of quantum statistical mechanics is now having a very quick development and follows in parallel the tracks of the older (classical) statistical mechanics theory: Gibbs states become K.M.S. states, generators of Feller processes become Lindblad generators of quantum dynamical semigroups,. . .. The important open problems in quantum statistical mechanics are those of return to equilibrium, recurrence, existence of invariant states, spectral gaps,. . .. It is well-known that the most remarkable answers to the corresponding problems in the classical theory were obtained with the help of Markov processes and stopping time theory.

But why are stopping times so difficult to handle in quantum theory? The first obstacle is physical and philosophical. It is very delicate (and sometime taboo) to associate an observable with the time when something happens in a quantum system. The main reason is that in order to observe such a time one should be continuously monitoring the system. This is of course very delicate in quantum mechanics as the system is definitely affected by the observation. The brutal continuous observation of a quantum system leads to surprising consequences such as freezing the system in the initial state (quantum Zeno effect). But on the other hand it is also true that the theory of continuous observation of quantum systems has made impressive progresses recently, both theoretically and experimentally (non-demolition measurement).

It is even more remarkable that, forgetting the physical constraints, it is very easy to mathematically associate an observable to the time when some event occurs in a quantum system. For example, when studying the quantum stochastic differential equations describing the dynamics of some quantum open systems (such as in quantum optics for example), one can exhibit abelian subalgebras of observables which are invariant under the dynamic.

This thus gives rise to commutative processes which can be realised (diagonalised) on some probability space. As a consequence, any classical stopping time associated to this process (exit times, hitting times,. . .) gives rise to a quantum stopping time when pulled back in the general setup.

The theory of quantum stopping times has been initiated by R.L. Hudson [9] in the framework of Fock space.

The basic idea is to say that a classical stopping time is a positive random variable (with value+∞admitted) which satisfies some adaptedness property with respect to a given filtration ofσ-fields. Thus a quantum stopping time is a quantum random variable (a self-adjoint operator on a spectral measure) which is positive, which admits the value+∞and which satisfies some adaptedness property on the Fock space. This theory has been developed by several authors in the same framework: [1–3,14], but also in the framework of filtered families ofσ-finite and finite von Neumann algebras: [4,6–8].

In this article, a stopping time is an increasing family of projections on a filtered Hilbert space, adapted to an increasing family of algebras. This approach thus covers all the preceding cases.

In the five first parts we study stopping times on filtered Hilbert spaces(H, (Ht)_t0), without really mentioning algebras and we define for a stopping timeT, the spacesHT,HT−, the propertyS < T for two stopping times and the notion of previsible stopping time. The last two notions being actually the real new ones with respect to the usual literature.

The first application of this first part is the case of the symmetric Fock space overL²(R⁺). It is well-known that any classical normal martingale (i.e. with angle bracketx, xt equal to t for allt ∈R⁺) which possesses the predictable representation property, admits its chaotic space to be naturally isomorphic to the symmetric Fock space overL²(R⁺). This the starting point of the connections between classical and quantum stochastic calculus.

It happens that all these classical martingales actually share another property: their natural filtration is quasi- left continuous (i.e. every accessible stopping time is predictable, or equivalently the jumps of these martingales are totally inaccessible). In Section 6 of this article, we show that this property is actually independent of the probabilistic interpretations of the Fock space, it is a completely intrinsic (and stronger) property of this space:

ΦT−=ΦT

for every predictable quantum stopping timeT, thus in particular for the classical predictable stopping times.

In the last part, we apply the preceding definitions and results to the cases of the antisymmetric Fock space over L²(R⁺)and the quasi-free representations of the CCR and CAR. We also prove that in all these cases the quasi-left continuity property is verified.

2. Quantum stopping times

A filtered Hilbert space is a complex separable Hilbert spaceHtogether with a family of orthogonal projections (Et)_t∈R+ with range(Ht)_t∈R+ satisfying

(i) H0C ands- limt→+∞Et=I (i.e.

t∈R⁺Ht=H).

(ii) EsEt=EtEs=Es for allst(i.e.Hs⊂Htforst).

(iii) s- limu→t

u>t E_u=E_t (i.e.

u>tHu=Ht).

