Arcwise Connectedness of Solution Sets of Lipschitzian Quantum Stochastic Differential Inclusions

In the framework of the Hudson - Parthasarathy quantum stochastic calculus, we employ some recent selection results to prove that the function space of the matrix elements of solutions to quantum stochastic differential inclusion (QSDI) is arcwise connected both locally and globally.


Introduction
This paper considers Quantum Stochastic Differential Inclusion(QSDI) in the framework of the Hudson and Parthasarathy formulation [11] of Quantum Stochastic Calculus. It has found applications in the study of quantum stochastic control theory [13] and often occurs as regularization of quantum stochastic differential equations with discontinuous coefficients.
In [3,4,8,9] some topological properties of solution sets of QSDI have been achieved. These were subject to some conditions on the coefficients of their inclusions.
There are some of the interesting motivations [1,2,13,14,15] for studying connectedness, path connectedness and arcwise connectedness of solution sets in the classical differential inclusions with their applications.This provides the possibility of moving from one solution to another.
However as established in [1,14,15] for the case of differential inclusions on finite dimensional Euclidian spaces, this work concerns the establishment of arcwise connectedness of solution sets of quantum stochastic differential inclusion in the integral form: where the driving process Q(t) is a martingale and H, F are sufficiently smooth ordinary functions.
We shall employ the various spaces of quantum stochastic processes introduced in [3,4,8].The remaining part of the work shall be arranged as follows; In section 2, some notations and fundamental structures shall be stated which shall be employed in the sequel. In section 3 some results and assumptions shall be stated and in section 4 the main result of this paper shall be established.

Notations and Fundamental Structures
In what follows, if N is a topological space, we denote by clos(N ), the collection of all non- To H also corresponds a Hilbert space Γ(H), called the Boson Fock space determined by H.
A natural dense subset of Γ(H) consists of the linear space generated by the set of exponential vectors in Γ(H) of the form where ⊗ 0 f = 1 and ⊗ n f is the n-fold tensor product of f with itself for n ≥ 1.
In what follows, ID is some pre-Hilbert space whose completion is R and γ is a fixed Hilbert The noncommutative stochastic processes discussed in the sequel are densely-defined linear operators on R ⊗ Γ(L 2 γ (IR + )); the inner product of this Hilbert space will be denoted by ⟨·, ·⟩. For each t > 0, the direct sum decomposition of Fock space.
Let IE, IE t and IE t , t > 0 be the linear spaces generated by the exponential vectors in Γ(L 2 γ (IR + )), Γ(L 2 γ ([0, t))) and Γ(L 2 γ ([t, ∞))), t > 0 respectively. Then we define where ⊗ denotes algebraic tensor product and 1 t (resp. 1 t ) denotes the identify map on We note that the spaces A t and A t , t > 0, may be naturally identified with subspaces of A.
In the foregoingĀ,Ā t andĀ t denote the completions of the locally convex spaces (A, τ w ), (A t , τ w ), A t , τ w ), t > 0 respectively we then note that {Ã t , t ∈ IR + } is a filtration ofÃ. For A, B ∈ clos(I C) and x ∈ I C , a complex number, define

Hausdorff topology: If
and Then ρ is a metric on clos(I C) and induces a metric topology on the space.
Let I ⊆ IR + . A stochastic process indexed by I is anÃ-valued map on I. A stochastic process X is called adapted if X(t) ∈Ã t for each t ∈ I. We write Ad(Ã) for the set of all adapted stochastic processes indexed by I.

Notation.
We write Ad(Ã) wac [resp. L p loc (Ã)] for the set of all weakly absolutely continuous (resp. locally absolutely p-integrable) members of Ad(Ã).

Stochastic Integrators
be the linear space of all measurable, locally bounded functions from IR + to γ [resp. to B(γ), the Banach space of Bounded endomorphisms of for g ∈ L 2 γ (IR + ). These are the annihilation, creation and gauge operators of quantum field theory.
For arbitrary f ∈ L ∞ γ,loc (IR + ) and π ∈ L ∞ B(γ),loc (IR + ), they give rise to the operator-valued maps t ∈ IR + , where χ I denotes the indicator function of the Borel set I ⊆ IR + .
The maps A f , A + f and ∧ π are stochastic processes, called the annihilation, creation and gauge processes, respectively, when their values are identified with their ampliations on R⊗Γ(L 2 γ (IR + )). These are the stochastic integrators in the Hudson and Parthasarathy [11]formulation of Boson quantum stochastic integration, which we shall adopt in the sequel.
as it is in the Hudson and Parthasarathy [11] formulation.

