Quantum Stochastic Differential Inclusions Research Articles

Viable solutions of lower semicontinuous quantum stochastic differential inclusions

We establish the existence and some properties of viable solutions of lower semicontinuous quantum stochastic differential inclusions within the framework of the Hudson–Parthasarathy formulations of quantum stochastic calculus. The main results here are accomplished by establishing a major auxiliary selection result. The results here extend the classical Nagumo viability theorems,valid on finite dimensional Euclidean spaces, to the present infinite dimensional locally convex space of noncommutative stochastic processes.

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

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.

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.

On topological properties of solution sets of non Lipschitzian quantum stochastic differential inclusions

In this paper, we establish results on continuous mappings of the space of the matrix elements of an arbitrary nonempty set of pseudo solutions of non Lipschitz quantum Stochastic differential inclusion (QSDI) into the space of the matrix elements of its solutions. we show that under the non Lipschitz condition, the space of the matrix elements of solutions is still an absolute retract, contractible, locally and integrally connected in an arbitrary dimension. The results here generalize existing results in the literature.

Extension of Continuous Selection Sets to Non-Lipschitzian Quantum Stochastic Differential Inclusion

We establish results on multifunction associated with a set of solutions of non-Lipschitz quantum stochastic differential inclusion (QSDI), which still admits a continuous selection from some subsets of complex numbers. The results here generalize existing results.

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

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

Topological Properties of Solution Sets of Lipschitzian Quantum Stochastic Differential Inclusions

In this paper, we establish results on continuous mappings of the space of the matrix elements of an arbitrary nonempty set of pseudo solutions of non Lipschitz quantum Stochastic differential inclusion (QSDI) into the space of the matrix elements of its solutions. we show that under the non Lipschitz condition, the space of the matrix elements of solutions is still an absolute retract, contractible, locally and integrally connected in an arbitrary dimension. The results here generalize existing results in the literature.

Continuous Interpolation of Solution Sets of Lipschitzian Quantum Stochastic Differential Inclusions

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.

Continuous Selections of Solution Sets of Lipschitzian Quantum Stochastic Differential Inclusions

A multifunction associated with the set of solutions of Lipschitzian quantum stochastic differential inclusion (QSDI) admits a selection continuous from some subsets of complex numbers to the space of the matrix elements of adapted weakly absolutely continuous quantum stochastic processes. In particular, the solution set map as well as the reachable set of the QSDI admit some continuous representations.

Error Estimates for Discretized Quantum Stochastic Differential Inclusions

This paper is concerned with the error estimates involved in the solution of a discrete approximation of a quantum stochastic differential inclusion (QSDI). Our main results rely on certain properties of the averaged modulus of continuity for multivalued sesquilinear forms associated with QSDI. We obtained results concerning the estimates of the Hausdorff distance between the set of solutions of the QSDI and the set of solutions of its discrete approximation. This extends the results of Dontchev and Farkhi Dontchev, A.L.; Farkhi, E.M. (Error estimates for discre‐ tized differential inclusions. Computing 1989, 41, 349–358) concerning classical differential inclusions to the present noncommutative quantum setting involving inclusions in certain locally convex space.

Exponential Formula for the Reachable Sets of Quantum Stochastic Differential Inclusions

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] for classical differential inclusions to the present noncommutative quantum setting.

CONSTRUCTION OF APPROXIMATE ATTAINABILITY SETS FOR LIPSCHITZIAN QUANTUM STOCHASTIC DIFFERENTIAL INCLUSIONS

We present a numerical method for constructing, with a specified accuracy attainability sets for Lipschitzian quantum stochastic differential inclusions. Results here generalize the Komarov-Pevchikh results concerning classical differential inclusions to the present noncommutative quantum setting involving unbounded linear operators on a Hilbert space. AMS Subject Classification (1991): 60H10, 60H20, 65L05, 81S25.

Quantum stochastic evolutions

Quantum stochastic differential inclusions of hypermaximal monotone type are studied, under very general conditions, by means of certain discrete schemes which approximate them. The existence of an evolution operator corresponding to each such inclusion is proved.

Quantum stochastic differential inclusions of hypermaximal monotone type

In continuation of our study of the existence of solutions of quantum stochastic differential inclusions, we first introduce and develop some aspects of the theory of maximal [resp. hypermaximal] monotone multifunctions, including the description of a number of properties of their resolvents and Yosida approximations, in the present noncommutative setting. Then, it is proved that, under a certain continuity assumption, a quantum stochastic differential inclusion of hypermaximal monotone type has a unique adapted solution which is obtained as the limit of the unique adapted solutions of a one-parameter family of Lipschitzian quantum stochastic differential equations. As examples, we show that a large class of quantum stochastic differential inclusions which satisfy the assumptions and conclusion of our main result arises as perturbations of certain quantum stochastic differential equations by some multivalued stochastic processes.

Lipschitzian quantum stochastic differential inclusions

Quantum stochastic differential inclusions are introduced and studied within the framework of the Hudson-Parthasarathy formulation of quantum stochastic calculus. Results concerning the existence of solutions of a Lipschitzian quantum stochastic differential inclusion and the relationship between the solutions of such an inclusion and those of its convexification are presented. These generalize the Filippov existence theorem and the Filippov-Wažewski relaxation theorem for classical differential inclusions to the present noncommutative setting.