Algebraic theory of quantum diffusions

Abstract

Quantum diffusions are quantum stochastic processes in the sense of Accardi, Frigerio and Lewis, in which evolution of elements of the initial algebra is governed by a system of autonomous quantum stochastic differential equations against the gauge, creation and annihilation processes. As a consequence of the quantum Itô formula the coefficients of these equations satisfy cohomological identities. A diffusion for which the coefficients differ in a cohomologically trivial sense from a given diffusion can be constructed by a perturbation procedure. Every quantum diffusion on the algebra of all bounded operators on a Hilbert space is characterised by a unitary process. In the commutative case certain "diffusions" with discontinuous sample paths are found; these are a feature of the zero temperature Fock quantum stochastic calculus used here and do not exist at finite temperature.