We write Ht− =

s<tHs and Et− is the orthogonal projection onto Ht−, t ∈ R⁺ (with the convention H0−=H0).

An operatorXonHis said to be adapted at time tif (i) E_u(DomX)⊂DomX, for allut, and

(ii) EuX=XEuon DomT, for allut.

A stopping time (or quantum stopping time)T on a filtered Hilbert space(H, (Et)_t∈R+)is a (right-continuous) spectral measure on R⁺∪ {+∞}with values in the set of orthogonal projectors onH, such that, for allt∈R⁺, the operatorT ([0, t])is adapted at timet.

In the following we adopt probabilistic-like notations: for every Borel subsetE⊂R⁺∪ {+∞}we write1T∈E

instead ofT (E). In the same way1Tt meansT ([0, t]),1T=t meansT ({t}), . . .. Note that in particular

1Tt=s- lim

ε→0 ε>0

1Tt+ε,

and

1T <t=s- lim

ε→0 ε>0

1Tt−ε.

In particular,1T <t is also adapted at timet, for allt∈R⁺.

Let us see briefly how this definition connects to the classical one. IfT is a classical stopping time on a filtered probability space(Ω,F, (Ft)_t∈R+, P ), then taking H=L²(Ω,F, P ),Ht =L²(Ω,Ft, P ) (if we assume that the filtration verifies the “usual conditions” of completeness and right-continuity) andE_t =E[·/Ft]makes up a filtered Hilbert space. The operatorsM1_Tt of multiplication by1(Tt )onHthen define a quantum stopping time onH.

Conversely, if T is a quantum stopping time on a filtered Hilbert space (H, (E_t)_t∈R+) then the operators 1Tt,t∈R⁺∪ {+∞}, are two by two commuting. Thus they simultaneously diagonalise on a probability space (Ω,F, P )to give rise to operators of multiplication by indicator functions of the form1(τt )for some random variableτ valued in R⁺∪ {+∞}. TakingFt to be theσ-field generated by the image ofHt intoL²(Ω,F, P ), makeτ being a classical stopping time.

Thus when considering one quantum stopping time (or a commuting family of quantum stopping times) leads to a theory which is exactly equivalent to the classical one. Of course the difference appears when considering several non-commuting stopping times onH. Each of them can be individually interpreted classically, but not together. They come from different probabilistic context and they are put together in the same context, exactly like observables in quantum mechanics.

A pointt∈R⁺is a continuity point for a quantum stopping timeT if1T=t =0. Note that asHis separable, then any stopping timeT admits an at most countable set of points which are not of continuity forT. Also note that iftis a continuity point forT then the maps→1Ts is strongly continuous att.

A stopping timeT is discrete (or simple) if there exists a finite setE= {0t₁< t₂<· · ·< t_n+∞} in R⁺∪ {+∞}such that1T∈E=I.

A sequence of stopping times(T_n)_n∈Nis said to converge to a stopping timeT ifs- lim_n→+∞1Tnt=1Tt for all continuity pointt ofT.

A stopping timeT is finite if1T=+∞=0.

Two stopping timesS,T satisfyST if1St1Ttfor allt∈R⁺(in the sense of comparison of projectors).

In particular

1Tt=1St1Tt=1Tt1St

for allt∈R⁺.

We begin by giving some elementary properties of stopping times. We omit the proofs because they are similar of these given in [6] for example.

Note that the stopping timeT given by 1Ts=

0 fors < t, I forst

is nothing but the deterministic timeT =tI, denotedtsimply.

LetE= {0=t0< t1< t2<· · ·< tn<+∞}be a partition of R⁺. Define the spectral measureTEby T_E({t_i})=T ([t_i−1, t_i[) fori=1, . . . , n,

T_E({+∞})=T ([t_n,+∞]).

ThenTEis clearly a quantum stopping time.

