Elect. Comm. in Probab. 5(2000)158–171
ELECTRONIC
COMMUNICATIONS in PROBABILITY
MILD SOLUTIONS OF QUANTUM STOCHASTIC
DIF-FERENTIAL EQUATIONS
FRANCO FAGNOLA
Dipartimento di Matematica, Universit`a di Genova, Via Dodecaneso 35, I-16146 Genova email: fagnola@dima.unige.it
STEPHEN J WILLS
School of Mathematical Sciences, University of Nottingham, University Park, Nottingham NG7 2RD
email: sjw@maths.nott.ac.uk
submitted September 12, 2000Final version accepted November 29, 2000 AMS 2000 Subject classification: 81S25; 60H20, 47D06
Quantum stochastic differential equation, stochastic differential equation, mild solution
Abstract
We introduce the concept of a mild solution for the right Hudson-Parthasarathy quantum stochastic differential equation, prove existence and uniqueness results, and show the corre-spondence between our definition and similar ideas in the theory of classical stochastic differ-ential equations. The conditions that a process must satisfy in order for it to be a mild solution are shown to be strictly weaker than those for it to be a strong solution by exhibiting a class of coefficient matrices for which a mild unitary solution can be found, but for which no strong solution exists.
1
Introduction
One of the main analytical difficulties in the theory of stochastic differential equations (both classical and quantum) arises whenever the coefficients driving the equation consist of un-bounded operators — a requirement that is largely unavoidable in the pursuit of interesting models. For example consider the linear SDE ([DIT],[DaZ]):
dXt=AXtdt+BXtdWt, X0=ξ (1.1)
whereAis the generator of a strongly continuous semigroup (St)t≥0 on some Hilbert spaceH, W is a Wiener process taking values inR (respectively some Hilbert spaceK, with covariance operator Q), and B is a linear map from DomB ⊂ H into H (resp. the Hilbert-Schmidt operators Q1/2(
K) → H). An obvious definition of solution for (1.1) is any process (Xt)t≥0
that satisfies the corresponding integral equation:
Xt=ξ+
Z t
0
AXsds+
Z t
0
BXsdWs,
159
in particular the two integrals on the right hand side must be well-defined, and for this to be true we must haveXt ∈DomA∩DomB almost surely. So if both A andB are unbounded then any study of (1.1) must incorporate an investigation of how well their domains match up. An alternative route (considered in Chapter 6 of [DaZ]) is to introduce the following weaker notion of solution: a processX is amild solution of (1.1) if it satisfies the following integral equation:
Xt=Stξ+
Z t
0
St−sBXsdWs. (1.2)
Note that for the above to make sense we no longer require that Xt lie in DomAa.s., only thatXt∈DomB a.s..
The purpose of this paper is to show that such ideas also have a role to play in the theory ofquantumstochastic differential equations, in particular when considering the right Hudson-Parthasarathy (HP) equation:
dUt= d
X
α,β=0
FβαUtdΛαβ(t), U0= 1. (R)
Here [Λα
β]dα,β=0 is the matrix of fundamental noise processes of HP quantum stochastic
cal-culus ([Mey],[Par]). Each component is a time-indexed family of operators acting on F, the symmetric Fock space overL2(
R+;C
d), and they divide into four distinct groups:
Time: Λ00(t) =t1, Annihilation: Λi0(t) =Ai(t)
Creation: Λ0j(t) =A
†
j(t), Conservation: Λij(t) =Nji(t)
(i, j = 1, . . . , d). Linear combinations of the creation and annihilation operators give re-alisations of Brownian motion; including the conservation processes leads to rere-alisations of (compensated) Poisson processes. The coefficient matrix [Fα
