Abstract. A recent characterisation of Fock-adapted contraction operator stochastic cocycles on a Hilbert space, in terms of their associated semigroups, yields a general principle for the construction of such cocycles by approximation of their stochastic generators. This leads to new existence results for quantum stochastic differential equations. We also give necessary and sufficient conditions for a cocycle to satisfy such an equation. In this paper we study the functional equation V0 = I, Vr+t = Vrσr(Vt) for all r, t ≥ 0. for a family of contractions on h ⊗ F adapted to the Fock operator filtration. Here F is the symmetric Fock space over L 2 (R+; k), h and k are fixed but arbitrary
Stationary quantum stochastic process j is introduced as a *-homomorphism embedding an involutive graded algebra
K~=⊕i=1∞Ki into a ring of (abelian) cohomologies of the one-parameter group
α consisting of *-automorphisms of certain operator algebra in a Hilbert space such that every x from
Ki is translated into an additive
i−α-cocycle j(x). It is shown that (noncommutative) multiplicative markovian cocycle defines a perturbation of the stationary quantum stochastic process in the sense of such definition. The
E0-semigroup
β~ on the von Neumann algebra
N associated with the markovian perturbation of K-flow j posseses the restriction
β~∣N0,N0⊂N, which is conjugate to the flow of Powers shifts
β associated with j. It yields for
β~ an analogue of the Wold decomposition for classical stochastic process on completely nondeterministic and deterministic parts. The examples of quantum stationary stochastic processes on the algebras of canonical commutation, anticommutation and square of white noise relations are considered. In the model situation of the space
L2(R) all markovian cocycles of the group of shifts are described up to unitary equivalence of perturbations.