  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 $\rho $ to the Gaussian states equivalent to
$\rho $.
