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Advances in Mathematics

Quantum Stochastic Differential Equations in Terms of Generalized Operators

Xiangjun Wang

Published Jan 1, 2003• 1 citation
OpenAlex

Abstract

Nonlinear quantum stochastic differential equations in term of generalized operators and their Wick products are considered. Existence and uniqueness of solutions are established. The continuity and continuous dependence on initial values of the solution are proved. Examples are illustrated.

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•Aug 14, 2016•8 citations
Many examples of quantum independent stationary increment processes are solutions of quantum stochastic differential equations. We give a common characterization of these examples by a quantum stochastic differential equation on an abstract *-bialgebra. Specializing this abstract *-bialgebra and the coefficients of the equation, we obtain the equations for the Unitary Noncommutative Stochastic processes of [12], the Quantum Wiener Process [2], the Azéma martingales [11] and for other examples. The existence and uniqueness of a solution of the general equation is shown. Assuming the boundedness of this solution, we prove that it is a continuous and stationary independent increment process.
Existence and uniqueness theorems for quantum stochastic differential equations with nontrivial initial conditions are proved for coefficients with completely bounded columns. Applications are given for the case of finite-dimensional initial space or, more generally, for coefficients satisfying a finite localisability condition. Necessary and sufficient conditions are obtained for a conjugate pair of quantum stochastic cocycles on a finite-dimensional operator space to strongly satisfy such a quantum stochastic differential equation. This gives an alternative approach to quantum stochastic convolution cocycles on a coalgebra.