On Markovian Cocycle Perturbations in Classical and Quantum Probability

Abstract

We introduce Markovian cocycle perturbations of the groups of transformations associated with the 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. It is namely the definition one needs because the Markovian perturbations of the Kolmogorov flows associated with the classical and quantum noises result in the perturbed group of transformations which can be decomposed in the sum of a part associated with deterministic stochastic processes lying in the past and a part associated with the noise isomorphic to the initial one. This decomposition allows to obtain 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 increaments it is constructed the model of Markovian perturbations describing all Markovian cocycles up to a unitary equivalence of the perturbations. Using this model we construct Markovian cocyclies transformating the Gaussian state $ρ$ to the Gaussian states equivalent to $ρ$.