Finally, for a stopping timeT, by a sequence of refiningT-partitions of R⁺ we mean a sequence(E_n)_n∈N of finite subsetsE_n= {0=t₀ⁿ< t₁ⁿ<· · ·< t_kⁿ<+∞}of R⁺such that:

(i) all thet_jⁿare continuity points forT,n∈N,j1;

(ii) E_n⊂E_n+1for alln∈N;

(iii) the diameterδ_n=sup{t_iⁿ₊₁−t_iⁿ; i∈N}ofE_ntends to 0 whenntends to+∞; (iv) supE_ntends to+∞asntends to+∞.

Note that, for any stopping timeT such a sequence always exists.

Now putTn=TE_n for alln∈N. Then we haveT0T1· · ·T and(Tn)_n∈Nconverges toT.

3. The spaceHT

Let(H, (Ht)_t∈R+)be a filtered Hilbert space andT be a stopping time onH. The spaceHtclassically interprets as the space of events occurring before timet(see the discussion in Section 2 about the connections with classical theory). Thus mimicking the classical definition ofFT, theσ-field of events anterior toT, that is

FT =

A∈F; (T t)∩A∈Ftfor allt∈R⁺ ,

we define in our quantum context: the space of events anterior to a stopping timeT is the space HT =

f ∈H; 1Ttf ∈Ht for allt∈R⁺ .

We denote byE_T the orthogonal projection ontoHT which is clearly a closed subspace ofH.

The spaceHT and the projectionE_T are already defined and studied in [6] for example, so some results are quite standards and we just enumerate them.

1) IfST thenHS⊂HT.

2) If(T_n)_n∈Nis a decreasing sequence of stopping times converging toT then HT =

n∈N

HTn

and(E_Tn)_n∈Ndecreases and converges toE_T. 3)HT = {f ∈H; 1T <tf∈Ht for allt∈R⁺}.

It can be proved that ifT is a discrete stopping time with_n

i=11T=ti=I, then one have E_T =

n

i=1

1T=t_iE_ti= n

i=1

E_ti1T=t_i, with the conventionE_∞=I.

As an easy consequence of the preceding results, we prove using the notations of sequence of refiningT- partitions, that the sequence(ET_n)n∈Nstrongly converges toET. In other words

ET =s- lim

n→+∞

Nn

i=1

1T∈[t_i−1,t_i[Et_i+1Tt_Nn

where the diameter of the partition{0=t₀< t₁<· · ·< tN_n}tends to 0 andtN_ntends to+∞.

4. The spaceHT−

As our definition ofHT in our setup seems to fit very well, we pursue the analogy with classical probability theory and define the space of event strictly anterior to a stopping timeT as the spaceHT− which is the closure of the subspace ofHgenerated byH0and{1T >tf; f ∈Ht, t∈R⁺}.

The different properties of these spaces are proved in [7] (The arguments are the same), so we just give here the principal results:

1) For every stopping timeT we haveHT−⊂HT.

2) IfSandT are two stopping times such thatST, thenHS−⊂HT−. 3) If(Tn)_n∈Nis an increasing sequence of stopping times converging toT then

HT−=

n

HTn−.

4) LetT be a finite stopping time (i.e.1T=+∞=0). Then the spectral integral

R⁺t1T∈dtdefines a self-adjoint operator onH, which we denote byT again.

IfT is any finite stopping time then the operatorT maps DomT ∩HT toHT. If moreoverT is bounded, it mapsHT−toHT−.

Proposition 1. If the filtration is continuous (i.e.Ht−=Ht for allt) thenHT−=HT for every discrete stopping timeT.

Proof. Letg∈(HT−)^⊥. We thus have, for allt∈R⁺, allf∈H, 1T >tE_tf, g =0, i.e. f,1T >tE_tg =0.

This means1T >tEtg=0 for allt∈R⁺. Now suppose thatT is discrete with_n

i=11T=ti=I. Ift < t₁then1T >t=I andE_tg=0 thus by the continuity hypothesisE_t1g=0.

If t_i t < t_i+1 then E_tg =1TtE_tg=_i

j=01T=tjE_tg. Now let t tend to t_i+1, this gives E_ti+1g= i

j=01T=t_jEt_i+1g. Thus in particular1T=t_i+1Et_i+1g=0.

