On The Solution Sets Of Semicontinuous Quantum Stochastic Di¤erential Inclusions

On The Solution Sets Of Semicontinuous Quantum Stochastic Di¤erential Inclusions

Michael Oluniyi Ogundiran

, Victor Folarin Payne

Received 1 June 2014

Abstract

The aim of this paper is to provide a uni…ed treatment of the existence of solution of both upper and lower semicontinuous quantum stochastic di¤erential inclusions. The quantum stochastic di¤erential inclusion is driven by operator- valued stochastic processes lying in certain metrizable locally convex space. The uni…cation of solution sets to these two discontinuous non-commutative stochastic di¤erential inclusions is established via the existence of directionally continuous selections.

1 Introduction

Existence results for the solutions of quantum stochastic di¤erential inclusions of Hud- son and Parthasarathy quantum stochastic calculus was established in [9]. The Topo- logical properties of solution sets for this Lipschitzian quantum stochastic di¤erential inclusions were established in [2]. The cases of coe¢ cients that are discontinuous mul- tivalued stochastic processes were established in [14, 15, 16]. In [15] the existence of solutions for upper semicontinuous was established via Fixed point theorem while in [14] the multivlaued stochastic processes possess minimal selections. The existence of solution for the case of Lower semicontinuous multivlaued stochastic processes were established in [16] via continuous selection of some prede…ned integral operators. The extension of quantum stochastic di¤erential inclusions to discontinuous cases was es- sentially to enhance further applications of the rich quantum stochastic calculus to quantum stochastic control theory and evolutions. The quantum stochastic di¤eren- tial inclusions considered in [9] have Lipschitzian coe¢ cients de…ned on certain locally convex space and in [10] more locally convex spaces were considered. By employing one of the locally convex spaces de…ned in [10], a uni…ed treatment of upper and lower semicontinuous cases in this work was established.

For classical di¤erential inclusions, the solution sets of upper and lower semicontin- uous di¤erential inclusions were considered via a directionally continuous selection in [4]. This directionally continuous selection which is a non-convex analogue of Michael selection was …rst considered in [3] for …nite dimensional case and for an arbitrary Ba- nach space in [6]. A more general case was established in [5], which shall be employed

Mathematics Sub ject Classi…cations: 81S25, 34A37.

yDepartment of Mathematics, Obafemi Awolowo University, Ile-Ife, Nigeria

zDepartment of Mathematics, University of Ibadan, Ibadan, Nigeria

135

in our work. The paracompact property of the locally convex space which is the do- main of our multivalued stochastic processes in this work guaranteed the existence of directional continuous selection which provides a link between upper and lower semi- continuous multifunctions. In the sequel, the work shall be as follows; in section 2, preliminaries on quantum stochastic di¤erential inclusions shall be given. In section 3, our main result shall be proved.

2 Preliminaries

In this section, we shall adopt the notations in [10]. Let Dbe some pre-Hilbert space whose completion is R; is a …xed Hilbert space and L²(R+)is the space of square integrable -valued maps onR+. The inner product of the Hilbert spaceR (L²(R+)) will be denoted by h:; :i and k : k the norm induced by h:; :i. Let E be linear space generated by the exponential vectors in Fock space (L²(R+)) and (D E)₁ be the set of all sequences = f ng¹n=1 and = f ng¹n=1 of members of D E, such that

1n=1j h n; x _ni j<1, 8x2 A, where A L⁺_w((D E)₁;R (L²(R+))). Then the family of seminormsfk:k ; ; 2(D E)₁g, where

kxk = X1 n=1

jh n; x _nij forx2 A;

generates a -weak topology, denoted by w [10]. The completion of (A; w) is denoted byAe:The underlying elements ofAeconsist of linear maps from(D E)₁ into R (L²(R+))having domains of their adjoints containing (D E)₁.

REMARK 1. By Theorem V.5 [18], we remark that the -weak topology w is metrizable since(D E)₁ has a countable base, henceAeis a paracompact space [13].