Stochastic Differential Inclusions
Definition: (i) By a multivalued stochastic process indexed by I ⊆ IR + , we mean a multifunction on I with values in Clos(Ã).
(ii) If Φ is a multivalued stochastic process indexed by I ⊆ IR + , then a selection of Φ is a stochastic process X : I →Ã with the property that X(t) ∈ Φ(t) for almost all t ∈ I.
(iii) A multivalued stochastic process Φ will be called We note that (1) the set of all locally absolutely p-integrable multivalued stochastic processes will be denoted by L p loc (Ã) mvs (2) For p ∈ (0, ∞) and I ⊆ IR + , L p loc (I ×Ã) mvs is the set of maps ,loc (IR + ) , 1 is the identity map on R ⊗ Γ(L 2 γ (IR + )), and M is any of the stochastic processes A f , A + g , A π and s −→ s1, s ∈ IR + .
Thus, we introduce stochastic integral (resp. differential) expressions as follows.
If Φ ∈ L 2 loc (I ×Ã) mvs and (t, X) ∈ I × L 2 loc (Ã), then we make the definition This leads to the following notion.
Definition: Let E, F.G, H ∈ L 2 loc (I ×Ã) mvs and (t 0 , x 0 ) be a fixed point of I ×Ã, then a relation of the form +G(s, X(s))dA + g (s) + H(s, X(s))ds); t ∈ I be called a stochastic integral inclusion with coefficient E, F, G and H initial data (t 0 , x 0 ). We shall sometimes write the foregoing inclusion as follows; This we refer to as stochastic differential inclusions with coefficients E, F, G and H and initial data (t 0 , X 0 ).
Definition: By a solution of (2.1) we mean a weakly absolutely continuous stochastic pro- +G(t, φ(t))dA + g (t) + H(t, φ(t))dt almost all t ∈ I, φ(t 0 ) = x 0 . (i) The existence of solution to a stochastic differential inclusion with Lipschitzian coefficients has been proved in [8].

Remarks
(ii) If M is a subset ofÃ, we write coM for the closed convex hull of M and if Φ : I ×Ã −→ Clos(Ã), we define (iii) Related to (2.1) is the following stochastic differential inclusion: To these functions, are associated the maps µIE, vF, σG, P and coP from I ×Ã into the set of multivalued sesquilinear forms on ID⊗IE defined by where

Inclusion (2.3) is a nonclassical ordinary differential inclusion and the map (η, ξ) → P (t, x)(η, ξ)
is a multivalued sesquilinear form on (ID⊗IE) 2 for (t, x) ∈ [0, T ] ×Ã. We refer the reader to [ 8,9,10] for the explicit forms of the map and the existence results for solutions of QSDI (1.1) of Lipschitz, hypermaximal monotone and of evolution types.

Preliminary Results and Assumptions
As in [3,4,8], As in the references above,we shall employ the space wac(Ã) which is the completion of the locally convex topological space (Ad(Ã) wac , τ ) of adapted weakly absolutely continuous stochastic processes Φ : [0, T ] →Ã whose topology τ is generated by the family of seminorms given by : We assume the following conditions in what follows: S (1) The coefficients E, F, G, H appearing in QSDI (1.1) are continuous.
S (2) The multivalued map (t, x) → P (t, x)(η, ξ) has nonempty and closed values as subsets of the field I C of complex numbers.
S (5) There exists a stochastic process Y : [0, T ] →Ã lying in Ad(Ã) wac such that for each  (5) above, it is well known that the set S (T ) (a) is not empty for arbitrary a ∈Ã (see [8,9,10]).
Definition : A space X is said to be arcwise connected if any two distinct points can be joined by an arc, that is a path f which is a homeomorphism between the unit interval and its