MILD SOLUTIONS OF QUANTUM STOCHASTIC DIF-FERENTIAL EQUATIONS

Quantum stochastic di(cid:11)erential equation, stochastic di(cid:11)erential equation, mild solution Abstract We introduce the concept of a mild solution for the right Hudson-Parthasarathy quantum stochastic di(cid:11)erential equation, prove existence and uniqueness results, and show the corre-spondence between our de(cid:12)nition and similar ideas in the theory of classical stochastic di(cid:11)er-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 coe(cid:14)cient matrices for which a mild unitary solution can be found, but for which no strong solution exists.


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 unbounded operators -a requirement that is largely unavoidable in the pursuit of interesting models. For example consider the linear SDE ( [DIT], [DaZ]): in particular the two integrals on the right hand side must be well-defined, and for this to be true we must have X t ∈ Dom A ∩ Dom B almost surely. So if both A and B 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 process X is a mild solution of (1.1) if it satisfies the following integral equation: Note that for the above to make sense we no longer require that X t lie in Dom A a.s., only that X t ∈ Dom B a.s.. The purpose of this paper is to show that such ideas also have a role to play in the theory of quantum stochastic differential equations, in particular when considering the right Hudson-Parthasarathy (HP) equation: Here [Λ α β ] d α,β=0 is the matrix of fundamental noise processes of HP quantum stochastic calculus ( [Mey], [Par]). Each component is a time-indexed family of operators acting on F , the symmetric Fock space over L 2 (R + ; C d ), and they divide into four distinct groups: . . . , d). Linear combinations of the creation and annihilation operators give realisations of Brownian motion; including the conservation processes leads to realisations of (compensated) Poisson processes. The coefficient matrix [F α β ] is made up of (unbounded) operators acting on another Hilbert space h, and the solution process U = (U t ) 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 t 0 X s dΛ α β (s)) is that t>0 Dom X t should contain a subspace of the form D E, the algebraic tensor product of a dense subspace D ⊂ h, and E ⊂ F, the linear span of the exponential vectors {ε(f ) : f ∈ L 2 (R + ; C d )}. Thus the first step in giving rigorous meaning to (R) must be to view each F α β as an operator on h ⊗ F, thereby giving meaning to the term F α β U . This can be done by first taking F α β 1, the algebraic ampliation of F α β with the identity operator on F , and then, making the further assumption that each F α β is closable, taking the closure of the resulting operator which throughout we will denote by F α β ⊗ 1. Then, as defined in [FW], a strong solution of (R) on D, a given dense subspace of h, is any process U such that the integral identity holds on D E, in particular for each integral to be well-defined we must have

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 U t should map D E into Dom F 0 0 ⊗ 1, the domain of the time coefficient. Instead we demand this behaviour from the smeared operator t 0 U s ds, so that a mild solution is a process U such that 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 operator C that behaves well with respect to the F α β . In particular in [FW] it was necessary to assume that Dom C 1/2 is contained in α,β≥0 Dom F α β in order to prove that (1.3) holds for D = Dom C 1/2 . Here, since we need only prove that (1.4) holds, it suffices to assume that Dom C 1/2 is contained in α+β>0 Dom F α β . That this is a significant weakening of the conditions imposed on C can be deduced (at least formally) by an application of the quantum Itô formula: for U to be a solution consisting of contractions it is necessary that F 0 0 have the same order of unboundedness as (F i 0 ) * F i 0 and F 0 j (F 0 j ) * for i, j = 1, . . . d, and that the conservation coefficients F i j be bounded (compare this with the classical equation (1.1), where again the Itô formula can be used to deduce that that time coefficient A is of the same order as B * B.) For example, when realising diffusion processes in the quantum setting F 0 0 is taken to be a second-order differential operator and F i 0 , F 0 j first-order. Thus a natural candidate for C when constructing a strong solution turns out to be ∂ 4 + 1 (where ∂ denotes the differentiation operator on L 2 (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 process U ( [F1], [ChF], [F2]).
As a final remark it should be noted that we have chosen to work with the HP calculus, in part because the majority of results on QSDEs have been obtained in this setting. Other calculi have been developed, for example in the boson Fock space case the recent reformulation by Attal and Lindsay ([AtL]) identifies maximal domains for quantum stochastic integrals of any densely defined process, so in particular if the domain of the process contains the exponential vectors then its AL integral is an extension of the HP one. However the idea of mild solutions should also play a role in the future study of QSDEs using these alternative calculi.

