Quantum stopping times and quasi-left continuity

Ann. I. H. Poincaré – PR 40 (2004) 497–512 www.elsevier.com/locate/anihpb

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 property S < 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 space Φ over L2 (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 over L2 (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 propriété S < 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 sur L2 (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

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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 of σ -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 of σ -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 )t
0 ), without really mentioning algebras and we define for a stopping time T , the spaces HT , HT − , the property S < 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 over L2 (R+ ). It is well-known that any classical normal martingale (i.e. with angle bracket x, xt equal to t for all t ∈ R+ ) which possesses the predictable representation property, admits its chaotic space to be naturally isomorphic to the symmetric Fock space over L2 (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 quasileft 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

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for every predictable quantum stopping time T , 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 space H together with a family of orthogonal projections (Et )t ∈R+ with range (Ht )t ∈R+ satisfying
(i) H0  C and s- limt →+∞ Et = I (i.e. t ∈R+ Ht = H). (ii) Es Et = Et Es = Es for all  s
t (i.e. Hs ⊂ Ht for s
t). (iii) s- lim u→t Eu = Et (i.e. u>t Hu = Ht ). u>t

We write Ht − = H0− = H0 ).


s
Hs and Et − is the orthogonal projection onto Ht − , t ∈ R+ (with the convention

An operator X on H is said to be adapted at time t if (i) Eu (Dom X) ⊂ Dom X, for all u  t, and (ii) Eu X = XEu on Dom T , for all u  t. 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 on H, such that, for all t ∈ R+ , the operator T ([0, t]) is adapted at time t. In the following we adopt probabilistic-like notations: for every Borel subset E ⊂ R+ ∪ {+∞} we write 1T ∈E instead of T (E). In the same way 1T t means T ([0, t]), 1T =t means T ({t}), . . . . Note that in particular 1T t = s- lim 1T t +ε , ε→0 ε>0

and 1T 0

In particular, 1T
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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 on H. 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 point t ∈ R+ is a continuity point for a quantum stopping time T if 1T =t = 0. Note that as H is separable, then any stopping time T admits an at most countable set of points which are not of continuity for T . Also note that if t is a continuity point for T then the map s → 1T s is strongly continuous at t. A stopping time T is discrete (or simple) if there exists a finite set E = {0
t1 < t2 < · · · < tn
+∞} in R+ ∪ {+∞} such that 1T ∈E = I . A sequence of stopping times (Tn )n∈N is said to converge to a stopping time T if s- limn→+∞ 1Tn t = 1T t for all continuity point t of T . A stopping time T is finite if 1T =+∞ = 0. Two stopping times S, T satisfy S
T if 1St  1T t for all t ∈ R+ (in the sense of comparison of projectors). In particular 1T t = 1St 1T t = 1T t 1St for all t ∈ 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 time T given by  0 for s < t, 1T s = I for s  t is nothing but the deterministic time T = tI , denoted t simply. Let E = {0 = t0 < t1 < t2 < · · · < tn < +∞} be a partition of R+ . Define the spectral measure TE by  TE ({ti }) = T ([ti−1 , ti [) for i = 1, . . . , n, TE ({+∞}) = T ([tn , +∞]). Then TE is clearly a quantum stopping time. Finally, for a stopping time T , by a sequence of refining T -partitions of R+ we mean a sequence (En )n∈N of finite subsets En = {0 = t0n < t1n < · · · < tkn < +∞} of R+ such that: (i) (ii) (iii) (iv)

all the tjn are continuity points for T , n ∈ N, j  1; En ⊂ En+1 for all n ∈ N; n the diameter δn = sup{ti+1 − tin ; i ∈ N} of En tends to 0 when n tends to +∞; sup En tends to +∞ as n tends to +∞.

