II. Quantum stopping times

! #"%$&(')*+-,

by Stéphane ATTAL & Agnès COQUIO

Prépublication de l’Institut Fourier n^o548 (2001)

.0/0/214352576060698;:=<?>=@BADC?@FEG>IHJ:K8ML2@NCDOI<?PIQJCRES:?@I5710@=C710>JP#QIAJT7U?/BA;<?OBVWEX.0/7YKQ

A. — 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, the propertyS<T for two stopping times, predictable stopping times,... . When applied to the particular case of the symmetric Fock spaceΦoverL²(Z

+)(the natural space for quantum stochastic calculus) we show an intrinsic quasi-left continuity property shared by all the probabilistic interpre- tations ofΦ. We finally apply these definitions to the case of a non commutative stochastic base([B\ ,Φ,([ t)_t

] 0,(M_t)_t

] 0)where^[B\ is a von Neumann algebra, generated by an increasing family of sub-von Neumann algebras([ t)_t

] 0,Φis a normal, faithfull state and (M_t)_t

] 0is a family of conditional expectations on([ t)_t

] 0. We show, in this context, that the fermionic Fock space overL²(Z

+), the quasi-free boson and fermion spaces are also quasi-left continuous.

I. 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.

The theory of quantum processes and quantum noises has had an impressive devel- opment 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 quan- tum open system, with the help of quantum noises, is one of the most remarkable applica- tion of quantum probability theory ([H-P]).

Keywords: quantum stopping times, Fock space, quantum stochastic calculus, non commutative stochastic base.

Math. classification: 46L53, 60G40, 81S25.

The theory of quantum statistical mechanics is now having a very quick develop- ment 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 gener- ators 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 corre- sponding 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 ob- stacle 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 af- fected 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 experi- mentally (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 even occurs in a quan- tum 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 (diag- onalised) on some probability space. As a consequence, any classical stopping time asso- ciated 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 ([Hud]) 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: [App], [P-S], [A-S], [Att], but also in the framework of filtered families ofσ-finite and finite von Neumann algebras: [BN1], [BN2], [B-L],. . ..

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 , _tt 0 , without really mentioning algebras and we define for a stopping timeT, the spaces _T,

T , the propertyS < T for two stopping times and the notion of previsible stopping time.

In the last two parts, we apply the preceding definitions and results to the cases of the symmetric Fock space, the antisymmetric Fock space overL² and the quasi- free representations of the CCR and CAR. We prove that in all these cases the quasi-left continuity property is verified.

II. Quantum stopping times

Afiltered Hilbert spaceis a complex separable Hilbert space together with a fam- ily of orthogonal projections E_t t with range _tt satisfying

i) ₀ ands lim

t

E_t I (i.e.

t

t ).

ii) E_sE_t E_tE_s E_sfor alls t(i.e. _s tfors t).

iii) s lim

u t u>t

E_u E_t(i.e.

u>t

u t).

We write _t

s<t

sandE_t is the orthogonal projection onto _t ,t ^!" (with the convention ₀ 0).

An operatorXon is said to beadapted at timetif i) Eu DomX DomX, for allu ^# t,

and

ii) E_uX X E_uon DomT, for allu ^# t.

Astopping time(orquantum stopping time)Ton a filtered Hilbert space , E_tt

is a (right-continuous) spectral measure on$&%&' )( with values in the set of orthog- onal projectors on , such that, for allt ^!* , the operatorT^,+0, t^- is adapted at time t.

In the following we adopt probabilistic-like notations: for every Borel subsetE &%"' )( we write 11_T Einstead ofT E. In the same way 11_T. tmeansT^/+0, t^- , 11_T0 t

meansT^1' t⁽ ,. . ..

Note that in particular

11_T. t s lim

ε 0

ε>0

11_T. t

ε,

and

11_T<t s lim

ε 0

ε>0

11_T. t ε. In particular, 11_T<tis also adapted at timet, for allt ^{! *} .

Let us see briefly how this definition connects to the classical one. IfT is a clas- sical stopping time on a filtered probability space Ω, ,

tt , P, then taking

L² Ω, , P, _t L² Ω, _t, P (if we assume that the filtration verifies the “usual con- ditions”) andE_t

+/

t^- makes up a filtered Hilbert space. The operators 11T t of multiplication by 11_T. t on then define a quantum stopping time on .

Conversely, ifTis a quantum stopping time on a filtered Hilbert space , E_tt

then the operators 11_T. t,t ^{! %&' )(} , are two by two commuting. Thus they simul- taneously diagonalise on a probability space Ω, , P to give rise to operators of multi- plication by indicator functions of the form 11_τ. t for some random variableτvalued in

% ' ( . Taking

t to be theσ-field generated by the image of _tintoL² Ω, , P , makeτbeing a classical stopping time.