β] is made up of (unbounded) operators acting on another Hilbert space h, and the solution process U = (Ut)t≥0 consists of contraction operators on the tensor product Hilbert space h⊗ F. In this paper we use the HP version of quantum stochastic calculus, and an essential part of the definition of the integral of an operator-valued process X against each of these noise processes (denoted
Rt
0XsdΛ
α
β(s)) is that
T
t>0DomXtshould contain a subspace of the formD ⊙ E, the algebraic tensor product of a dense subspace D ⊂ h, and E ⊂ F, the linear span of the exponential vectors{ε(f) :f ∈L2(
R+;C
d)}. Thus the first step in giving rigorous meaning to (R) must be to view eachFα
β as an operator onh⊗ F, thereby giving meaning to the termF α
βU. This can be done by first takingFα
β ⊙1, the algebraic ampliation ofFβα with the identity operator onF, and then, making the further assumption that eachFα
β is closable, taking the closure of the resulting operator which throughout we will denote byFα
β ⊗1. Then, as defined in [FW], a strong solution of (R) onD, a given dense subspace of h, is any process U such that the integral identity
Ut= 1 + d
X
α,β≥0
Z t
0
(Fβα⊗1)UsdΛβα(s)
holds onD ⊙ E, in particular for each integral to be well-defined we must have
[
t>0
Ut(D ⊙ E)⊂ \
α,β≥0
160 1 INTRODUCTION
Clearly this corresponds to the notion of strong solution given above for the equation (1.1). In this paper we introduce a weaker notion of solution which, as in the classical case above, removes the restriction that Utshould map D ⊙ E into DomF00⊗1, the domain of the time
coefficient. Instead we demand this behaviour from the smeared operatorR0tUsds, so that a mild solution is a processU such that
[
t>0
Ut(D ⊙ E)⊂ \
α+β>0
DomFα
β ⊗1, (1.4)
[
t>0
Z t
0
Usds(D ⊙ E)⊂DomF00⊗1,
and
Ut= 1 + (F00⊗1)
Z t
0
Usds+ d
X
α+β>0
Z t
0
(Fβα⊗1)UsdΛβα(s).
We show how this relates to the classical notion of mild solution in Proposition 2.1. Moreover in Section 3 we show that this distinction between strong and mild solutions is nontrivial by exhibiting a class of matrices F for which it is possible to construct a mild unitary solution of (R), but for which no strong solution can exist.
The other main result of this paper (Theorem 2.3) is a general method for constructing mild solutions of (R). This is a modification of the method developed in [FW] for obtaining strong solutions, and both rely on the introduction of a positive self-adjoint operatorCthat behaves well with respect to the Fα
β. In particular in [FW] it was necessary to assume that DomC1/2 is contained in Tα,β≥0DomFα
β in order to prove that (1.3) holds for D= DomC1/2. Here, since we need only prove that (1.4) holds, it suffices to assume that DomC1/2 is contained in
T
α+β>0DomFβα. That this is a significant weakening of the conditions imposed onCcan be deduced (at least formally) by an application of the quantum Itˆo formula: forU to be a solution consisting of contractions it is necessary that F0
0 have the same order of unboundedness as
(Fi
0)∗F0i and Fj0(Fj0)∗ fori, j = 1, . . . d, and that the conservation coefficientsFji be bounded (compare this with the classical equation (1.1), where again the Itˆo formula can be used to deduce that that time coefficientAis of the same order asB∗B.) For example, when realising
diffusion processes in the quantum settingF0
0 is taken to be a second-order differential operator
and Fi
0, Fj0 first-order. Thus a natural candidate forC when constructing a strong solution turns out to be ∂4+ 1 (where ∂ denotes the differentiation operator on L2(
R)), but if we only require a mild solution then we may replace this by ∂2+ 1, which is the same reference
operator used to study the conservativity of the quantum Markov semigroup associated to the processU ([F1],[ChF],[F2]).