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, if S and T are closable operators on Hilbert spaces H and K respectively then S ⊗ T will denote the closure of the operator S T whose domain is the inner product space Dom S Dom T . So in particular if S ∈ B(H) and T ∈ B(K), then S ⊗ T is the unique continuous extension of S T from H K to the Hilbert space H ⊗ K. At times we will follow the trends prevelant in the literature and identify bounded operators with their ampliations whenever this causes no confusion.
2 Mild solutions of the right HP equation

Quantum stochastic calculus
Fix a Hilbert space h, called the initial space, and a number d ≥ 1, the number of dimensions of quantum noise. Let H = h ⊗ F, the Hilbert space tensor product of the initial space and A crucial ingredient of the HP quantum stochastic calculus is that all of the processes considered are adapted, a property that is defined through the continuous tensor product factorisation property of Fock space: for each t > 0 let the spaces F t and F t are viewed as subspaces of F by tensoring with the appropriate vacuum vector ε(0). Let D be a dense subspace of h. An operator process on D is a family X = (X t ) t≥0 of operators on H satisfying: for all u ∈ h, v ∈ D, f, g ∈ M and t > 0 -condition (iii) is the adaptedness condition. Any process that satisfies the further condition The stochastic integrals t 0 X s dΛ α β (s) are defined for any stochastically integrable process X in [HuP], where [Λ α β ] d α,β=0 are the fundamental noise processes defined with respect to the standard basis of C d . The integral has domain D E, and for all u ∈ h, v ∈ D, f, g ∈ M and t > 0 (2.1) Here f 1 , . . . , f d are the components of the C d -valued function f , and by convention f 0 ≡ 1 and f α = f α . If Y is another process that is stochastically integrable on some subspace D , then, putting Finally we have the estimate

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 = [F α β ], a matrix of operators on h, although we shall encounter the left equation: and in either case we shall only be concerned with contraction process solutions, that is pro- Associated to any such matrix F of operators is the following subspace of h: Given a dense subspace D ⊂ h and a matrix of operators G, a contraction process V is a strong solution to (L) on D for the operator matrix G if Because V is assumed to be a contraction process the matrix of processes [V t G α β ] is well-defined on D, and so to check that (Li) holds it is sufficient to check that the maps t → V t G α β ξ are strongly measurable for all α, β ≥ 0 and ξ ∈ D E. However, note that as soon as (Lii) is shown to hold we know more, since then t → V t η is strongly continuous for all η ∈ 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 of U . If T is a closable operator on h then T 1, the algebraic tensor ampliation with the identity operator on F , is again closable -see Section 1 of [FW] for a discussion of these matters. We will always assume that each component F α β of the stochastic generator in (R) is closable, and denote by F ⊗ 1 the matrix [F α β ⊗ 1] of closed operators on H. With D as above, a contraction process U is a strong solution of (R) on D for the operator matrix F if: The new notion of solution that we are introducing in this paper is the following: the process U is a mild solution of (R) on D for the operator matrix F if: 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 (F 0 0 ⊗ 1) t 0 U s ds 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 F 0 0 is the generator of a strongly continuous contraction semigroup on h. We also justify our terminology since (2.4) below contains stochastic convolution terms analogous to those appearing in (1.2).