For a …xed Hilbert space , the spaces L^p_loc(Ae), L¹_;loc(R+) and L^p_loc(I Ae) are adopted as in [10]. For a topological space N, let clos(N) be the collection of all nonempty closed subsets of N; we shall employ the Hausdor¤ topology onclos(Ae)as de…ned in [9]. Moreover, for A; B2clos(C)and x2C, a complex number, we de…ne the Hausdor¤ distance, (A; B)as

d(x; B) inf

y2Bjx yj, (A; B) sup

x2A

d(x; B), and (A; B) max( (A; B); (B; A)):

Then is a metric onclos(C)and induces a metric topology on the space.

DEFINITION 1.

(a) By a multivalued stochastic process indexed by I = [0; T] R+, we mean a multifunction onI with values inclos(Ae).

(b) If is a multivalued stochastic process indexed byI R+, then a selection of is a stochastic processX:I!Aewith the property thatX(t)2 (t)for almost allt2I.

(c) A multivalued stochastic process will be called (i) adapted if (t) Aetfor each t2R+;

(ii) measurable if t 7! d (x; (t)) is measurable for arbitrary x 2 Ae, ; 2 D E;

(iii) locally absolutelyp-integrable ift7!k (t)k , t 2R+, lies inL^p_loc(Ae)for arbitrary ; 2D E.

(d) The set of all absolutely p-integrable multivalued stochastic processes will be denoted by L^p_loc(Ae)_mvs and for p 2 (0;1), L^p_loc(I Ae)_mvs is the set of maps :I A !e clos(Ae)such thatt7! (t; X(t)),t2I lies in L^p_loc(Ae)mvs for every X2L^p_loc(Ae):

Consider stochastic processesE; F; G; H2L²_loc(I Ae)and(0; x₀)be a …xed point in [0; T] Ae. Then, a relation of the form

X(t)2x₀+ Z t

0

(E(s; X(s))d (s) +F(s; X(s))dA_f(s) +G(s; X(s))dA⁺_g(s) +H(s; X(s))dsfort2[0; T]

will be called a stochastic integral inclusion with coe¢ cientsE; F;andGandH:

The stochastic di¤erential inclusion corresponding to the integral inclusion above is dX(t)2E(t; X(t))d (t) +F(t; X(t))dAf(t)

+G(t; X(t))dA⁺_g(t) +H(t; X(t))dt;

X(0) =x0 for almost allt2[0; T]:

(1)

LetP: [0; T] A !e 2^{sesq(D E)2} be sesquilinear form valued stochastic process de…ned in [9] in terms ofE; F; G; Hby using the matrix elements in Hudson and Parthasarathy quantum stochastic calculus [12], it was established that problem (1) is equivalent to

d

dth ; X(t) i 2P(t; X(t))( ; );

X(0) =x0for almost allt2[0; T]:

(2) As explained in [9], the map Pis such that:

P(t; x)( ; )6=Pe(t;h ; x i)

for some complex-valued multifunctionPede…ned onI Cfort2I,x2Ae, ; 2D E: The notion of solution of (1) or equivalently (2) is de…ned as follows:

DEFINITION 2. By a solution of (1) or equivalently (2), we mean a stochastic process'2Ad(Ae)_wac\L²_loc(Ae)such that

d'(t)2E(t; '(t))d (t) +F(t; '(t))dAf(t)

+G(t; '(t))dA⁺_g(t) +H(t; '(t))dtfor almost allt2I;

'(t₀) ='₀;

or equivalently

d

dth ; '(t) i 2P(t; '(t))( ; ):

'(t0) ='₀;

for arbitrary ; 2 (D E)₁, almost all t 2 I. A multivalued stochastic process :I A !e 2^Aeis said to be lower semicontinuous if for every open setV Ae, ¹(V) is open. Also, :I A !e 2^Aeis said to be upper semicontinuous if, for everyx2Ae and >0, there exists >0such that

d ((t1; x);(t2; y))< =) (t2; y) B( (t1; x); );

where

d ((t1; x);(t2; y)) = max n

jt1 t2j;kx yk o and

B( (t1; x); ) = n

(t; z)2I Ae:jt t1j< and kz xk <

o :