Note that, for any stopping time T such a sequence always exists. Now put Tn = TEn for all n ∈ N. Then we have T0  T1  · · ·  T and (Tn )n∈N converges to T . 3. The space HT Let (H, (Ht )t ∈R+ ) be a filtered Hilbert space and T be a stopping time on H. The space Ht classically interprets as the space of events occurring before time t (see the discussion in Section 2 about the connections with classical theory). Thus mimicking the classical definition of FT , the σ -field of events anterior to T , that is   FT = A ∈ F ; (T
t) ∩ A ∈ Ft for all t ∈ R+ ,

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we define in our quantum context: the space of events anterior to a stopping time T is the space   HT = f ∈ H; 1T t f ∈ Ht for all t ∈ R+ . We denote by ET the orthogonal projection onto HT which is clearly a closed subspace of H. The space HT and the projection ET are already defined and studied in [6] for example, so some results are quite standards and we just enumerate them. 1) If S
T then HS ⊂ HT . 2) If (Tn )n∈N is a decreasing sequence of stopping times converging to T then  HT = HTn n∈N

and (ETn )n∈N decreases and converges to ET . 3) HT = {f ∈ H; 1T
n 

1T =ti Eti =

i=1

n 

Eti 1T =ti ,

i=1

with the convention E∞ = I . As an easy consequence of the preceding results, we prove using the notations of sequence of refining T partitions, that the sequence (ETn )n∈N strongly converges to ET . In other words ET = s- lim

n→+∞

Nn 

1T ∈[ti−1 ,ti [ Eti + 1T
tNn

i=1

where the diameter of the partition {0 = t0 < t1 < · · · < tNn } tends to 0 and tNn tends to +∞. 4. The space HT − As our definition of HT 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 time T as the space HT − which is the closure of the subspace of H generated by H0 and {1T >t f ; 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 time T we have HT − ⊂ HT . 2) If S and T are two stopping times such that S
T , then HS− ⊂ HT − . 3) If (Tn )n∈N is an increasing sequence of stopping times converging to T then HT − = HTn − . n

4) Let T be a finite stopping time (i.e. 1T =+∞ = 0). Then the spectral integral operator on H, which we denote by T again. If T is any finite stopping time then the operator T maps Dom T ∩ HT to HT . If moreover T is bounded, it maps HT − to HT − .

R+

t1T ∈dt defines a self-adjoint

Proposition 1. If the filtration is continuous (i.e. Ht − = Ht for all t) then HT − = HT for every discrete stopping time T .

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Proof. Let g ∈ (HT − )⊥ . We thus have, for all t ∈ R+ , all f ∈ H, 1T >t Et f, g = 0,

i.e.

f, 1T >t Et g = 0.

This means 1T >t Et g = 0 for all t ∈ R+ . Now suppose that T is discrete with ni=1 1T =ti = I . If t < t1 then 1T >t = I and Et g = 0 thus by the continuity hypothesis Et1 g = 0.  If ti
t < ti+1 then Et g = 1T t Et g = ij =0 1T =tj Et g. Now let t tend to ti+1 , this gives Eti+1 g = i Et g = 0. j =0 1T =tj Eti+1 g. Thus in particular 1T =ti+1  i+1 All together we have proved that ET g = i 1T =ti Eti g = 0 and thus g ∈ (HT )⊥ . 2

5. Strictly smaller stopping times We wish to give a correct meaning to the relation S < T for two quantum stopping times, S and T on H. If S is  a discrete stopping time with ni=1 1S=si = I and if T is any stopping time, it is then natural to say that S < T if and only if 1S=si = 1T >si 1S=si

for all i.

Note that this in particular implies S
T and we have S < T if and only if n 

1T >si 1S=si = I

i=1

or else if and only if n 

1S=si 1T >si = I.

i=1

This motivates the general definition. Two stopping times S and T on H are said to satisfy S < T if and only if one has that the expression Nn 

1T >ti 1S∈[ti−1 ,ti [

i=1

weakly converges to I when {ti , i = 1, . . . , Nn } follows a sequence of refining S-partitions of R+ . Note that S < T implies S
T for if S < T then 1T t

Nn 

1T >ti 1S∈[ti−1 ,ti [ 1St

i=1

converges weakly to 1T t 1St , but (1) also equals 1T t

Nn 

1T >ti 1S∈[ti−1 ,ti [

i=1

which weakly converges to 1T t .

