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Journal of mathematical extension.

Existence and Uniqueness of Solutions of a Class of Evolution Quantum Stochastic Dierential Equations

Sheila Amina Bishop, E. O. Ayoola

Published Apr 20, 2020• 0 citations
OpenAlex

Abstract

We study the properties of the existence and uniqueness of solu-tions of a class of evolution quantum stochastic differential equations(QSDEs) dened on a locally convex space whose topology is gen-erated by a family of seminorms dened via the norm of the rangespace of the operator processes. These solutions are called strong solutions in comparison with the solutions of similar equations denedon the space of operator processes where the topology is generated bythe family of seminorms dened via the inner product of the rangespace. The evolution operator generates a bounded semigroup. Weshow that under some more general conditions, the unique solutionis stable. These results extend some existing results in the literatureconcerning strong solutions of quantum stochastic differential equa-tions.

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We study the properties of the existence and uniqueness of solu-tions of a class of evolution quantum stochastic differential equations(QSDEs) dened on a locally convex space whose topology is gen-erated by a family of seminorms dened via the norm of the rangespace of the operator processes. These solutions are called strong solutions in comparison with the solutions of similar equations denedon the space of operator processes where the topology is generated bythe family of seminorms dened via the inner product of the rangespace. The evolution operator generates a bounded semigroup. Weshow that under some more general conditions, the unique solutionis stable. These results extend some existing results in the literatureconcerning strong solutions of quantum stochastic differential equa-tions.
We study the properties of the existence and uniqueness of solu-tions of a class of evolution quantum stochastic differential equations(QSDEs) dened on a locally convex space whose topology is gen-erated by a family of seminorms dened via the norm of the rangespace of the operator processes. These solutions are called strong solutions in comparison with the solutions of similar equations denedon the space of operator processes where the topology is generated bythe family of seminorms dened via the inner product of the rangespace. The evolution operator generates a bounded semigroup. Weshow that under some more general conditions, the unique solutionis stable. These results extend some existing results in the literatureconcerning strong solutions of quantum stochastic differential equa-tions.
We establish further results concerning the existence, uniqueness and stability of weak solutions of quantum stochastic differential equations (QSDEs). Our results are achieved by considering a more general Lipschit condition on the coefficients than our previous considerations in [1]. We exhibit a class of Lipschitzian QSDEs in the formulation of this paper, whose coefficients are only continuous on the locally convex space of the weak solution.Journal of the Nigerian Association of Mathematical Physics Vol. 8 2004: pp. 5-8
We study stochastic evolution equations describing the dynamics of open quantum systems. First, using resolvent approximations, we obtain a suf- ficient condition for regularity of solutions to linear stochastic Schrodinger equa- tions driven by cylindrical Brownian motions applying to many physical systems. Then, we establish well-posedness and norm conservation property of a wide class of open quantum systems described in position representation. Moreover, we prove Ehrenfest-type theorems that describe the evolution of the mean value of quantum observables in open systems. Finally, we give a new criterion for the existence and uniqueness of weak solutions to non-linear stochastic Schrequations. We apply our results to physical systems such as fluctuating ion traps and quantum measurement processes of position.
In the calculus of Hudson and Parthasarathy we show the existence of a solution to the quantum stochastic evolution equation U ( t ) = I + ∫ 0 t U ( s ) ( L 1 d Λ ( s ) + L 2 d A ( s ) + L 3 d A † ( s ) + L 4 d s ) for a class of unbounded operators L1, …, L4 in the initial space, and show that the necessary and sufficient condition that U be a unitary process is the same as in the case that L1, …, L4 are bounded. Applications of the vacuum conditional expectation to U give rise to strongly continuous semigroups and their generators in the initial space, and to ultraweakly continuous completely positive semigroups and their generators in the von Neumann algebra of the initial space.
We establish the existence of viable solutions of quantum stochastic differential equations (QSDE) within the framework of Hudson and Parthasarathy formulation of quantum stochastic calculus.The main results are established for QSDE whose coefficients are Lipschitzian and quasicompact.This work extends the Nagumo viability results for classical differential equations on finite dimensional Euclidean spaces.Viable solutions of QSDE in the present formulation take values in some closed subsets of the infinite dimensional locally convex spaces of non-commutative stochastic processes.