Letmeas(J)be the Lebesgue measure of a setJ R,tis a point of density forJ if lim!0

meas(J\[t ; t+ ])

2 = 1:

It follows from the previous works; [15], [16] and [14], that ifE; F; G; H2L²_loc(I Ae) are upper semicontinuous (resp. lower semicontinuous) then the equivalent sesquilinear form valued stochastic processPis upper semicontinuous (resp. lower semicontinuous).

We consider a topology ⁺ onI Aestronger than the usual metric topology ofI Ae: A topology ⁺ is said to satisfy a property(P):

(P) For every pair of setsA B withAclosed andBopen (in the original topology) there exists a setC closed-open with respect to the topology ⁺ such thatA

C B

Let I = [a; b] and I Ae, the following set de…ned in [7] is a basis of open neighbourhoods for a topology ⁺ on stronger than the metric one, and satis…es propertyP: For every(t; x)2 and >0,

V(t; x; ) = n

(s; y)2 :t s < t+ and ky xk M(s t) o

:

Moreover, each setV(t; x; )is closed-open in the topology ⁺:

3 Main Results

The following Lemma shall be employed in the proof of the main result.

LEMMA 1. LetX(:)be a Caratheodory solution of upper (lower) semicontinuous quantum stochastic di¤erential inclusion

d

dth ; X(t) i 2 (t; X(t))( ; )on[a; b]:

Assume thatJ is the set of timest2[a; b]such that (i)

d

dth ; X(t) i 2 (t; X(t))( ; ):

(ii) If there exists a sequence tk;strictly decreasing tot, with d

dth ; X(t_k) i 2 d

dth ; X(t) iand d

dth ; X(t_k) i 2 (t_k; X(t_k))( ; ) for anykand ; 2(D E)₁.

Thenmeas(J) =b a:

PROOF. Let J1 be the set of times where (i) holds. Since X is a caratheodory solution, then meas(J1) = b a: Fix any > 0, since _dt^dh ; X(t) i is measurable, by Lusin’s theorem there exists a weakly continuous stochastic process u such that h ; u(t) i = _dt^dh ; X(t) i for every t in a set J2 J1 with meas(J2) > b a : Clearly (ii) holds at every t2J2 which is a point of density forJ2:Hence meas(J) meas(J2)> b a , since was arbitrary, the lemma is proved.

The following is an adaptation of Theorem 1 in [5] to our non commutative setting.

THEOREM 1. Suppose the following hold:

(i) For almost all t 2 I and ; 2 D E, the maps X ! (t; X)( ; ), 2 f E; F; G; Hgare non-empty lower semicontinuous multivalued stochastic processes.

(ii) For almost all t2I and ; 2D E, the mapst! (t; X)( ; )are closed.

(iii) ⁺ is a topology onI Aewith property (P).

Then the sesquilinear form valued multifunction, (t; X(t))!P(t; X(t))( ; ) P(t; X(t))( ; ) = ( E)(t; X(t))( ; ) + ( F)(t; X(t))( ; )

+ ( G)(t; X(t))( ; ) +H(t; X(t))( ; ) admits a ⁺-continuous selection.

PROOF.P is non-empty since each of 2 f E; F; G; Hg is non-empty then P is a non-empty lower semicontinuous sesquilinear form-valued multifunction. We shall employ a similar procedure as in the proof of Theorem 3.2 in [5] to construct a ⁺- continuous -approximate selections P ofP, hence by inductive hypothesis we obtain a ⁺-continuous selectionP ofP:Let >0 be …xed, sinceX !P(t; X)( ; )is lower

semicontinuous, for every X(t) 2 Ae, we choose point y _;X(t) 2 P(t; X(t))( ; ) and neighbourhood UX ofX(t)such that

inf

y ;P(t)2P(t;X(t⁰))( ; )jy _;X(t) y _;P(t)j< forX(t⁰)2U_X: (3) Now, let (V ) ₂ be a local …nite open re…nement of (U_X)_X(t)2Ae, with V U_X , and let (W ) ₂ be another open re…nement such that cl(W ) V for all 2 . By property (P), for each , we can choose a setZ , clopen w.r.t. ⁺, such that

cl(W ) int(Z ) cl(Z ) V : (4)