(1)

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Note that if S and T commute, that is if for all s and t in R+ , 1Ss 1T t = 1T t 1Ss , then for all sequence  n of refining partitions of R+ , Rn = N i=1 1T >ti 1S∈[ti−1 ,ti [ converges strongly. In fact, in this case, (Rn )n
0 is an increasing sequence of projections. Proposition 1. Let S and T be two stopping times. Then the following assertions are equivalent. (i)  S ti converges weakly to I . Nn (iii) i=1 1T ti 1S∈[ti−1 ,ti [ converges weakly to 0 and 1S=+∞ = 0. Nn (iv) i=1 1S∈[ti−1 ,ti [ 1T ti converges weakly to 0 and 1S=+∞ = 0. Proof. Assumption (i) implies that Nn  i=1

1S∈[ti−1 ,ti [ −

Nn 

1T ti 1S∈[ti−1 ,ti [ converges weakly to I.

i=1

That is Nn 

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

i=1

thus Nn 

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

i=1

thus 1S=+∞ = 0. All the others parts of the proof are obvious. 2 Proposition 2. If S and T are two stopping times on H such that S < T then HS ⊂ HT − . Proof. For all f ∈ H, the quantity Nn 

1T >ti 1S∈[ti−1 ,ti [ Eti f

i=1

belongs to HT − , but it is also equal to Nn 

1T >ti 1S∈[ti−1 ,ti [ ESE f,

i=1

where E = {ti , i = 1, . . . , Nn } (with the notation SE of the Section 2) which converges weakly to ES f . Thus ES f belongs to HT − . 2 Proposition 3. (i)
If (Tn )n∈N is an increasing sequence of stopping times converging to T and with Tn < T for all n, then HT − = n∈N HTn . (ii)  If (Tn )n∈N is a decreasing sequence of stopping times converging to T and with Tn > T for all n, then HT = n∈N HTn − . Proof. (i)
We have H
Tn ⊂ HT − for all n ∈ N, thus HT − = n HTn − ⊂ HTn .




n∈N HTn

⊂ HT − . But by the result 3) of Section 4 we have

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   (ii) We have HT ⊂ HTn − for all n ∈ N, thus HT ⊂ n∈N HTn − . But n∈N HTn − is included in n HTn which is equal to HT 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 time T by putting 1T t = Et ,

t ∈ R+ .

Let us then compute HT and HT − . We have     HT = f ∈ H; 1T t f ∈ Ht for all t = f ∈ H; Et f ∈ Ht for all t and thus HT = H. We have 1T >t Et f = (I − Et )Et f = 0 for all t ∈ R+ . Thus HT − = H0 . Now let Tn be defined by 1Tn t = Et +1/n , t ∈ R+ . Then Tn is a stopping time again and the sequence (Tn )n∈N is increasing and converging to T . We obviously have Tn < T for all n. Thus HTn = HTn − = H0 . We end this section with two definitions, which follow from the classical corresponding definitions. A stopping time T is previsible if there exists an increasing sequence of stopping times (Tn )n∈N which converges to T and such that Tn < T for all n ∈ N. A filtered Hilbert space (H, (Et )t ∈R+ ) is quasi-left continuous if HT = HT − for every previsible stopping time T . The pathological example above shows that a filtered Hilbert space is never quasi-left continuous. We actually have to enlarge our definitions. Let (H, (Et )t ∈R+ ) be a filtered Hilbert space.  Let U be a closed subalgebra of B(H) and (Ut )t ∈R+ an increasing family of closed subalgebras of U such that t ∈R Ut generated U , and which satisfies XEu = Eu X for all X ∈ Ut , all u  t. We define a (Ut )t ∈R+ -stopping time T to be a spectral measure on R+ ∪ {+∞}, valued in H and such that 1T t belongs to Ut for all t. Then note that all what has been proved before remains valid for any (Ut )t ∈R+ -stopping times. Examples. (1) If Ut = {X ∈ B(H); ∀u  t, Eu X = XEu } then we recover the case studied in the previous sections. (2) If U is a von Neumann algebra acting on an Hilbert space H and (Ut )t ∈R+ is an increasing family of von Neumann subalgebras which generates U . Assume there exists a unit vector Ω ∈ H which is cyclic and separating for U and a family (Mt )t ∈R+ of normal ω-invariant conditional expectations Mt : U → Ut , where ω(·) = Ω, ·Ω. We denote by Ht the closure of Ut Ω in H and by Et the orthogonal projection onto Ht . We then have Et (XΩ) = Mt (X)Ω and thus for all u  t, all X ∈ Ut , we have Eu X = XEu (indeed Eu XAΩ = Mu (XA)Ω = XMu (A)Ω = XEu AΩ). Thus our definitions covers the case of stopping times in von Neumann algebras such as studied in [6] or [5]. (3) Let Φ be the symmetric Fock space on L2 (R+ ; C): Φ = Γs (L2 (R+ ; C)). If we define