1.1 Tensor product conventions 161
1.1
Tensor product conventions
We maintain the conventions concerning tensor products used in [FW], namely that the symbol
⊙is used to denote algebraic tensor products, whereas⊗is reserved for the Hilbert space tensor product of Hilbert spaces and their vectors. Moreover, ifS and T are closable operators on Hilbert spacesHandKrespectively thenS⊗T will denote the closure of the operatorS⊙T
whose domain is the inner product space DomS⊙DomT. So in particular if S ∈ B(H) and
T ∈ B(K), thenS⊗T is the unique continuous extension ofS⊙T fromH⊙Kto the Hilbert spaceH⊗K. At times we will follow the trends prevelant in the literature and identifybounded operators with their ampliations whenever this causes no confusion.
2
Mild solutions of the right HP equation
2.1
Quantum stochastic calculus
Fix a Hilbert spaceh, called theinitial space, and a numberd≥1, the number of dimensions of quantum noise. LetH=h⊗ F, the Hilbert space tensor product of the initial space and
F= Γ(L2(
R+;C
d)), the symmetric Fock space overL2(
R+;C
d). Put
M =L
2(
R+;C
d)∩L∞
loc(R+;C
d) and E = Lin{ε(f) :f ∈ M},
whereε(f) = ((n!)−1/2f⊗n) is the exponential vector associated to the test functionf. Since the subspace M is dense in L
2(
R+;C
d), it follows that E is a dense subspace of F. The elementary tensoru⊗ε(f) will (usually) be abbreviated touε(f) below.
A crucial ingredient of the HP quantum stochastic calculus is that all of the processes consid-ered are adapted, a property that is defined through the continuous tensor product factorisation property of Fock space: for eacht >0 let
Ft= Γ(L2([0, t[;C
d)), Ft= Γ(L2([t,∞[;
C d)).
Then F =Ft⊗ Ft via continuous linear extension of the isometric mapε(f) 7→ε(f|[0,t[)⊗ ε(f|[t,∞[); the spacesFtandFtare viewed as subspaces ofFby tensoring with the appropriate vacuum vectorε(0).
LetDbe a dense subspace ofh. Anoperator process onDis a familyX = (Xt)t≥0of operators onHsatisfying:
(i) D ⊙ E ⊂Tt≥0DomXt,
(ii) t7→ huε(f), Xtvε(g)iis measurable,
(iii) Xtvε(g|[0,t[)∈h⊗ Ft, andXtvε(g) = [Xtvε(g|[0,t[)]⊗ε(g|[t,
∞[),
for all u∈ h, v ∈ D, f, g∈M andt > 0 — condition (iii) is the adaptedness condition. Any process that satisfies the further condition
(iv) t7→Xtvε(g) is strongly measurable andR0tkXsvε(g)k2ds <∞ ∀t >0,
is calledstochastically integrable onD.
The stochastic integralsR0tXsdΛαβ(s) are defined for any stochastically integrable processX in [HuP], where [Λα
162 2 MILD SOLUTIONS OF THE RIGHT HP EQUATION
standard basis of C
d. The integral has domainD ⊙ E, and for all u∈
h, v∈ D, f, g∈M and
t >0
huε(f),
Z t
0
XsdΛαβ(s)vε(g)i=
Z t
0
fβ(s)gα(s)huε(f), X
svε(g)ids. (2.1)
Here f1, . . . , fd are the components of the C
d-valued function f, and by conventionf0 ≡ 1
and fα =fα. If Y is another process that is stochastically integrable on some subspace D′, then, putting IX
t =
Rt
0XsdΛ
α
β(s) andItY =
Rt
0YsdΛ
µ ν(s),
hItXuε(f), ItYvε(g)i=
Z t
0
n
fν(s)gµ(s)hIsXuε(f), Ysvε(g)i
+fα(s)gβ(s)hXsuε(f), IsYvε(g)i+bδβνfα(s)gµ(s)hXsuε(f), Ysvε(g)i
o
ds (2.2)
foru∈ D, v∈ D′, f, g∈
M, and wherebδ∈Md+1(C) is the Evans delta matrix defined by
b
δαβ =
(
1, 1≤α=β≤d
0, otherwise .