Then (Z ) is a local …nite ⁺ clopen covering ofAe. Let be a well-ordering of the set , de…ne for each 2 ,

=Z n [

<

Z :

Set O = ( ); 2 . By well-ordering, everyx2Aebelongs to exactly one set where = minf 2 :x2Z g. Hence,O is a partition ofAe. Moreover, sinceZ is locally …nite(wrt and therefore wrt ⁺), the setsS

< Z are ⁺ clopen. Hence O is a ⁺ clopen disjoint covering ofAesuch that,fcl( )gre…nes(V ) . By setting y _; =y ;X andP(t; X(t))( ; ) =y ;X ,8 2 ;we have ⁺continuous function P , which by (3), satis…es

inf

y ;P(t)2P(t;X(t))( ;)jP(t; X(t))( ; ) y ;P(t)j< :

Therefore, there exists an -approximate selection P of P. Since was arbitrarily chosen,thus we have a ⁺-continuous selection P ofP.

LetP :I A !e sesq(D E)²₁be sesquilinear form -valued directionally continuous map as de…ned above. The upper semicontinuous, convex valued regularization of P, corresponding to a given ; 2(D E)₁ is de…ned as

R(t; x)( ; ) =\

>0

co n

P(s; y)( ; ) :jt sj< and kx yk <

o

: (5)

THEOREM 2. Let be a closed subset ofI Ae, and letP:I A !e 2^{sesq(D E)1)} be a bounded, lower semicontinuous multifunction. Then there exists an upper semi- continuous map R : ! 2^{sesq(D E)1)} with compact convex values such that every Caratheodory solution of

d

dth ; X(t) i 2P(t; X(t))( ; ) (6)

is also a solution of

d

dth ; X(t) i 2R(t; X(t))( ; ): (7)

PROOF. LetX(t)be a Caratheodory solution of _dt^dh ; X(t) i 2R(t; X(t))( ; )on [a; b]:De…neJ [a; b]to be the set of timestsuch that

(i) _dt^dh ; X(t) i 2R(t; X(t))( ; ).

(ii) There exists a sequence of timest_kstrictly decreasing totsuch that _dt^dh ; X(t_k) i 2 R(t_k; X(t_k))( ; )and _dt^dh ; X(t_k) i ! _dt^dh ; X(t) i.

By Lemma 1 above, J has a full measure in [a; b]: We claim that _dt^dh ; X(t) i= P(t; X(t))( ; )for everyt2J:Assume on the contrary thatt2J but

d

dth ; X(t) i P(t; X(t))( ; ) = >0: (8) Using the directional continuity of P at the point(t; X), choose >0 such that

jP(s; y)( ; ) P(t; X(t))( ; )j<

2 (9)

whenever t s < t+ , k y X(t)k M(s t): Let tk ! t be a sequence with properties stated in (ii), then there exists klarge enough so that0< tk t < and

d

dth ; X(t_k) i d

dth ; X(t) i <

2: (10)

The boundedness assumptionjP(t; X)( ; )j< Limplies that R(t; X)( ; ) B(0; L) for all (t; X): Our solutionX(t) is therefore Lipschitz continuous with constantL:In particular,

kX(tk) X(t)k L(tk t)< M(tk t):

Then we conclude that

R(t_k; X(t_k)) B P(t; X(t))( ; );

2 : (11)

Hence

d

dth ; X(t_k) i P(t; X(t))( ; )

2: (12)

Comparing we obtain a contradiction, which proves that the caratheodory solutions of d

dth ; X(t) i=P(t; X(t))( ; )

and (7) coincide. Now since P is bounded we can assume P(t; X)( ; ) B(0; L) for some constant L and all(t; X) 2 : Choose M > L and letP be ⁺-continuous selection of P; by Theorem 1 above, such aP exists. Then if R is the regularization multivalued stochastic process as de…ned above, R is upper semicontinuous compact convex-valued [1]. Let now X(:)be a Caratheodory solution of (7) on [a; b]sinceP is a selection ofP, thenX(:)is also a solution of (6).