Φt ] = Γs L2 ([0, t]; C) and Φ[t = Γs L2 ([t, +∞[; C) , we then have the well-known “continuous tensor product” property of Fock spaces: Φ  Φt ] ⊗ Φ[t and we can consider Φt ] as a subspace of Φ (see next section for more details).

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In the framework of quantum stochastic calculus [11] a bounded operator H on Φ is said to be adapted at time t if it is of the form H = K ⊗ I for some K : Φt ] → Φt ] . By considering the algebras Ut of t-adapted bounded operators, t ∈ R+ , our set up covers all the theory of quantum stopping times on the Fock space Φ. We extend our definitions into: • A (Ut )t ∈R+ -stopping time T is previsible if there exists a sequence (Tn )n∈N of (Ut )t ∈R+ -stopping times such that (Tn )n∈N converges to T and Tn < T for all n ∈ 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 time T we have HT − = 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 = Γs (L2 (R+ ; C)). This space can be advantageously understood as the space L2 (P) where P is the set of finite subsets of R+ equipped with the Guichardet symmetric measure. That is, an element f of Φ = L2 (P) is a measurable function f : P → C such that   ∞ 2  2   

 2     f {s1 , . . . , sn } 2 ds1 · · · dsn < ∞. f (σ ) dσ = f (∅) + f  = n=1 0
P

We then have the following properties: (i) If we write Φt ] (respectively Φ[t ) for the subspace of Φ made of those f such that f (σ ) = 0 unless σ ⊂ [0, t] (respectively σ ⊂ [t, +∞[), then the mapping Φt ] ⊗ Φ[t → Φ f ⊗ g → h, with h(σ ) = f (σ ∩ [0, t])g(σ ∩ [t, +∞[), extends to a unitary operator. We thus identify Φ to Φt ] ⊗ Φ[t for all t ∈ R+ . (ii) For all f ∈ Φ, if we define Dt f by

[Dt f ](σ ) = f σ ∪ {t} 1σ ⊂[0,t ] we then have that Dt f belongs to Φ for a.a.t and ∞  2 2   f  = f (∅) + Dt f 2 dt. 0

(iii) If (gt )t ∈R+ is a family of elements of Φ such that (a) gt ∈ Φt ] for all t, (b) t → gt is measurable, ∞ (c) 0 gt 2 dt < ∞, then (gt )t ∈R+ is said to be Ito-integrable. In this case we write ∞ gt dχt 0

for the element h of Φ given by  0 h(σ ) = gtn ({t1 , . . . , tn−1 })

if σ = ∅, if σ = {t1 < t2 < · · · < tn }.

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This element h of Φ is called the Ito integral of (gt )t ∈R+ and we have ∞ h2 =

gt 2 dt. 0

If we denote by 1 the vacuum of Φ, that is the element of Φ given by  1 if σ = ∅, 1(σ ) = 0 otherwise, and by Et the orthogonal projection from Φ onto Φt ] , t ∈ R+ \{0}, we then easily have the following theorem, cf. [2] for details. Theorem 1. For every f ∈ Φ, the family (Dt f )t ∈R+ is Ito integrable and we have ∞ f = f (∅)1 +

Dt f dχt .