Finally we have the estimate
kItXuε(f)k2≤2 exp(νf(t))
Z t
0
kXsuε(f)k2dνf(s) (2.3)
for all u∈ D, f ∈M and t >0, whereνf(t) =
Rt
0(1 +kf(s)k2)ds. This implies in particular
that the map t7→R0tXsdΛαβ(s)ξis continuous for allξ∈ D ⊙ E.
2.2
The right and left equations; notions of solution
As stated in the introduction our main concern in this paper is the right HP equation (R) determined by F = [Fα
β], a matrix of operators on h, although we shall encounter the left equation:
dVt= d
X
α,β=0
VtGαβdΛαβ(t), V0= 1, (L)
and in either case we shall only be concerned with contraction process solutions, that is pro-cessesU orV such thatkUtk ≤1 orkVtk ≤1 for allt. If eachFβα is densely defined thenF∗ will denote the adjoint matrix [(Fβ
α)∗]. Associated to any such matrix F of operators is the following subspace ofh:
Dom [F] := \ α,β≥0
DomFβα.
Given a dense subspaceD ⊂hand a matrix of operatorsG, a contraction processV is astrong solution to (L)on Dfor the operator matrixGif
(Li) D ⊂Dom [G] and each process (VtGαβ)t≥0 is stochastically integrable onD;
2.2 The right and left equations; notions of solution 163
BecauseV is assumed to be a contraction process the matrix of processes [VtGαβ] is well-defined on D, and so to check that (Li) holds it is sufficient to check that the maps t 7→ VtGαβξ are strongly measurable for allα, β ≥ 0 andξ ∈ D ⊙ E. However, note that as soon as (Lii) is shown to hold we know more, since thent7→Vtη is strongly continuous for allη∈h⊗ F. For the right equation (R) the situation is more delicate, as noted in [FW], since we must now pay attention to the image ofU. IfTis a closable operator onhthenT⊙1, the algebraic tensor ampliation with the identity operator onF, is again closable — see Section 1 of [FW] for a discussion of these matters. We will always assume that each componentFα
β of the stochastic generator in (R) is closable, and denote byF⊗1 the matrix [Fα
β ⊗1] of closed operators onH. WithD as above, a contraction processU is a strong solution of (R) on Dfor the operator matrixF if:
(Ri) St>0Ut(D ⊙ E) ⊂Dom [F⊗1], and each of the processes (Fα
β ⊗1)Ut is stochastically integrable onD;
(Rii) Ut= 1 +Pα,β≥0
Rt
0(F
α
β ⊗1)UsdΛβα(s) onD ⊙ E.
The new notion of solution that we are introducing in this paper is the following: the process
U is amild solution of (R)onD for the operator matrixF if: (Mi) St>0Ut(D ⊙ E)⊂Tα+β>0DomFα
β ⊗1, and each of the processes (Fβα⊗1)Utis stochas-tically integrable on D;
(Mii) The mapt7→Utξis strongly measurable for allξ∈ H, and
Rt
0Usds(D ⊙E)⊂DomF 0 0⊗1
for allt >0;
(Miii) Ut= 1 + (F00⊗1)
Rt
0Usds+
P
α+β>0
Rt
0(Fβα⊗1)UsdΛβα(s) onD ⊙ E.
Note. The operatorR0tUsdsin the HP quantum stochastic calculus is defined by
Rt
0Usdsξ:=
Rt
0Usξ ds — the Bochner integral of a vector-valued function rather than the integral of an
operator-valued function.
It is easy to see that any strong solution is also a mild solution. Also (2.3) implies that any strong solution must consist of a strongly continuous family of operators, but the presence of the term (F0
0 ⊗1)
Rt
0Usds seems at first glance only to imply that a mild solution is
weakly continuous. The next result allows us to improve on this by giving an alternative characterisation of mild solutions, once we make the reasonable assumption that F0
0 is the
generator of a strongly continuous contraction semigroup onh. We also justify our terminology since (2.4) below contains stochastic convolution terms analogous to those appearing in (1.2).