Note. The operator
Proposition 2.1 Let U be a contraction process, F = [F α β ] a matrix of closable operators, and D ⊂ h a dense subspace. Suppose further that F 0 0 is the generator of a strongly continuous one-parameter semigroup of contractions (P t ) t≥0 on h. The following are equivalent : (i) U is a mild solution of (R) on D for this F .
(ii) (Mi) holds, and the integral identity Moreover if Dom [F * ] is a core for (F 0 0 ) * and either of the above hold then U is the unique mild solution on D for this F .
Then for all s ∈ [0, t], (2.1) implies (2.5) and since P * is absolutely continuous. Moreover, since each of the Bochner integrals that appears in (2.5) is a.e. differentiable, we have for a.a. s ∈ [0, t]. Now the identity (2.5), with u replaced by −K * u, appears on the right hand side of the above, and so cancellations give for a.a. s ∈ [0, t]. The processes (P t−s (F α β ⊗ 1)U s ) 0≤s≤t are clearly stochastically integrable for all α + β > 0, and so integrating over [0, t] and applying (2.1) gives as required.
(ii ⇒ i): Let u ∈ Dom K * , v ∈ D and f, g ∈ M . Then (2.1) applied to (2.4) gives (2.6) Again the function t → uε(f ), U t vε(g) is absolutely continuous, and so for a.a. t, since on differentiating (2.6) appears with u replaced by K * u. Integrating this over [0, t], and using (2.1) and the stochastic integrability assumptions on the non-time coefficients, yields and since Dom K * E is a core for K * ⊗ 1 = (K ⊗ 1) * , we see that t 0 U s ds vε(g) ∈ Dom K ⊗ 1 and so U 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 That is, the matrix elements uε(f ), U * t vε(g) satisfy the same integral identity satisfied by any strong solution to this equation, but we do not demand that t → U * t (F α β ) * uε(f ) is strongly measurable, and hence stochastically integrable. However, if Dom [F * ] is a core for K * = (F 0 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 remark 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 that t 0 U s ds is well-defined, i.e. that the map t → U t ξ is 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 (U t ) t≥0 is strongly continuous.

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 in F is the time coefficient F 0 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 matrix G = [G α β ] and S ∈ B(h), let θ G (S) denote the form defined by . Also, let ι(S) denote the identity form defined by

Theorem 2.2 Assume that the initial space h is separable, and let F be an operator matrix satisfying the following:
(i) F 0 0 is the generator of a strongly continuous one-parameter semigroup of contractions.
Then there is a contraction process U that satisfies (R) mildly on h.
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 (F 0 0 ) * for the operator matrix F * . Then (2.1) gives and the result follows since Dom (F 0 0 ) * E is a core for (F 0 0 ⊗ 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 operator C with 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 (f : where b 1 , b 2 are positive constants that do not depend on . In [FW] we must also check that (Ri) holds, and so we also demand that Dom C 1/2 ⊂ Dom [F ], forcing C to be "as unbounded as" (F 0 0 ) 2 . An appropriate choice for f in this case is f (x) = x(1 + x) −2 . However we are now only looking for mild solutions, and so to satisfy (Mi) it is enough to assume Dom C 1/2 ⊂ α+β>0 Dom F α β . Thus C will be of the same order as F 0 0 , and so if (F α β ) * f (C) 1/2 is to be bounded for all α, β ≥ 0, in particular for α = β = 0, then we must use higher powers of the resolvent; a reasonable choice is to set Theorem 2.3 Let U be a contraction process and F a matrix of closable operators with F 0 0 the generator of a strongly continuous one-parameter semigroup of contractions. Suppose that C is a positive self-adjoint operator on h, and δ > 0 and b 1 , b 2 ≥ 0 are constants such that the following hold : (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) on D for the operator matrix F * .
(iv) Dom [F ] is dense in h, and for all 0 < < δ the form θ F (C ) satisfies the inequality Then U is a mild solution to the right equation (R) on Dom C 1/2 for the operator matrix F .

Remark.
Since F 0 0 is the generator of a strongly continuous one-parameter semigroup of contractions on h, the fact that D is assumed to be a core for (F 0 0 ) * implies that there is at most one (strong) solution U * to (L) on D for this F * by the result of Mohari ([Moh], cf. the proof of our Proposition 2.1). The uniqueness of U * implies that it is a Markovian cocycle and hence U * and U 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 form θ F (C ) is bounded, and so the inequality in (iv) holds on all of h. Also the integral identity 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 Itô formula (2.2), the Gronwall Lemma and the Spectral Theorem to conclude that Then by (iii) we have U t (Dom C 1/2 E) ⊂ α+β>0 Dom F α β ⊗ 1, and it is straightforward to show that the processes {(F α β ⊗ 1)U t } are stochastically integrable for α + β > 0. So for all u ∈ D, v ∈ Dom C 1/2 and f, g ∈ M since U * is a (strong) solution of (L) for F * , and so by what we have shown so far The result follows once more because D E is a core for (F 0 0 ⊗ 1) * .