(2)

0

For all t

∈ R+ \{0}

we have t

Et f = f (∅)1 +

Ds f dχs .

(3)

0

We have the isometry formula 2  f  = f (∅) +

∞ Ds f 2 ds.

2

(4)

0

We put U to be the algebra B(Φ) of bounded operators on Φ and, for all t ∈ R+ , Ut is the algebra of t-adapted bounded operators on Φ in the sense of Hudson–Parthasarathy, that is the algebra of bounded operators H on Φ of the form H =k⊗I on Φt ] ⊗ Φ[t , for some bounded operator k on Φt ] . Note that, for all t ∈ R+ , all u  t and all H ∈ Ut we have Eu H = H Eu . Clearly, (Φ, (Et )t ∈R+ ) is a filtered Hilbert space and from now on we define stopping times on Φ as being affiliated to the family (Ut )t ∈R+ . In particular (Et )t ∈R+ is not a stopping time on Φ. The following theorem is proved in [3], Proposition 6. Theorem 2. Let T be a stopping time on Φ. Then for all f ∈ Φ we have ∞ ET f = f (∅)1 +

1T >s Ds f dχs . 0

Corollary 3. Let (Tn )n∈N be any sequence of stopping times on Φ converging to T . Then (ETn )n∈N converges strongly to ET .

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Proof. Indeed, we have by Theorem 2 and Theorem 1(4), ∞ 2  ET f − ETn f  = (1T >s − 1Tn >s )Ds f  ds 2

0

which converges to 0. 2 The Fock space Φ admits 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. Let T be a previsible stopping time on Φ and (Tn )n∈N an increasing sequence of stopping times converging to T with Tn < T for all n ∈ N.
By Proposition 3 in Section 5 we know that ΦT − =
n∈N ΦTn . But if f ∈ ΦT we have
f = ET f = limn→+∞ ETn f (by Corollary 3 in Section 6) and thus f ∈ n∈N ΦTn . This proves that ΦT ⊂ n∈N ΦTn . Thus ΦT − = ΦT . 2 We are now going to discuss some interesting examples of stopping times on Φ: 1) Projection on chaoses. For every n ∈ N, we denote by Cn the space of f ∈ Φ such that f (σ ) = 0 unless #σ = n. It is a closed subspace of Φ and we have  Cn . Φ= n∈N

The space Cn is called the nth chaos of Φ. We denote by Qn the orthogonal projection from Φ onto is

n

i=0 Ci ,

[Qn f ](σ ) = f (σ )1#σ n and by Qn,t the operator [Qn,t , f ](σ ) = f (σ )1#(σ ∩[0,t ])n. The operator Qn,t is t-adapted and equal to Qn|Φt] ⊗ I|Φ[t . It is an orthogonal projection also and Qn,t
Qn,s if s
t. We define a stopping time Tn by putting  1Tn >t = Qn,t , 1Tn =+∞ = Qn . We clearly have Tn
Tn+1 for all n ∈ N. Note that for all s, t ∈ R+ we have 1Tn s 1Tn+1 t = 1Tn+1 t 1Tn s . We also have ∞ ETn f = f (∅)1 +

∞ 1Tn >t Dt f dχt = f (∅)1 +

0

∞ Qn Dt f dχt = f (∅)1 + Qn+1

0

Dt f dχt = Qn+1 f. 0

that

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 n Thus ΦTn = n+1 i=0 Ci . i=0 Ci . But note that 1Tn >t Et f = Qn Et f and thus ΦTn − = In particular the Tn ’s are not previsible. 2) Jumping times of the Poisson process. For this example only we refer to quantum stochastic integration on Φ (cf. [2] or [13] for details) and we consider the reader very familiar with it. Let (at+ )t ∈R+ , (at− )t ∈R+ and (at0)t ∈R+ be the usual creation, annihilation and conservation processes on Φ. Let Nt = at+ + at− + at0 + tI be the Poisson process on Φ. We define a family of stopping times (Tn )n∈N by  T0 = 0,

