Continuous Interpolation of Solution Sets of Lipschitzian Quantum Stochastic Differential Inclusions

Given any ﬁnite set of trajectories of a Lipschitzian quantum stochastic di ﬀ erential inclusion (QSDI), there exists a continuous selection from the complex-valued multifunction associated with the solution set of the inclusion, interpolating the matrix elements of the given trajectories. Furthermore, the di ﬀ erence of any two of such solutions is bounded in the seminorm of the locally convex space of solutions. distrib-uted under Attribution the original is properly cited.


Introduction
Establishment of continuous selections from the solution set multifunctions of differential inclusions defined on finite dimensional Euclidean spaces and their applications have been considered by many authors (see, e.g., Aubin and Cellina [1], Repovs and Semenov [2], Smirnov [3] and the references they contain). However, in the context of quantum stochastic differential inclusions (QSDI), research in these subjects has not enjoyed a comparable attention. In addition, theoretical and numerical aspects of QSDI have not enjoyed significant development in comparison with the classical cases although there are some recent results along these directions (see, e.g., [4][5][6][7][8][9][10]). This situation remains in spite of the numerous practical problems in quantum dynamical systems, quantum open systems, quantum measurement theory, quantum optics, and quantum stochastic control theory for which methods of quantum stochastic inclusions are applicable. In particular, it is well known that discontinuous quantum stochastic differential equations can be numerically and theoretically treated by reformulating them as regularized inclusions (see [8][9][10][11][12][13]).

Journal of Applied Mathematics and Stochastic Analysis
Our present research effort in this field is motivated by the need to further explore the properties of solution spaces of quantum stochastic differential inclusions. The present paper is, therefore, concerned with the establishment of the existence of continuous selections from the complex valued multifunctions associated with the solution sets of QSDI interpolating the matrix elements of a given finite set of trajectories that start from distinct points. This work is a continuation of our work in [4] extending the case of a single trajectory to the case of a finite set of trajectories. In addition, we show that the difference of any two solutions of QSDI (1.1) below, that correspond to two distinct initial values is bounded in the seminorm of the locally convex space of solutions.
In what follows, we will be concerned with quantum stochastic differential inclusion in the integral form, given by (1.1) We will employ in this paper the various spaces of quantum stochastic processes introduced in the works of Ekhaguere [8] and Ayoola [4]. As usual, our work is accomplished within the framework of the Hudson and Parthasarathy [11] formulation of quantum stochastic calculus employing the notations and the QSDI setup due to [8]. Corresponding to a pre-Hilbert space D with completion and the Boson Fock space Γ(L 2 γ (R + )) with the dense subspace E generated by exponential vectors, we follow the fundamental concepts and structures as in the references by employing the locally convex space Ꮽ of noncommutative stochastic processes whose topology is generated by the family of seminorms { x ηξ = | η,xξ |, x ∈ Ꮽ, η,ξ ∈ D⊗E}. The underlying elements of Ꮽ consists of linear maps from D⊗E into ⊗ Γ(L 2 γ (R + )) having domains of their adjoints containing D⊗E. In particular, the spaces L P loc ( Ꮽ), L ∞ γ,loc (R + ), L P loc (I × Ꮽ) for a fixed Hilbert space γ are being adopted as in the above references. In the foregoing setup, the integral appearing in (1.1) is a set-valued quantum stochastic integral as defined in [8]. The coefficients E, F, G, H are elements of L 2 loc ([0,T] × Ꮽ) mvs , where Ꮽ is a locally convex space and (0,a) ∈ [0,T] × Ꮽ is a fixed point. The maps f , g, π appearing in (1.1) lie in some suitable function spaces. The integrators ∧ π , A + g , and A f are the gauge, creation, and annihilation processes associated with the basic field operators of quantum field theory.
As in Ayoola [4][5][6][7] we will consider the equivalent form of (1.1) given by d dt η,X(t)ξ ∈ P t,X(t) (η,ξ), (1.2) Inclusion (1.2) is a nonclassical ordinary differential inclusion and the map (η,ξ)→ P(t,x)(η,ξ) is a multivalued sesquilinear form on (D⊗E) 2 for (t,x) ∈ [0,T] × Ꮽ. We refer the reader to the works of Ekhaguere [8][9][10]  The rest of the paper is organised as follows: In Section 2, we outline some fundamental definitions, notations and results needed for the establishment of the main result. Section 3 is devoted to the main results of the paper.

Preliminary results and assumptions
As in [4,8], we let clos(ᏺ) denote the family of all nonempty closed subsets of a topological space ᏺ. For ᏺ ∈ { Ꮽ, C}, we adopt the Hausdorff topology on clos(ᏺ) as explained in the references above. We denote by d(x,A), the distance from a point x ∈ C to a set A ⊆ C. For A,B ∈ clos(C), ρ(A,B) denote the Hausdorff distance between the sets.
For a real number δ > 0, we let B(x,δ) denote the open ball of radius δ around a point x ∈ C. As in the references above,we shall employ the space wac( Ꮽ) which is the completion of the locally convex topological space (Ad( Ꮽ) wac ,τ) of adapted weakly absolutely continuous stochastic processes Φ : [0,T]→ Ꮽ whose topology τ is generated by the family of seminorms given by Associated with space wac( Ꮽ), we will employ the space wac( Ꮽ)(η,ξ) consisting of absolutely continuous complex valued functions η,Φ(·)ξ := Φ ηξ (·) : [0,T]→C, where Φ ∈ wac( Ꮽ) and for arbitrary pair of points η,ξ ∈ D⊗E. We will also denote by S (T) (a), the subset of wac( Ꮽ) consisting of the set of solutions of QSDI (1.1) corresponding to the initial value a ∈ Ꮽ and write We assume the following conditions in what follows.
( (2) ) The multivalued map (t,x)→P(t,x)(η,ξ) has nonempty and closed values as subsets of the field C of complex numbers.
( (6) ) The initial point a lies in a set A ⊆ Ꮽ such that the set of complex numbers A(η,ξ) := { η,aξ : a ∈ A} is compact in C. For points a i ∈ A, i = 1,2,..., and Y i ∈ S (T) (a i ), we employ the notation a ηξ,i := η,a i ξ and Y ηξ,i (·):= η,Y i (·)ξ where a→S (T) (a) is the multivalued solution map of QSDI (1.1) corresponding to the initial value x = a.

Journal of Applied Mathematics and Stochastic Analysis
Under the conditions above, it is well known that the set S (T) (a) is not empty for arbitrary a ∈ Ꮽ (see Ekhaguere [8][9][10]).
Next, we recall from [4] a useful result in what follows.

5)
for some set E ⊆ I with measure μ(E) < .

Main results
The main result of this paper is established by adapting to the present quantum stochastic calculus, a line of argument employed in the work of Broucke and Arapostathis [14], concerning classical differential inclusions where multifunctions take values in finite dimensional Euclidean spaces. In what follows, we establish a continuous selection that interpolates a finite number of trajectories extending the case of a single trajectory established in [4].  let {p j } M j=1 be a partition of unity subordinated to it (see Ayoola [4] for the existence of such partition of unity).
We remark that by the definition of the covering, each a ηξ,i belongs to exactly one member of the subcovering since for each k = i, the inequality |a ηξ,k − a ηξ,i | < δ(a ηξ,i ) is invalid.