Proposition 2.1 Let U be a contraction process, F = [Fα
β] a matrix of closable operators, andD ⊂ha dense subspace. Suppose further thatF
0
0 is the generator of a strongly continuous
one-parameter semigroup of contractions(Pt)t≥0 on h. The following are equivalent: (i) U is a mild solution of (R)on Dfor this F.
(ii) (Mi)holds, and the integral identity
Ut=Pt+
X
α+β>0
Z t
0
Pt−s(Fβα⊗1)UsdΛβα(s) (2.4)
164 2 MILD SOLUTIONS OF THE RIGHT HP EQUATION that appears in (2.5) is a.e. differentiable, we have
d
side of the above, and so cancellations give
2.3 Existence results 165
for a.a.t, since on differentiating (2.6) appears with ureplaced byK∗u. Integrating this over
[0, t], and using (2.1) and the stochastic integrability assumptions on the non-time coefficients, yields
huε(f),(Ut−1)vε(g)i=h(K∗⊙1)uε(f),
Z t
0
Usds vε(g)i
+ X
α+β>0
huε(f),
Z t
0
(Fβα⊗1)UsdΛβα(s)vε(g)i,
and since DomK∗⊙ Eis a core forK∗⊗1 = (K⊗1)∗, we see thatRt
0Usds vε(g)∈DomK⊗1
and soU is a mild solution as required.
Finally, for the uniqueness part, if U is a mild solution to (R) on D for this F then it is easy to check that the adjoint process U∗ is a weak solution of the adjoint left equation
dU∗
t =Ut∗(Fαβ)∗dΛβα(t) on Dom [F∗]. That is, the matrix elements huε(f), Ut∗vε(g)i satisfy the same integral identity satisfied by any strong solution to this equation, but we do not demand that t7→U∗
t(Fβα)∗uε(f) is strongly measurable, and hence stochastically integrable. However, if Dom [F∗] is a core for K∗ = (F0
0)∗ then there is at most one weak solution by
Mohari’s uniqueness result for the left HP equation ([Moh], Proposition 3.6; see also the re-mark after Proposition 2.2 of [FW]), which thus guarantees the uniqueness of the mild solution
to (R).
Remark. For the proof (ii⇒i) we need to know thatR0tUsdsis well-defined, i.e. that the map
t7→Utξis strongly measurable for all ξ∈h⊗ F. But this is immediate from (2.4) — indeed this identity together with (2.3) can be used to show that (Ut)t≥0isstrongly continuous.
2.3
Existence results
We now establish two existence results for mild solutions of (R). For the first we make the very strong assumption that the only unbounded term inF is the time coefficientF0
0 (as happens
in our example in Section 3), and so condition (Mi) becomes a triviality to verify. However the most interesting examples from a probabilistic or physical point of view do not satisfy this assumption — the creation and annihilation coefficients will typically be unbounded — and so Theorem 2.3 below shows how to adapt Theorem 2.3 of [FW] in order to be able to check that (Mi) holds in these cases.
The statements of both results involve form inequalities: given any operator matrixG= [Gα β] andS∈ B(h), let θG(S) denote the form defined by
θG(S)((uγ),(vγ)) = d
X
α,β=0
n
huα, SGαβvβi+hGβαuα, Svβi+ d
X
i=1
hGiαuα, SGiβvβi
o
,
with domainLdγ=0Dom [G]. Also, letι(S) denote the identity form defined by
ι(S)((uγ),(vγ)) =X α≥0
huα, Svαi,
166 2 MILD SOLUTIONS OF THE RIGHT HP EQUATION
Theorem 2.2 Assume that the initial space his separable, and let F be an operator matrix satisfying the following:
(i) F0
0 is the generator of a strongly continuous one-parameter semigroup of contractions.
(ii) Fα
β ∈ B(h)wheneverα+β >0. (iii) θF∗(1)≤0 on
Ld
γ=0Dom [F∗].