Thus when considering one quantum stopping time (on 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 stop- ping times on . 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 con- text, exactly like observables in quantum mechanics.

A pointt ^! is acontinuity pointfor a quantum stopping timeTif 11_T0 t 0.

Note that as is 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 forTthen the maps 11_T. sis strongly continuous att.

A stopping timeT isdiscreteif there exists a finite setE ' 0 t₁ < t₂ < <

t_n ⁽ in ^{% ' (} such that 11_T E I.

A sequence of stopping times T_nn is said toconverge to a stopping timeT if s lim

n

11Tn^. t 11T^. tfor all continuity pointtofT. A stopping timeTisfiniteif 11_T0

0.

Two stopping timesS,TsatisfyS Tif 11S^. t ^# 11T t for allt ^!&* (in the sense of comparison of projectors). In particular

11_T. t 11_S. t11_T. t 11_T. t11_S. t

for allt ^! .

LetSandTbe two stopping times on . We define the stopping timesS Tand S Ton by, for allt ^! ,

11_S T^. t orthogonal projection onto Ran 11_S. t Ran 11_T. t

11_S T^. t orthogonal projection onto Ran 11_S. t Ran 11_T. t.

P1. — For any two stopping timesS,Ton we have i) S T SandS T T.

ii) S T ^# SandS T ^# T.

iii) IfUis any stopping time satisfyingU SandU Tthen one hasU S T. iv) IfUis any stopping time satisfyingU ^# SandU ^# Tthen one hasU ^# S T.

Proof.

i) We have Ran 11_S T^. t Ran 11_S. t ^% Ran 11_T. t. One concludes easily. The proof ofii)is identical.

iii)IfU SandU Tthen Ran 11U^. t Ran 11S^. t^% Ran 11T^. t. Thus Ran 11U^. t

Ran 11S^. t Ran 11T^. t Ran 11S T^. t. One concludes easily. The proof ofiv)is identical.

Note that the stopping timeTgiven by 11_T. s

0 fors<t I fors ^# t is noting but the deterministic timeT t I, denotedtsimply.

Finally, for a stopping timeT, by asequence of refiningT-partitions of^$ we mean a sequence E_nn of finite subsetsE_n ' 0 t₀ⁿ <t₁ⁿ < <t_kⁿ < ⁽ of such that:

i) all thet_jⁿare continuity points forT,n^! , j ^# 1;

ii) E_n E_n

1for alln ^! ; iii) the diameterδ_n sup^' t_iⁿ

1 t_iⁿ; i ^{! (} ofE_ntends to 0 whenntends to ; iv) supEntends to asntends to .

Note that, for any stopping timeTsuch a sequence always exists.

P2. — For every stopping timeT there exists a sequence T_nn of discrete stopping times such thatT₀ ^# T₁ ^{# #} Tand T_nn converges toT.

Proof. — LetE ' 0 t₀ < t₁ < t₂ < < t_n < ⁽ be a partition of . Define the spectral measureT_Eby

T_E ^' t_i⁽ $ T^/+t_i 1, t_i⁺ fori 1, . . . , n, T_E ^{' )(} T^/+t_n, ^-.

ThenT_Eis clearly a quantum stopping time.

Now let Enn be a sequence of refiningT-partitions of^* and putTn TEnfor alln ^! . Let us first check thatTn ^# Tn

1 ^# Tfor alln ^! . For allt ^! , lett

n i₀ be given byt

n

i0 max^' ti ^! En; ti t⁽ . AsEn En

1we havet

n i0 t

n

1

i0 tand

11_Tn. t 11

T0,t_iⁿ 0

, 11_Tn

1^. t 11

T0,t_i^{n 1} 0

. This provesTn ^# Tn

1 ^# T.

Let us now prove the convergence. Letf ^! andt ^{! *} . We have

11_Tn. tf 11_T. tf

2

11T<t_iⁿ 0

f 11_T. tf

2

11Tt_iⁿ 0 ,tf

2,

which converges to

11_T0 tf

2whenntends to . Thus it converges to 0 iftis a conti- nuity point forT.

III. The space

Let , _tt be a filtered Hilbert space andT be a stopping time on . The space _tclassically interprets as the space of events occurring before timet(see the dis- cussion in section II about the connections with classical theory). Thus mimicking the classical definition of _T, theσ-field of events anterior toT, that is

T ^' A ^!

; T t A ^!

t for all t ^{! (} ,

we define in our quantum context: thespace of events anterior to a stopping timeTis the space

T ^' f ^! ; 11_T. tf ^! _t for all t ^{! (} .