t 1Tn >t = I − 0 (1Tn >s − 1Tn−1 >s ) dNs , n  1. Indeed, straightforward applications of the quantum Ito formula show that the family (1Tn >t )t ∈R+ is a decreasing family of projectors, adapted at time t. Thus they define a stopping time Tn . More straightforward applications of the quantum Ito formula show that, for all s, t ∈ R+ , all n, m ∈ N 1Tn s 1Tm t = 1Tm t 1Tn s , and that Tn
Tm for n
m. One can even be more precise. Proposition 5. For all t ∈ R+ , the self-adjoint operator Nt admits a spectrum equal to N and the spectral projection onto the eigenspace associated to n ∈ N is 1Tn t 1Tn+1 >t . Proof. Consider, for t ∈ R+ , n ∈ N∗ , the operators Xtn = 1Tn t 1Tn+1 >t . The family (Xtn )n∈N∗ is family of two by two orthogonal projections whose sum is equal to I (for 1Tn >t converges strongly to I when n tends to +∞). Thus (Xtn )n∈N is a spectral measure. Furthermore, t Xtn

= 1Tn+1 >t − 1Tn >t =

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

0

0

Thus ∞ 

nXtn

=

t  ∞

n=1

t Xsn dNs

=

0 n=0

I dNs = Nt . 0

Details are left to the reader.

2

Proposition 6. For all n  1, we have Tn < Tn+1 . Proof. Let R=

N−1 

1Tn ∈[ti ,ti+1 [ 1Tn+1 ∈[ti ,ti+1 [ =

i=0

N−1 

1Tn+1 ∈[ti ,ti+1 [ 1Tn ∈[ti ,ti+1 [ .

i=0

From the identity n−1    ( ε(f ), 1Tn >t ε(g) = j =0

t 0

 k(s) ds)j − t f (s) ds  ε(f ), ε(g) , e 0 j!

n−1

Xs − Xsn dNs .

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509

 where k(s) = (1 + f (s))(1 + g(s)) and ε(h)(σ ) = s∈σ h(s) for all σ ∈ P, all h ∈ L2 (R+ ; C), we can prove that R converges strongly to 0 when the partition refines and that 1Tn =+∞ = 0. 2 Proposition 7. We have ΦTn = ΦTn − for all n ∈ N∗ , but Tn is not previsible, for any n ∈ N∗ . Proof. Let f ∈ ΦT⊥n − , we have 1Tn >t Et f = 0 for all t ∈ R+ . But also (see [2]) we have t 1Tn >t Et f = f (∅)1 +



1Tn−1 >s Ds f − (1Tn >s − 1Tn−1 >s )Es f dχs

0

t −

(1Tn >s − 1Tn−1 >s )(Ds f + Es f ) ds. 0

Thus, for a.a.t. 1Tn−1 >t Dt f = (1Tn >t − 1Tn−1 >t )Et f and (1Tn >t − 1Tn−1 >t )(Dt f + Et f ) = 0. In particular, 1Tn >t Dt f = 0 for a.a.t. and thus ETn f = 0 (Theorem 2 in Section 6). This proves ΦTn = ΦTn − . It is proved in [3] that if, for all t ∈ R+ t xt =

t ms dχs +

0

as ds 0

∞

∞ with, for all s ∈ R+ , ms and as belong to Φs] , 0 ms 2 ds < ∞ and 0 as  ds < ∞, then for every stopping time T , the limit (over refining partitions as usual)  1T ∈[ti ,ti+1 [ xti+1 xT = lim i

exists and is equal to ∞ xT =

∞ 1T >s ms dχs +

0

1T >s as ds. 0

Suppose Tn is previsible and let (Sp )p∈N be a sequence of stopping times converging to Tn and with Sp < Tn for all p. We know that  1Sp ∈[ti ,ti+1 [ 1Tn >ti+1 1 i (n)

converges to 1. But xt (n) x Sp

= 1Tn >t 1 = 1 −

t

(n) 0 (xs

(n−1)

− xs

)(dχs + ds) thus

+∞

=1− 1Sp >s xs(n) − xs(n−1) (dχs + ds). 0

But 1Sp >s (xs(n) − xs(n−1)) = 1Sp >s 1 − 1Sp >s 1Tn−1 >s 1. This quantity converges to

to +∞.

