Infinite Dimensional Analysis Quantum Probability and Related Topics

Further results on the arcwise connectedness of solution sets of discontinuous quantum stochastic differential inclusions

Dauda Alani Dikko

Published Sep 1, 2020• 1 citation

Abstract

In the framework of the Hudson–Parthasarathy quantum stochastic calculus, we employ a recent generalization of the Michael selection results in the present noncommutative settings to prove that the function space of the matrix elements of solutions to discontinuous quantum stochastic differential inclusion (DQSDI) is arcwise connected.

DOI: 10.1142/s0219025720500228

Related papers

Arcwise Connectedness of Solution Sets of Lipschitzian Quantum Stochastic Differential Inclusions

Dauda Alani Dikko, E. O. Ayoola

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.

Journal of Physics Conference SeriesMar 1, 20172 citations

FURTHER RESULTS ON THE EXISTENCE OF CONTINUOUS SELECTIONS OF SOLUTION SETS OF QUANTUM STOCHASTIC DIFFERENTIAL INCLUSIONS

E. O. Ayoola

ABSTRACT. We prove that the map that associates to the initial value the set of solutions to the Lipschitzian Quantum Stochastic Differential Inclusion (QSDI) admits a selection which is continuous from the locally convex space of stochastic processes to the space of adapted and weakly absolutely continuous solutions. As a corollary, the reachable set multifunction admits a continuous selection. In the framework of the Hudson-Parthasarathy formulation of quantum stochastic calculus, these results are achieved subject to some compactness conditions on the set of initial values and on some coefficients of the inclusion.

Dynamic Systems and ApplicationsJan 1, 20081 citation

On the existence of continuous selections of solution and reachable sets of quantum stochastic differential inclusions

E. O. Ayoola

We prove that the map that associates to the initial value the set of solutions to the Lipschitzian Quantum Stochastic Differential Inclusion (QSDI) admits a selection continuous from the locally convex space of stochastic processes to the adapted and weakly absolutely continuous space of solutions. As a corollary, we show that the reachable sets admit some continuous selections. In the framework of the Hudson - Parthasarathy formulations of quantum stochastic calculus, our results are achieved subject to some compactness conditions on the set of initial values and on some coefficients of the inclusion. The results here compliment similar results in our previous work in [3] where continuous selections defined on the set of the matrix elements of initial values were established. JONAMP Vol. 11 2007: pp. 71-82

Journal of the Nigerian Association of Mathematical PhysicsJun 13, 20080 citations

Continuous Interpolation of Solution Sets of Lipschitzian Quantum Stochastic Differential Inclusions

E. O. Ayoola, John O. Adeyeye

Given any finite set of trajectories of a Lipschitzian quantum stochastic differential 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 difference of any two of such solutions is bounded in the seminorm of the locally convex space of solutions.

Journal of Applied Mathematics and Stochastic AnalysisJan 23, 20070 citations

Exponential Formula for the Reachable Sets of Quantum Stochastic Differential Inclusions

E. O. Ayoola

We establish an exponential formula for the reachable sets of quantum stochastic differential inclusions (QSDI) which are locally Lipschitzian with convex values. Our main results partially rely on an auxilliary result concerning the density, in the topology of the locally convex space of solutions, of the set of trajectories whose matrix elements are continuously differentiable. By applying the exponential formula, we obtain results concerning convergence of the discrete approximations of the reachable set of the QSDI. This extends similar results of Wolenski[20] Wolenski, P.R. 1990. The exponential formula for the reachable set of a Lipschitz differential inclusion. SIAM J. Control Optim., 28(5): 1148–1161. [Crossref], [Web of Science ®] , [Google Scholar] for classical differential inclusions to the present noncommutative quantum setting.

Stochastic Analysis and ApplicationsJan 5, 200317 citations

LIPSCHITZIAN QUANTUM STOCHASTIC DIFFERENTIAL EQUATIONS AND THE ASSOCIATED KURZWEIL EQUATIONS

E. O. Ayoola

Kurzweil or generalized differential equations associated with Lipschitzian quantum stochastic differential equations (QSDEs) are introduced and studied. This is accomplished within the framework of the Hudson-Parthasarathy formulations of quantum stochastic calculus. Results concerning the equivalence of these classes of equations satisfying the Caratheodory conditions are presented. It is further shown that the associated Kurzweil equation may be used to obtain a reasonably high accurate approximate solutions of the QSDEs. This generalize analogous results for classical initial value problems to the noncommutative quantum setting involving unbounded linear operators on a Hilbert space. Numerical examples are given.

Stochastic Analysis and ApplicationsJun 30, 200110 citations