On the Generators of Quantum Stochastic Flows

✍ Scribed by J.Martin Lindsay; K.R Parthasarathy

• Publisher: Elsevier Science
• Year: 1998
• Tongue: English
• Weight: 381 KB
• Volume: 158
• Category: Article
• ISSN: 0022-1236
• DOI: 10.1006/jfan.1997.3194

✦ Synopsis

A time-indexed family of V-homomorphisms between operator algebras ( j t : A Ä B) t # I is called a stochastic process in quantum probability. When E C : B Ä C is a conditional expectation onto a subalgebra, the composed process (k t =E C b j t ) t # I is no longer V-homomorphic, but is completely positive and contractive. In some situations, the filtered process k may be described by a stochastic differential equation. The central aim of this paper is to study completely positive processes k which admit a differential description through a stochastic equation of the form dk t =k t b % :

; d4 ; : (t), in which 4 is the matrix of basic integrators of finite dimensional quantum stochastic calculus, and % is a matrix of bounded linear maps on the algebra. The structure required of the matrix %, for complete positivity of the process, is obtained. The stochastic generators of contractive, unital, and V-homomorphic processes are also studied. These results are applied to the equation dV t =l : ; V t d4 ; : (t) in which l is a matrix of bounded Hilbert space operators.