Then there is a contraction process U that satisfies (R)mildly on h. Proof. Since θF∗(1) ≤ 0 on
Ld
γ=0Dom [F∗] and h is separable we can apply Theorem 3.6 of [F1] to show the existence of a contraction process U∗ that is a strong solution to (L) on
Dom (F0
0)∗ for the operator matrixF∗. Then (2.1) gives
huε(f),(Ut−1)vε(g)i=
X
α,β≥0
Z t
0
fα(s)gβ(s)h(Fβα)∗uε(f), Usvε(g)ids
for all u∈Dom (F0
0)∗,v ∈handf, g ∈M. NowF α
β ⊗1∈ B(H) wheneverα+β >0, so we have
huε(f),hUt−1−
X
α+β>0
Z t
0
(Fβα⊗1)UsdΛβα(s)
i
vε(g)i=h(F00)∗uε(f),
Z t
0
Usds vε(g)i,
and the result follows since Dom (F0
0)∗⊙ E is a core for (F00⊗1)∗.
In order to be able to deal with unbounded coefficients in the next theorem we follow the ideas of [FW] and introduce a positive self-adjoint operatorCwith which we can gain some control. Both the next result and Theorem 2.3 of [FW] make the same basic assumptions, namely that we hypothesise the existence of a family of (continuous) bounded maps (fǫ: [0,∞[→[0,∞[)ǫ>0
such thatfǫ(x)↑xasǫ↓0 for allx∈[0,∞[, and which satisfy
θF(fǫ(C))≤b1ι(fǫ(C)) +b21, and
(Fβα)∗fǫ(C)1/2 is bounded∀α, β≥0,
where b1, b2 are positive constants that do not depend on ǫ. In [FW] we must also check that (Ri) holds, and so we also demand that DomC1/2 ⊂ Dom [F], forcing C to be “as
unbounded as” (F0
0)2. An appropriate choice for fǫ in this case is fǫ(x) = x(1 +ǫx)−2. However we are now only looking for mild solutions, and so to satisfy (Mi) it is enough to assume DomC1/2 ⊂ T
α+β>0DomFβα. Thus C will be of the same order as F00, and so if
(Fα
β)∗fǫ(C)1/2 is to be bounded for all α, β≥ 0, in particular forα =β = 0, then we must use higher powers of the resolvent; a reasonable choice is to set
Cǫ:=C(1 +ǫC)−4 ∀ǫ >0.
Theorem 2.3 Let U be a contraction process and F a matrix of closable operators with F0 0
the generator of a strongly continuous one-parameter semigroup of contractions. Suppose that
2.3 Existence results 167
(i) There is a dense subspace D ⊂h that is a core for(F
0
0)∗, and the adjoint process U∗ is
a strong solution to (L)onD for the operator matrixF∗.
(ii) For each 0 < ǫ < δ there is a dense subspace Dǫ ⊂ D such that (Cǫ)1/2(Dǫ) ⊂ D and (Fα
β)∗(Cǫ)1/2|Dǫ is bounded for allα, β≥0.
(iii) DomC1/2⊂T
α+β>0DomFβα.
(iv) Dom [F]is dense in h, and for all0< ǫ < δ the form θF(C
ǫ)satisfies the inequality
θF(Cǫ)≤b1ι(Cǫ) +b21
on Dom [F].
Then U is a mild solution to the right equation (R) onDomC1/2 for the operator matrixF.
Remark. Since F0
0 is the generator of a strongly continuous one-parameter semigroup of
contractions onh, the fact thatD is assumed to be a core for (F
0
0)∗ implies that there is at
most one (strong) solutionU∗ to (L) onDfor thisF∗ by the result of Mohari ([Moh], cf. the
proof of our Proposition 2.1). The uniqueness ofU∗ implies that it is a Markovian cocycle
and henceU∗ andU are both strongly continuous.