All together we have proved thatETg=

i1T=t_iEt_ig=0 and thusg∈(HT)^⊥. 2

5. Strictly smaller stopping times

We wish to give a correct meaning to the relationS < T for two quantum stopping times,SandT onH. IfSis a discrete stopping time withn

i=11S=s_i=I and ifT is any stopping time, it is then natural to say thatS < T if and only if

1S=si=1T >si1S=si for alli.

Note that this in particular impliesST and we haveS < T if and only if n

i=1

1T >s_i1S=s_i=I

or else if and only if n

i=1

1S=s_i1T >s_i=I.

This motivates the general definition.

Two stopping timesSandT onHare said to satisfyS < T if and only if one has that the expression

Nn

i=1

1T >ti1S∈[ti−1,ti[

weakly converges toI when{t_i, i=1, . . . , N_n}follows a sequence of refiningS-partitions of R⁺. Note thatS < T impliesST for ifS < T then

1Tt N_n

i=1

1T >ti1S∈[ti−1,ti[1St (1)

converges weakly to1Tt1St, but (1) also equals

1Tt Nn

i=1

1T >ti1S∈[t_i−1,ti[

which weakly converges to1Tt.

Note that ifS andT commute, that is if for allsandt in R⁺,1Ss1Tt=1Tt1Ss, then for all sequence of refining partitions of R⁺,R_n=Nn

i=11T >ti1S∈[ti−1,ti[converges strongly. In fact, in this case,(R_n)_n0is an increasing sequence of projections.

Proposition 1. LetSandT be two stopping times. Then the following assertions are equivalent.

(i) S < T. (ii) _Nn

i=11S∈[t_i−1,t_i[1T >t_iconverges weakly toI. (iii) _Nn

i=11Tt_i1S∈[t_i−1,t_i[converges weakly to 0 and1S=+∞=0.

(iv) Nn

i=11S∈[t_i−1,t_i[1Tt_i converges weakly to 0 and1S=+∞=0.

Proof. Assumption (i) implies that

Nn

i=1

1S∈[t_i−1,ti[−

Nn

i=1

1Tt_i1S∈[t_i−1,ti[converges weakly toI.

That is

N_n

i=1

1Tti1S∈[ti−1,ti[converges weakly to −1S=+∞

thus

Nn

i=1

1Tt_i1S∈[t_i−1,t_i[1S=+∞converges weakly to −1S=+∞

thus1S=+∞=0. All the others parts of the proof are obvious. 2

Proposition 2. IfSandT are two stopping times onHsuch thatS < T thenHS⊂HT−. Proof. For allf ∈H, the quantity

Nn

i=1

1T >t_i1S∈[t_i−1,t_i[Et_if

belongs toHT−, but it is also equal to

Nn

i=1

1T >ti1S∈[t_i−1,ti[E_SEf,

whereE= {t_i, i=1, . . . , N_n}(with the notationS_Eof the Section 2) which converges weakly toE_Sf. ThusE_Sf belongs toHT−. 2

Proposition 3. (i) If(T_n)_n∈N is an increasing sequence of stopping times converging toT and withT_n< T for alln, thenHT−=

n∈NHTn.

(ii) If(Tn)_n∈N is a decreasing sequence of stopping times converging to T and with Tn> T for alln, then HT =

n∈NHT_n−.

Proof. (i) We haveHTn⊂HT−for alln∈N, thus

n∈NHTn⊂HT−. But by the result 3) of Section 4 we have HT−=

nHT_n−⊂ HT_n.

(ii) We haveHT ⊂HT_n−for alln∈N, thusHT ⊂

n∈NHT_n−. But

n∈NHT_n−is included in

nHT_n which is equal toHT by result 2) of Section 3. This proves (ii). 2

Let us study a rather pathological example. Consider a filtered Hilbert spaces(H, (Et)_t∈R+). Then the spectral measure(Et)_t∈R+itself defines a quantum stopping timeT by putting

1Tt=E_t, t∈R⁺.

Let us then computeHT andHT−. We have HT =

f ∈H; 1Ttf ∈Ht for allt

=

f∈H; Etf∈Ht for allt and thusHT =H.

We have1T >tEtf=(I−Et)Etf =0 for allt∈R⁺. ThusHT−=H0.

