We introduce Markovian cocycle perturbations of the groups of transformations associated with classical and quantum stochastic processes with stationary increments, which are characterized by a localization of the perturbation to the algebra of events of the past. The Markovian cocycle perturbations of the Kolmogorov flows associated with the classical and quantum noises result in the perturbed group of transformations which can be decomposed into the sum of two parts. One part in the decomposition is associated with a deterministic stochastic process lying in the past of the initial process, while another part is associated with the noise isomorphic to the initial one. This construction can be considered as some analog of the Wold decomposition for classical stationary processes excluding a nondeterministic part of the process in the case of the stationary quantum stochastic processes on the von Neumann factors which are the Markovian perturbations of the quantum noises. For the classical stochastic process with noncorrelated increments, the model of Markovian perturbations describing all Markovian cocycles up to a unitary equivalence of the perturbations has been constructed. Using this model, we construct Markovian cocycles transforming the Gaussian state ρ to the Gaussian states equivalent to ρ.
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.
We develop fully noncommutative Feynman–Kac formulae by employing quantum stochastic processes. To this end, we establish some theory for perturbing quantum stochastic flows on von Neumann algebras by multiplier cocycles. Multiplier cocycles are constructed via quantum stochastic differential equations whose coefficients are driven by the flow. The resulting class of cocycles is characterized under alternative assumptions of separability or Markov regularity. Our results generalize those obtained using classical Brownian motion on the one hand, and results for unitarily implemented flows on the other.
Important information about stationary flows (B I ) I Î I is contained in moments of the corresponding process. We consider the case where we have an order invariance of the moments of the increments, in the sense that such moments do not change if we shift the increments against each other, as long as we do not change the relative position of the intervals. We then show that for large classes of quantum processes one can make precise statements about the time behaviour of their moments. Keywords: Von Neumann algebras, Stationary flows and quantum processes, order invariant, Trace and moments.
Quantum Markovian master equations with generally unbounded generators, having physically relevant symmetries, such as Weyl, Galilean or boost covariance, are characterized. It is proven in particular that a fully Galilean covariant zero spin Markovian evolution reduces to the free motion perturbed by a covariant stochastic process with independent stationary increments in the classical phase space. A general form of the boost covariant Markovian master equation is discussed and a formal dilation to the Langevin equation driven by quantum Boson noises is described.