1Tn >s 1 − 1Tn−1 >s 1 when p tends

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Finally, 1=1−

∞ (1Tn >s 1 − 1Tn−1 >s 1)(dχs + ds). 0

As +∞ (1Tn >s 1 − 1Tn−1 >s 1)(dχs + ds) lim 1Tn >t 1 = 0 = 1 −

t →+∞

0

we have a contradiction. 2

7. Applications to (strictly non-Fock) quasifree boson or fermion quantum stochastic theories In the case of quasifree boson and fermion quantum stochastic theories, we have a family (U
t )t ∈R+ of von Neumann algebras acting on a Hilbert space H and such that Us ⊂ Ut for all s
t. We put U∞ = t ∈R+ Ut . We also suppose that there exists a cyclic and separating unit vector Ω for U∞ in H, and that there exists a family (Mt )t ∈R+ of normal, ω-invariant, conditional expectations Mt : U∞ → Ut where ω is the vector state associated to Ω. We denote by Ht the closure of Ut Ω in H and by Et the orthogonal projection from H to Ht . We have Et XΩ = Mt (X)Ω for all X ∈ U∞ . Furthermore since Ht is invariant under Ut , it follows that Et belongs to Ut . This setup includes the Ito–Clifford (fermion) theory and the quasi-free CAR and CCR theories. In the former case, ω is a tracial state. In all these three cases we have a representation theorem for the elements of H, see [4–6,10,12]. This representation implies that if (Tn )n is a sequence of (Ut )t ∈R+ -stopping times converging to T , then ETn converges strongly to ET (Corollary 3.4 of [6], Theorem 3.7 of [4]). Thus, as in Theorem 4 in Section 6 the filtered space (H, (Et )t ∈R+ , (Ut )t ∈R+ ) is quasi-left continuous.  In fact, in [4,6,7] and others, one defines MT as the strong limit of i 1T ∈[ti−1 ,ti [ Mti and one proves that MT is an orthogonal projection on L2 (U∞ ). It is easy to see that, for X ∈ L2 (U∞ ), we have MT (X)Ω = ET XΩ. Thus, using the isometry between L2 (U∞ ) and H (given  by X → XΩ) makes the study of ET or MT equivalent. T this limit, then clearly Actually, in [4] and [6], MT is the limit of i Mti 1T ∈[ti−1 ,ti [ , but if one denotes by M T (X∗ )∗ . MT (X) = M The case of tracial state. Let us suppose now that ω is tracial. This is for example the case of the Ito–Clifford theory and of the CAR algebra over L2 (R+ ) where ω is the gauge-invariant quasi-free state given by

1 ω b∗ (f )b(g) = 2

+∞ f¯(s)g(s) ds. 0

Proposition 1. Let S, T be two (Ut )t
0 -stopping  times such that S
T . Let (Sn )n∈N be an increasing family of partitions of R+ and RSn be defined by RSn = ti ∈Sn 1T ∈[ti−1 ,ti [ 1S∈[ti−1 ,ti [ . Then the sequence (RSn )n∈N is always strongly convergent. Proof. We have

  ω (RSn − RSm )∗ (RSn − RSm ) = Ω, (RSn − RSm )∗ (RSn − RSm )Ω



    = ω RS∗ n RSn + ω RS∗ m RSm − Ω, RS∗ n RSm Ω − Ω, (RSm )∗ RSn Ω .