Now letTnbe defined by1T_nt =Et+1/n,t∈R⁺. ThenTnis a stopping time again and the sequence(Tn)n∈N

is increasing and converging toT. We obviously haveTn< T for alln. ThusHTn=HTn−=H0. We end this section with two definitions, which follow from the classical corresponding definitions.

A stopping timeT is previsible if there exists an increasing sequence of stopping times(Tn)n∈Nwhich converges toT and such thatTn< T for alln∈N.

A filtered Hilbert space (H, (Et)_t∈R+)is quasi-left continuous if HT =HT− for every previsible stopping timeT.

The pathological example above shows that a filtered Hilbert space is never quasi-left continuous. We actually have to enlarge our definitions.

Let(H, (E_t)_t∈R+)be a filtered Hilbert space. LetUbe a closed subalgebra ofB(H)and(Ut)_t∈R+an increasing family of closed subalgebras ofU such that

t∈RUt generatedU, and which satisfies XE_u=E_uX

for allX∈Ut, allut.

We define a(Ut)_t∈R+-stopping timeT to be a spectral measure on R⁺∪ {+∞}, valued inHand such that1Tt

belongs toUt for allt. Then note that all what has been proved before remains valid for any(Ut)_t∈R+-stopping times.

Examples.

(1) IfUt= {X∈B(H); ∀ut,E_uX=XE_u}then we recover the case studied in the previous sections.

(2) IfU is a von Neumann algebra acting on an Hilbert spaceHand(Ut)_t∈R+ is an increasing family of von Neumann subalgebras which generatesU. Assume there exists a unit vectorΩ∈Hwhich is cyclic and separating forU and a family(Mt)_t∈R+of normalω-invariant conditional expectationsMt:U →Ut, whereω(·)= Ω,·Ω. We denote byHt the closure ofUtΩ inHand byEt the orthogonal projection ontoHt. We then have

Et(XΩ)=Mt(X)Ω

and thus for allut, allX∈Ut, we haveE_uX=XE_u(indeedE_uXAΩ=M_u(XA)Ω=XM_u(A)Ω=XE_uAΩ).

Thus our definitions covers the case of stopping times in von Neumann algebras such as studied in [6] or [5].

(3) LetΦbe the symmetric Fock space onL²(R⁺;C):Φ=Γ_s(L²(R⁺;C)). If we define Φ_t]=Γ_s

L²([0, t];C)

and Φ_[t=Γ_s

L²([t,+∞[;C) ,

we then have the well-known “continuous tensor product” property of Fock spaces:

ΦΦ_t]⊗Φ_[t

and we can considerΦt]as a subspace ofΦ(see next section for more details).

In the framework of quantum stochastic calculus [11] a bounded operatorH onΦis said to be adapted at time tif it is of the formH=K⊗I for someK:Φt]→Φt].

By considering the algebrasUt oft-adapted bounded operators, t∈R⁺, our set up covers all the theory of quantum stopping times on the Fock spaceΦ.

We extend our definitions into:

•A(Ut)_t∈R+-stopping timeT is previsible if there exists a sequence(T_n)_n∈Nof(Ut)_t∈R+-stopping times such that(Tn)_n∈Nconverges toT andTn< T for alln∈N.

• A filtered Hilbert space (H, (Et)_t∈R+) together with a family(Ut)_t∈R+ is quasi-left continuous if for all previsible(Ut)_t∈R+-stopping timeT we haveHT−=HT.

6. The Fock space case

We here just recall few facts about the symmetric Fock space. Details can be found in [2] or [13].

The Fock space Φ is the symmetric Fock space over L²(R⁺;C): Φ =Γ_s(L²(R⁺;C)). This space can be advantageously understood as the spaceL²(P) whereP is the set of finite subsets of R⁺ equipped with the Guichardet symmetric measure. That is, an elementf ofΦ=L²(P)is a measurable functionf:P→C such that

f²=

P

f (σ )²dσ=f (∅)²+ ∞ n=1

0<s₁<···<s_n

f

{s1, . . . , sn}²ds1· · ·dsn<∞. We then have the following properties:

(i) If we writeΦt](respectivelyΦ_[t) for the subspace ofΦmade of thosef such thatf (σ )=0 unlessσ ⊂ [0, t] (respectivelyσ⊂ [t,+∞[), then the mapping

Φt]⊗Φ_[t→Φ f⊗g→h,

withh(σ )=f (σ ∩ [0, t])g(σ ∩ [t,+∞[), extends to a unitary operator. We thus identifyΦ toΦ_t]⊗Φ_[t for all t∈R⁺.

