ON THE GENERATORS OF QUANTUM STOCHASTIC FLOWS

A time-indexed family of -homomorphisms between operator algebras (j( t):A --> B)(t is an element of I) is called a stochastic process in qu antum probability. When E-C:B --> C is a conditional expectation onto a subalgebra, the composed process (k(t) = E(C)circle j(t))(t is an el ement of I) is no longer -homomorphic, but is completely positive and contractive. Ln some situations, the filtered process k may be descri bed by a stochastic differential equation. The central aim of this pap er is to study completely positive processes k which admit a different ial description through a stochastic equation of the form dk(t) = kt c ircle theta(beta)(alpha) d Lambda(alpha)(beta)(t), in which Lambda is the matrix of basic integrators of finite dimensional quantum stochast ic calculus, and theta is a matrix of bounded linear maps on the algeb ra. The structure required of the matrix theta, for complete positivit y of the process, is obtained. The stochastic generators of contractiv e, unital, and -homomorphic processes are also studied. These results are applied to the equation dV(t) = l(beta)(alpha)V(t) d Lambda(alpha )(beta)(t) in which l is a matrix of bounded Hilbert space operators. (C) 1998 Academic Press.