Proof. To prove this result it is possible to recycle almost all of the argument used in the proof of Theorem 2.3 of [FW]. In particular the formθF(Cǫ) is bounded, and so the inequality in (iv) holds on all ofh. Also the integral identity
Ut∗(Cǫ)1/2= (Cǫ)1/2+
X
α,β≥0
Z t
0
Us∗(Fαβ)∗(Cǫ)1/2dΛβα(s)
holds on h⊙ E, and since all of the terms appearing above are bounded we may take the adjoint of this expression and apply the quantum Itˆo formula (2.2), the Gronwall Lemma and the Spectral Theorem to conclude that
Ut(DomC1/2⊙ E)⊂DomC1/2⊗1.
Then by (iii) we haveUt(DomC1/2⊙ E)⊂T
α+β>0DomFβα⊗1, and it is straightforward to show that the processes{(Fα
β ⊗1)Ut} are stochastically integrable forα+β >0. So for all
u∈ D,v∈DomC1/2andf, g∈
M
huε(f),(Ut−1)vε(g)i=
X
α,β≥0
Z t
0
fα(s)gβ(s)h(Fβα)∗uε(f), Usvε(g)ids
sinceU∗ is a (strong) solution of (L) for F∗, and so by what we have shown so far
huε(f),hUt−1−
X
α+β>0
Z t
0
(Fα
β ⊗1)UsdΛβα(s)
i
vε(g)i=h(F0
0)∗uε(f),
Z t
0
Usds vε(g)i.
The result follows once more becauseD ⊙ E is a core for (F0
168 3 A MILD SOLUTION THAT CANNOT BE A STRONG SOLUTION
3
A mild solution that cannot be a strong solution
We now show that for a certain class of operator matricesF one can construct a unitary mild solution of (R) onh, but that there isno strong solution forany choice of domainD ⊂h. To achieve this we need the following general result from semigroup theory:
Lemma 3.1 Let(Tr)r≥0be a strongly continuous one-parameter semigroup on a Hilbert space
H, and denote its generator by Z. Letu∈H, then
u∈DomZ⇐⇒ sup r∈]0,1]
kr−1(Tr−1)uk<∞.
Proof. One implication is obvious, so assume that supr∈]0,1]kr−1(Tr−1)uk <∞. SinceT is strongly continuous the closed subspaceH0= Lin{Tru:r≥0}ofHis separable. Let (ek)k≥1
be a basis of H0, and (rn)n≥1 a sequence in ]0,∞[ with limnrn = 0. Then for each k the sequence {hek, rn−1(Trn−1)ui} is bounded. A diagonalisation argument allows us to find a
subsequence (sm)m≥1of (rn)n≥1and numbers vk ∈C such that
lim m→∞hek, s
−1
m(Tsm−1)ui=vk ∀k≥1.
Fatou’s Lemma implies
X
k≥1
|vk|2≤lim inf m→∞
X
k≥1
|hek, s−m1(Tsm−1)ui|
2= lim inf
m→∞ ks −1
m(Tsm−1)uk
2<∞
and so the series Pk≥1vkek converges to an element v∈H0, satisfying
lim m→∞hw, s
−1
m(Tsm−1)ui=hw, vi ∀w∈H.
It then follows by Theorem 1.24 of [Dav] thatu∈DomZ, withZu=v.
Lethbe any separable Hilbert space, H an unbounded self-adjoint operator on hand choose
u0∈houtside the domain ofH. Letd= 1, so that we are working with only one dimension of quantum noise, and define an operator matrixF by
F0
0 =iH−
1 2E, F
1
0 =E=−F10, F11= 0,
whereEis the orthogonal projection onto the one-dimensional subspace spanned byu0. Writ-ten as matrices we have
F =
iH−12E −E
E 0
, F∗=
−iH−12E E
−E 0
,
with Dom [F] = Dom [F∗] = DomH. Standard results from perturbation theory for
semi-groups show that F0
0 is the generator of a contraction semigroup — a perturbation of the
one-parameter unitary group with Stone generatorH. IndeedF0
0 generates a strongly
contin-uous one-parametergroup onh, denoted (Pt)t∈
R, with kPtk ≤1 whenevert≥0.