(ii) For allf ∈Φ, if we defineD_tf by [D_tf](σ )=f

σ∪ {t} 1σ⊂[0,t]

we then have thatD_tf belongs toΦfor a.a.t and f²=f (∅)²+

∞

0

Dtf²dt.

(iii) If(g_t)_t∈R+is a family of elements ofΦ such that (a)gt∈Φt]for allt,

(b)t→gt is measurable, (c)_∞

0 g_t²dt <∞,

then(g_t)_t∈R+is said to be Ito-integrable. In this case we write ∞

0

g_tdχ_t

for the elementhofΦgiven by h(σ )=

0 ifσ= ∅,

gt_n({t₁, . . . , tn−1}) ifσ= {t₁< t₂<· · ·< tn}.

This elementhofΦis called the Ito integral of(gt)_t∈R+and we have h²=

∞

0

g_t²dt.

If we denote by1the vacuum ofΦ, that is the element ofΦ given by 1(σ )=

1 ifσ= ∅, 0 otherwise,

and byE_t the orthogonal projection fromΦ ontoΦ_t],t∈R⁺\{0}, we then easily have the following theorem, cf. [2] for details.

Theorem 1. For everyf ∈Φ, the family(Dtf )_t∈R+is Ito integrable and we have f=f (∅)1+

∞

0

D_tf dχ_t. (2)

For allt∈R⁺\{0}we have

E_tf =f (∅)1+ t

0

D_sf dχ_s. (3)

We have the isometry formula

f²=f (∅)²+ ∞

0

Dsf²ds. (4)

We putU to be the algebraB(Φ)of bounded operators onΦ and, for allt∈R⁺,Ut is the algebra oft-adapted bounded operators onΦin the sense of Hudson–Parthasarathy, that is the algebra of bounded operatorsHonΦof the form

H=k⊗I

onΦ_t]⊗Φ_[t, for some bounded operatork onΦ_t]. Note that, for allt∈R⁺, allut and all H∈Ut we have E_uH=H E_u.

Clearly,(Φ, (E_t)_t∈R+)is a filtered Hilbert space and from now on we define stopping times onΦ as being affiliated to the family(Ut)_t∈R+.

In particular(E_t)_t∈R+is not a stopping time onΦ. The following theorem is proved in [3], Proposition 6.

Theorem 2. LetT be a stopping time onΦ. Then for allf ∈Φ we have ETf =f (∅)1+

∞

0

1T >sDsf dχs.

Corollary 3. Let(Tn)_n∈N be any sequence of stopping times onΦ converging to T. Then(ET_n)_n∈N converges strongly toET.

Proof. Indeed, we have by Theorem 2 and Theorem 1(4), ETf−ET_nf²=

∞

0

(1T >s−1T_n>s)Dsf²ds

which converges to 0. 2

The Fock spaceΦadmits several probabilistic interpretations (cf. [2] or [13]) in terms of the Brownian motion, the compensated Poisson process or the Azema martingales. All these classical martingales have in common that their canonical space and filtration is quasi-left continuous.

The theorem to come proves that this property is actually intrinsic to the Fock space structure, and does not depend on any classical probabilistic interpretation of it.

Theorem 4. The filtered Fock space(Φ, (Et)_t∈R+,(Ut)_t∈R+)is quasi-left continuous.

Proof. LetT be a previsible stopping time onΦand(T_n)_n∈Nan increasing sequence of stopping times converging toT withT_n< T for alln∈N.