A mild solution that cannot be a strong solution
We now show that for a certain class of operator matrices F one can construct a unitary mild solution of (R) on h, but that there is no strong solution for any choice of domain D ⊂ h. To achieve this we need the following general result from semigroup theory: Proof. One implication is obvious, so assume that sup r∈]0,1] r −1 (T r − 1)u < ∞. Since T is strongly continuous the closed subspace H 0 = Lin {T r u : r ≥ 0} of H is separable. Let (e k ) k≥1 be a basis of H 0 , and (r n ) n≥1 a sequence in ]0, ∞[ with lim n r n = 0. Then for each k the sequence { e k , r −1 n (T rn − 1)u } is bounded. A diagonalisation argument allows us to find a subsequence (s m ) m≥1 of (r n ) n≥1 and numbers v k ∈ C such that and so the series k≥1 v k e k converges to an element v ∈ H 0 , satisfying It then follows by Theorem 1.24 of [Dav] that u ∈ Dom Z, with Zu = v.
Let h be any separable Hilbert space, H an unbounded self-adjoint operator on h and choose u 0 ∈ h outside the domain of H. Let d = 1, so that we are working with only one dimension of quantum noise, and define an operator matrix F by where E is the orthogonal projection onto the one-dimensional subspace spanned by u 0 . Written as matrices we have Standard results from perturbation theory for semigroups show that F 0 0 is the generator of a contraction semigroup -a perturbation of the one-parameter unitary group with Stone generator H. Indeed F 0 0 generates a strongly continuous one-parameter group on h, denoted (P t ) t∈R , with P t ≤ 1 whenever t ≥ 0. From now on we will identify the bounded operators E and P t with their ampliations E ⊗ 1 and P t ⊗ 1, recalling that the generator of the one-parameter group (P t ⊗ 1) t∈R is F 0 0 ⊗ 1.
It is easy to check that the form inequality θ F * (1) ≤ 0 holds on Dom H ⊕ Dom H, and so by Theorem 2.2 there is a contraction process U that satisfies (R) mildly on h for this F , that is on h E, and all of the terms in the above make sense! In fact we have that θ F (1) = 0 and θ F * (1) = 0, so that F * satisfies the formal conditions for the process U to consist of unitary operators. That this is indeed the case follows from Section 5 of [F1] and Example 3.2 in [BhS].
That U cannot be a strong solution to (R) for any domain D will be proved using the following lemmas.

Lemma 3.2 The identity
Proof. For any stochastically integrable process X on D and S ∈ B(h) we have for all α, β ≥ 0 and t > 0, and so the result is immediate from Proposition 2.1. Proof. For each r > 0 set S r = r −1 (P r − 1), then by Lemma 3.2 we have Now Im ξ, η ≥ − 1 2 ξ 2 − 1 2 η 2 for any ξ, η ∈ H, and so if we fix T ≥ 0 then Finally note that sup r∈]0,1] S r u 0 2 = ∞ by Lemma 3.1 and choice of u 0 , and if u, u 0 = 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 each t ∈ R the operator P t defines, by restriction, a bijective map of Dom F 0 0 ⊗ 1 onto itself. Thus the unitary process U cannot be a strong solution of (R) for this operator matrix F and any dense subspace D ⊂ h, since for all u ∈ D satisfying u, u 0 = 0 we have by Lemma 3.4 that U t uε(f ) / ∈ Dom F 0 0 ⊗ 1, and the set of such u is dense in D. But U is a mild solution of (R) on h, and so the uniqueness of mild solutions (Proposition 2.1) together with the elementary fact that any strong solution to (R) is necessarily a mild solution shows that there are no strong solutions to (R) for this F and any choice of domain D.