We denote byE_Tthe orthogonal projection onto _T which is clearly a closed subspace of

.

P3.

1) IfS Tthen _S T.

2) If Tnn is a decreasing sequence of stopping times converging toTthen

T

n

Tn

and E_Tnn decreases and converges toE_T.

3) _T ' f ^! ; 11_T<tf ^! tfor allt ^{! (} . Proof.

1) Iff ^! _Sthen 11_T. tf 11_T. t11_S. tf. By hypothesis 11_S. tf belongs to _tand thus so does 11_T. t11_S. tf (by adaptedness). Thusf ^! _T.

2) We haveT Tnand thus T n

Tn. Now iff belongs to

n

Tn andt is a continuity point forT, then 11_T. tf lim

n

11_Tn. tf and thus 11_T. tf belongs to _t. Now, iftis not a continuity point forTwe can find a sequence t_nn of continuity points forT which decreases and converges tot. Thus 11_T. tnf belongs to _tn and 11_T. tf

nlim

11_T. tnf belongs to

n

tn t.

3) As 11_T<t 11_T<t11_T. t and 11_T<t is adapted at timet, we clearly have _T

' f ^! ; 11_T<tf ^! _t for allt ^{! (} . Now, if 11_T<tf ^! _t for allt, then 11_T. tf

nlim

11_T<t

1

nf which belongs to

n

t

1 n

t.

T4. — LetTbe a discrete stopping time with

n

i⁰ 1

11_T0 ti I. We then have

E_T

n

i⁰ 1

11_T0 tiE_ti

n

i⁰ 1

E_ti11_T0 ti,

with the conventionE

I.

Proof. — IfTis a discrete stopping time with values^' 0 t₁ < <t_n ⁾⁽

then the operator

P

n

i⁰ 1

11_T0 tiE_ti (withE

I)

is easily seen to be an orthogonal projector. Iff belongs to RanP, ift ^{! $} then 11T^. tf

n

i 1 ti t

11T⁰ t_iEt_if Et n

i 1 ti t

11T⁰ t_iEt_if

E_t11_T. tf

Thusf belongs to _T. That is, RanP T. Now iff belongs to _Tthen P f

n

i⁰ 1

11_T0 tiE_tif

n

i⁰ 1

E_ti11_T0 tif

n

i⁰ 1

11_T0 tif (for 11_T0 tif ^! _tiby hypothesis)

f . Thus RanP TandP E_T.

T5. — LetT be a quantum stopping time on . Let T_n n be a se- quence of discrete stopping times converging toT, such as in Proposition 2, then the se- quence E_Tnn strongly converges toE_T. In other words

E_T s lim

n

Nn

i⁰ 1

11_Tt_{i 1},t_iE_ti 11_T t_Nn

where the diameter of the partition^' 0 t₀ <t₁ < <t_Nn⁽ tends to0andt_Nntends to

.

Proof. — This is now an easy consequence of Proposition 2, Proposition 3, part 2) and Theorem 4.

P6.

1) IfSandTare two discrete stopping times then E_S T E_S E_T.

2) IfSandTare two stopping times then E_S T E_S E_T.

Proof.

1) We always have _{S T} S T. Now letSandT be discrete and be the union of their support. Suppose

*

' 0 t₁ < <t_N⁽ . Letf ^! _S

T. We haveE_Sf 0 but

E_Sf 11_S0

f

N

i⁰ 0

11_S0 tiE_tif

11_S0

f 11_S. tNE_tN

N 1

i⁰ 0

11_S. ti E_ti

1 E_ti f

f

N

i⁰ 0

11_S. t_i Et_i

1 Et_i f with the conventiontN

1 .

In the same way

E_S Tf f

N

i⁰ 0

11_S T^. ti E_ti

1 E_ti f . But

11_S T^. t_i s lim

n

1 n

n 1

k⁰ 0

11_S. t_i11_T. t_i

k

thus

E_S Tf f lim

n

1 n

n 1

k⁰ 0 N

i⁰ 0

11_S. ti11_T. ti k E_ti

1 E_ti f . Note that

N

i⁰ 0

11_S. ti E_ti

1 E_ti

N

i⁰ 0

11_T. tj E_tj

1 E_tj$

N

i⁰ 0

11_S. ti11_T. ti E_ti

1 E_ti. In the same way

N

i⁰ 0

11_S. ti11_T. ti k E_ti

1 E_ti$ N

i⁰ 0

11_S. ti E_ti

1 E_ti

N

j⁰ 0

11_T. tj E_tj

1 E_tj

k

I E_S I E_T

k

.