By Proposition 3 in Section 5 we know that ΦT−=

n∈NΦT_n. But if f ∈ΦT we have f =ETf = limn→+∞ETnf (by Corollary 3 in Section 6) and thusf ∈

n∈NΦTn. This proves thatΦT ⊂

n∈NΦTn. Thus Φ_T−=Φ_T. 2

We are now going to discuss some interesting examples of stopping times onΦ:

1) Projection on chaoses. For everyn∈N, we denote by Cn the space of f ∈Φ such thatf (σ )=0 unless

#σ=n. It is a closed subspace ofΦand we have

Φ=

n∈N

Cn.

The spaceC_nis called thenth chaos ofΦ. We denote byQ_nthe orthogonal projection fromΦonto_n

i=0C_i, that is

[Qnf](σ )=f (σ )1#σn

and byQ_n,tthe operator

[Qn,t, f](σ )=f (σ )1#(σ∩[0,t])n. The operatorQ_n,t ist-adapted and equal to

Qn|Φ_t]⊗I_|Φ_[t.

It is an orthogonal projection also andQ_n,tQ_n,s ifst. We define a stopping timeT_nby putting 1Tn>t=Q_n,t,

1Tn=+∞=Q_n.

We clearly haveT_nT_n+1for alln∈N. Note that for alls, t∈R⁺we have 1Tns1Tn+1t=1Tn+1t1Tns.

We also have

ET_nf =f (∅)1+ ∞

0

1T_n>tDtf dχt=f (∅)1+ ∞

0

QnDtf dχt=f (∅)1+Qn+1

∞

0

Dtf dχt=Qn+1f.

ThusΦT_n=_n+1

i=0Ci. But note that1T_n>tEtf =QnEtf and thusΦT_n−=_n

i=0Ci. In particular theTn’s are not previsible.

2) Jumping times of the Poisson process. For this example only we refer to quantum stochastic integration onΦ (cf. [2] or [13] for details) and we consider the reader very familiar with it.

Let(a_t⁺)_t∈R+,(a_t⁻)_t∈R+and(a⁰_t)_t∈R+be the usual creation, annihilation and conservation processes onΦ. Let Nt=a_t⁺+a⁻_t +a_t⁰+tI be the Poisson process onΦ. We define a family of stopping times(Tn)_n∈Nby

T0=0, 1T_n>t=I−t

0(1T_n>s−1T_n−1>s) dNs, n1.

Indeed, straightforward applications of the quantum Ito formula show that the family(1T_n>t)_t∈R+is a decreasing family of projectors, adapted at timet. Thus they define a stopping timeT_n.

More straightforward applications of the quantum Ito formula show that, for alls, t∈R⁺, alln, m∈N 1Tns1Tmt=1Tmt1Tns,

and thatT_nT_mfornm.

One can even be more precise.

Proposition 5. For allt∈R⁺, the self-adjoint operatorNtadmits a spectrum equal to N and the spectral projection onto the eigenspace associated ton∈N is

1Tnt1T_n+1>t.

Proof. Consider, fort∈R⁺,n∈N^∗, the operators Xⁿ_t =1T_nt1T_n+1>t.

The family(Xⁿ_t)_n∈N∗is family of two by two orthogonal projections whose sum is equal toI (for1Tn>t converges strongly toI whenntends to+∞). Thus(Xⁿ_t)_n∈Nis a spectral measure.

Furthermore,

Xⁿ_t =1Tn+1>t−1Tn>t= t

0

(−1Tn+1>s+1Tn>s+1Tn>s−1Tn−1>s) dN_s= t

0

X_sⁿ⁻¹−X_sⁿ dN_s.

Thus ∞ n=1

nX_tⁿ= t

0

∞ n=0

Xⁿ_sdN_s= t

0

I dN_s=N_t.

Details are left to the reader. 2

Proposition 6. For alln1, we haveT_n< T_n+1. Proof. Let

R=

N−1 i=0

1T_n∈[t_i,t_i+1[1T_n+1∈[t_i,t_i+1[=

N−1 i=0

1T_n+1∈[t_i,t_i+1[1T_n∈[t_i,t_i+1[. From the identity

ε(f ),1Tn>tε(g)

=

n−1

j=0

(_t

0k(s) ds)^j j! e⁻

_t

0f (s) ds

ε(f ), ε(g) ,