From now on we will identify the bounded operatorsE and Pt with their ampliationsE⊗1 andPt⊗1, recalling that the generator of the one-parameter group (Pt⊗1)t∈RisF
169
It is easy to check that the form inequalityθF∗(1)≤0 holds on DomH⊕DomH, and so by
Theorem 2.2 there is a contraction processU that satisfies (R) mildly onhfor thisF, that is
Ut= 1 + (F00⊗1)
θF∗(1) = 0, so thatF∗ satisfies the formal conditions for the processU to consist of unitary
operators. That this is indeed the case follows from Section 5 of [F1] and Example 3.2 in [BhS]. ThatUcannot be a strong solution to (R) for any domainDwill be proved using the following lemmas.
Proof. For any stochastically integrable processX onDandS∈ B(h) we have (S⊗1)
for allu∈[0, t]. Integrating over this interval gives
−θ(0) =−
170 3 A MILD SOLUTION THAT CANNOT BE A STRONG SOLUTION
Lemma 3.4 For eacht >0letIt denote the operator
P−tUt= 1 +
Z t
0
P−sEUsdA†s−
Z t
0
P−sEUsdAs
with domain h⊙ E, and letu∈h. Then
hu, u0i 6= 0 =⇒Ituε(f)∈/ DomF00⊗1 ∀f ∈M.
Proof. For eachr >0 setSr=r−1(Pr−1), then by Lemma 3.2 we have
SrIt=SrIu+
Z t
u
SrP−sEUsdA†s−
Z t
u
SrP−sEUsdAs
for all 0≤u≤t. Applying the quantum Itˆo formula (2.2) (with initial spaceh⊗ Fu) gives
kSrItuε(f)k2=kSrIuuε(f)k2+ 4
Z t
u
Imf(s)ImhSrIsuε(f), SrP−sEUsuε(f)ids
+
Z t
u
kSrP−sEUsuε(f)k2ds
Now Imhξ, ηi ≥ −12kξk2−1 2kηk
2 for anyξ, η∈ H, and so if we fixT ≥0 then
kSrItuε(f)k2≥ kSrIuuε(f)k2−8kf|[0,T]k2∞
Z t
u
kSrIsuε(f)k2ds
+1 2
Z t
u
kSrP−sEUsuε(f)k2ds
for all 0≤u≤t≤T. Applying Lemma 3.3 gives
kSrItuε(f)k2≥ 1 2e
−ktZ t
0
kSrP−sEUsuε(f)k2ds
wherek= 8kf|[0,T]k2∞. ButSrandP−scommute, and sincePsis a contraction for eachs≥0 we have
kP−sξk ≥ kPskkP−sξk ≥ kξk ∀ξ∈ H, s∈[0, t].
Thus
kSrItuε(f)k2≥1 2e
−kt
Z t
0
kSrEUsuε(f)k2ds
=1 2e
−ktkS ru0k2
Z t
0
kEUsuε(f)k2ds.
Finally note that supr∈]0,1]kSru0k2 =∞by Lemma 3.1 and choice of u0, and ifhu, u0i 6= 0 then the integral on the right hand side above is strictly positive; the result follows by another
application of Lemma 3.1.
Conclusion. Note that for eacht∈R the operatorPtdefines, by restriction, a bijective map of DomF0
REFERENCES 171
for this operator matrix F and any dense subspace D ⊂ h, since for all u ∈ D satisfying
hu, u0i 6= 0 we have by Lemma 3.4 thatUtuε(f)∈/ DomF00⊗1, and the set of suchuis dense
inD. ButU is a mild solution of (R) onh, and so the uniqueness of mild solutions (Proposi-tion 2.1) together with the elementary fact that any strong solu(Proposi-tion to (R) is necessarily a mild solution shows thatthere are no strong solutionsto (R) for thisF andanychoice of domainD.
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