This gives

ES Tf f lim

n

1 n

n 1

k⁰ 0

I ES I ET k

f

f ⁾E_S E_T f

0. This proves 1).

2) We always have _S T S T. Now letf belong to _S T. For allt ^! we have

11_S T^. tf f 11_S T>tf

f lim

n

1 n

n 1

k⁰ 0

11_S>t11_T>t

kf .

But

11_S>t11_T>t

k

I 11_S. t I 11_T. t k

I Lt

whereL_tsatisfiesL_tf ^! _t. Thus

1 n

n 1

k⁰ 0

11_S>t11_T>t

kf f g

withg ^! _t. This shows 11_S T^. tf ^! _tand proves 1).

IV. The space

As our definition of _T in our setup seems to fit very well, we pursue the analogy with classical probability theory and define thespace of event strictly anteriorto a stopping timeT as the space _T which is the closure of the subspace of generated by ₀and

' 11_T>tf ; f ^! _t, t ^{! (} .

P7. — For every stopping timeTwe have _T T.

Proof. — First of all, iff ^!& ₀then 11_T. tf 11_T. tE₀f 11_T. tE_tf E_t11_T. tf and thus 11_T. tf belongs to _t, that is ₀ T.

Now iff ^! _t let us considerg 11_T>tf. We then have 11_T. sg

0 ifs t

11_T t ,sf ifs >t. In particular 11_T. sgbelongs to _sfor alls. Thusgbelongs to _T.

We have proved that all the generators of _T are elements of _T, which is closed.

Thus _T T.

P8. — If the filtration is continuous (i.e. _t t for allt) then

T T for every discrete stopping timeT.

Proof. — Letg ^! _T . We thus have, for allt ^{! *} , allf ^! , 11_T>tE_tf , g 0, i.e. f ,11_T>tE_tg 0.

This means 11_T>tE_tg 0 for allt ^! . Now suppose thatT is discrete with

n

i⁰ 1

11_T0 ti I. Ift < t₁then 11_T>t I and E_tg 0 thus by the continuity hypothesisE_t1g 0.

Ift_i t <t_i

1thenE_tg 11_T. tE_tg

i

j⁰ 0

11_T0 t_jE_tg. Now letttend tot_i

1, this givesEt_i

1g

i

j⁰ 0

11_T0 t_jEt_i

1g. Thus in particular 11_T0 t_i

1Et_i

1g 0.

All together we have proved thatETg

i

11_T0 t_iEt_ig 0 and thusg ^!& T .

P9. — IfSandTare two stopping times such thatS T, then _S T .

Proof. — The space _S is generated by ₀and the 11_S>tEtf,f ^! ,t ^! . But 11_S>t 11_S>t11_T>t and thus 11_S>tE_tf 11_S>t11_T>tE_tf 11_T>tE_t11_S>tf ^! _T . This proves

S T .

P10. — If T_nn is an increasing sequence of stopping times con- verging toTthen

T

n

Tn .

Proof. — By Proposition 9, each space _Tn is a subspace of _T , thus

n Tn T . Conversely, if f belongs to and ift is a continuity point forT,

then 11_T>tE_tf lim

n

11_Tn>tE_tf which belongs to n Tn . Iftis not a continuity point

forT, then choose a decreasing sequence tn n of continuity points forT, converging to t. We conclude by the identity

11_T>tE_tf lim

n

11_T>tnE_tf . LetT be a finite stopping time (i.e. 11_T0

0). Then the spectral integral

t11_T dtdefines a self-adjoint operator on , which we denote byTagain.

P 11. — IfT is any finite stopping time then the operator T maps DomT T to _T.

If moreoverTis bounded, it maps _T to _T .

Proof. — Iff belongs to DomT Tthen T f lim

i

t_i11_Tt_i,t_i

1f

(where the limits is taken as usual under a sequence of partitions whose diameter tends to 0) and thus, as 11_T. s

i

t_i11_Tti,ti

1f belongs to _s, we have 11_T. sT f ^!& _sand thus T f ^! T. This proves the first part.

IfTis bounded, we have

T11_T>tEtf lim

i

ti11_T

t_i,t_i

111_T>tEtf

where each term in the sum is an element of _T . ThusT11_T>tE_tf belongs to _T . This provesT _T T .

V. Strictly smaller stopping times

We wish to give a correct meaning to the relationS<Tfor two quantum stopping times,SandT on . IfSis a discrete stopping time with

n

i⁰ 1

11_S0 si I and ifT is any stopping time, it is then natural to say thatS<Tif and only if

11_S0 si 11_T>si11_S0 si for all i .

Note that this in particular impliesS Tand we haveS<Tif and only if

n

i⁰ 1

11_T>si11_S0 si I