References

[1] J. P. Aubin and A. Cellina, Di¤erential Inclusions. Set-Valued Maps and Viability Theory. Berlin-Heidelberg-New York-Tokyo, Springer-Verlag 1984.

[2] E. O. Ayoola, Topological properties of solution sets of Lipschitzian quantum stochastic di¤erential inclusions, Acta Appl. Math., 100(2008), 15–37.

[3] A. Bressan, Directionally continuous selections and di¤erential inclusions, Funk- cial. Ekvac., 31(1988), 459–470.

[4] A. Bressan, Upper and Lower Semicontinuous Di¤erential Inclusions: A Uni…ed Approach. Nonlinear Controllability and Optimal Control, 21–31, Monogr. Text- books Pure Appl. Math., 133, Dekker, New York, 1990

[5] A. Bressan and G. Colombo, Selections and representations of multifunctions in paracompact spaces, Studia Math., 102(1992), 209–216.

[6] A. Bressan and A. Cortesi, Directionally continuous selections in Banach spaces and di¤erential inclusions, Nonlinear Anal., 13(1989), 987–992.

[7] A. Cernea and V. Staicu, Directionally continuous selections and Nonconvex evo- lution inclusions, Set-Valued Anal., 11(2003), 9–20.

[8] K. Deimling, Multivalued Di¤erential Equations. de Gruyter Series in Nonlinear Analysis and Applications, 1. Walter de Gruyter & Co., Berlin, 1992.

[9] G. O. S. Ekhaguere, Lipschitzian quantum stochastic di¤erential inclusions, Inter- nat. J. Theoret. Phys., 31(1992), 2003–2027.

[10] G. O. S. Ekhaguere, Topological solutions of noncommutative stochastic di¤eren- tial equations, Stoch. Anal. Appl., 25(2007), 961–993.

[11] A. Guichardet, Symmetric Hilbert Spaces and Related Topics. In…nitely Divisible Positive De…nite Functions. Continuous Products and Tensor Products. Gaussian and Poissonian stochastic processes. Lecture Notes in Mathematics, Vol. 261.

Springer-Verlag, Berlin-New York, 1972.

[12] R. L. Hudson and K. R. Parthasarathy, Quantum Ito’s formula and stochastic evolutions, Comm. Math. Phys., 93(1984), 301–323.

[13] J. L. Kelley, General Topology. D. Van Nostrand Company, Inc., Toronto-New York-London, 1955.

[14] M. O. Ogundiran, Upper semicontinuous Quantum Stochastic Di¤erential Inclu- sions, Int. J. Pure and Appl. Math., 85(2013), 609–627.

[15] M. O. Ogundiran and E. O. Ayoola, Upper semicontinuous quantum stochastic dif- ferential inclusions via Kakutani-Fan …xed point theorem, Dynam. Systems Appl., 21(2012), 121–132.

[16] M. O. Ogundiran and E. O. Ayoola, Lower semicontinuous Quantum stochastic Di¤erential Inclusions, Eur. J. Math. Sci., 2(2013), 1–16.

[17] K. R. Parthasarathy, An Introduction to Quantum Stochastic Calculus, Mono- graphs in Mathematics, 85. Birkhäuser Verlag, Basel, 1992.

[18] M. Reed and B. Simon, Methods of Modern Mathematical Physics. I. Functional analysis. Second edition. Academic Press, Inc., New York, 1980.