S. Attal, A. Coquio / Ann. I. H. Poincaré – PR 40 (2004) 497–512

511

We claim that if S ⊂ S  then RS Ω, RS  Ω = Ω, RS  Ω. Indeed,  RS  = 1T ∈[ti−1 ,ti [ 1S∈[ti−1 ,ti [ t i ∈S 

and RS∗ RS  = =

  s j ∈S t i ∈S 



1S∈[sj−1 ,sj [ 1T ∈[sj−1 ,sj [ 1T ∈[ti−1 ,ti [ 1S∈[ti−1 ,ti [ 

1S∈[sj−1 ,sj [ 1T ∈[ti−1 ,ti [ 1S∈[ti−1 ,ti [

sj ∈S ti ;sj−1 ti−1
thus, by traciality

 ω RS∗ RS  =





ω 1T ∈[ti−1 ,ti [ 1S∈[ti−1 ,ti [ = ω(RS  ).

sj ∈S ti ;sj−1 ti−1
This gives



ω (RSn − RSm )∗ (RSn − RSm ) = ω RS∗ n RSn − ω RS∗ m RSm and (RSn )n∈N is converging in L2 (U∞ ). Let R be the limit. We have RSn ∈ U∞ and RSn 
1 for all n. Therefore RSn converges to R strongly and R belongs to U∞ . 2 In this context we are thus always able to say if two stopping times such that S
T are such that S < T or not. Indeed, this is the case if and only if R = 0 and 1S=+∞ = 0. Remark. If we want the following property to be satisfied: S
and T
R

⇒

S
we need to define S < T by a strong convergence of

 i

1T >ti 1S∈[ti−1 ,ti [ to I .

Indeed, in this case    1R∈[ti−1 ,ti [ 1S∈[ti−1 ,ti [ = 1R∈[ti−1 ,ti [ 1T ∈[ti−1 ,ti [ 1T ∈[ti−1 ,ti [ 1S∈[ti−1 ,ti [ . i

i

i

But the second sum in the right hand side converges strongly to 0 and the first one is bounded by 1. This gives the claim.

References [1] D. Applebaum, The strong Markov property for fermion Brownian motion, J. Funct. Anal. 65 (1986) 273–291. [2] S. Attal, Extensions of quantum stochastic calculus, in: S. Attal, J.M. Lindsay (Eds.), Quantum Probability Summer School, Grenoble, World Scientific, 1998. [3] S. Attal, K.B. Sinha, Stopping semimartingales on Fock space, in: Quantum Probability Communications, vol. X, World Scientific, 1998, pp. 171–186. [4] C. Barnett, T. Lyons, Stopping non-commutative processes, Math. Proc. Camb. Philos. Soc. (1986) 99–151. [5] C. Barnett, R.-F. Streater, I.F. Wilde, The Ito–Clifford integral, J. Funct. Anal. 48 (1982) 172–212. [6] C. Barnett, I. Wilde, Random times and time projections, Proc. Amer. Math. Soc. 110 (2) (1990). [7] C. Barnett, I. Wilde, Random times, predictable processes and stochastic integration in finite von Neumann algebra, Proc. London Math. Soc. 3 (1993) 355–383. [8] C. Barnett, B. Thakrar, Times projections in a von Neumann algebra, J. Operator Theory 18 (1987) 19–31.

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[9] R.L. Hudson, The strong Markov property for canonical Wiener process, J. Funct. Anal. 34 (1979) 266–281. [10] R.L. Hudson, J.M. Lindsay, A non-commutative martingale representation theorem for non-Fock quantum Brownian motion, J. Funct. Anal. 61 (1985) 202–221. [11] R.L. Hudson, K.R. Parthasarathy, Quantum Ito’s formula and stochastic evolutions, Comm. Math. Phys. 93 (1984) 301–323. [12] J.M. Lindsay, Fermion martingales, Probab. Theory Related Fields 71 (1986) 307–320. [13] P.-A. Meyer, Quantum Probability for Probabilists, in: Lecture Notes in Math., vol. 1538, Springer, 1993. [14] K.R. Parthasarathy, K.B. Sinha, Stop times in Fock space stochastic calculus, Probab. Theory Related Fields 75 (1987) 317–349.