Stochastic semigroups: their construction by perturbation and approximation

The main object of the paper is to present a criterion for the minimal semigroup associated with the Kolmogorov differential equations to be stochastic in `1. Our criterion uses a weighted `1space. As an abstract preparation we present a perturbation theorem for substochastic semigroups which generalizes known results to the case of ordered Banach spaces which need not be AL-spaces We also consider extensions of Kolmogorov’s equations to spaces of measures. In an appendix we present a version of the Miyadera perturbation theorem for positive semigroups on ordered Banach spaces.