On Markovian perturbations of the group of unitary operators associated with a stochastic process with stationary increments

We introduce “Markovian” cocycle perturbations of the group of unitary operators associated with a stochastic process with stationary increments, which are characterized by a localization of the perturbation to the algebra of past events. The definition we give is necessary because the Markovian perturbation of the group associated with a stochastic process with noncorrelated increments results in the perturbed group for which there exists a stochastic process with noncorrelated increments associated with it. On the other hand, some “deterministic” stochastic process lying in the past can also be associated with the perturbed group. 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 Gaussian measures to the equivalent Gaussian measures.